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Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures. Together with geometry, analysis, topology, combinatorics, and number theory, algebra is one of the main branches of pure mathematics. .The part of algebra called elementary algebra is often part of the curriculum in secondary education and introduces the concept of variables representing numbers.^ In IEEE 754, NaNs are often represented as floating-point numbers with the exponent e max + 1 and nonzero significands.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ On the other hand, the VAX TM reserves some bit patterns to represent special numbers called reserved operands .
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

Statements based on these variables are manipulated using the rules of operations that apply to numbers, such as addition. This can be done for a variety of reasons, including equation solving. .Algebra is much broader than elementary algebra and studies what happens when different rules of operations are used and when operations are devised for things other than numbers.^ The IEEE standard uses denormalized 18 numbers, which guarantee (10) , as well as other useful relations.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ Several different representations of real numbers have been proposed, but by far the most widely used is the floating-point representation.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ Rather than using all these digits, floating-point hardware normally operates on a fixed number of digits.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

Addition and multiplication can be generalized and their precise definitions lead to structures such as groups, rings and fields.

Contents

History

While the word algebra comes from the Arabic language (al-jabr, الجبر literally, restoration) and much of its methods from Arabic/Islamic mathematics, its roots can be traced to earlier traditions, most notably ancient Indian mathematics, which had a direct influence on Muhammad ibn Mūsā al-Khwārizmī (c. 780-850). He learned Indian mathematics and introduced it to the Muslim world through his famous arithmetic text, Kitab al-jam’wal tafriq bi hisab al-Hindi (Book on Addition and Subtraction after the Method of the Indians).[1][2] .He later wrote al-Kitāb al-muḫtaṣar fī ḥisāb al-jabr wa-l-muqābala (The Compendious Book on Calculation by Completion and Balancing), which established algebra as a mathematical discipline that is independent of geometry and arithmetic.^ An optimizer that believed floating-point arithmetic obeyed the laws of algebra would conclude that C = [ T - S ] - Y = [( S + Y )- S ] - Y = 0, rendering the algorithm completely useless.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

[3]
The roots of algebra can be traced to the ancient Babylonians,[4] who developed an advanced arithmetical system with which they were able to do calculations in an algorithmic fashion. .The Babylonians developed formulas to calculate solutions for problems typically solved today by using linear equations, quadratic equations, and indeterminate linear equations.^ Consider the problem of solving a system of linear equations, .
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ There is another solution using trap handlers called over/underflow counting that avoids both of these problems [Sterbenz 1974].
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ If an instruction that combines two single precision operands to produce a double precision product was only useful for the quadratic formula, it wouldn't be worth adding to an instruction set.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

By contrast, most Egyptians of this era, as well as Greek and Chinese mathematicians in the first millennium BC, usually solved such equations by geometric methods, such as those described in the Rhind Mathematical Papyrus, Euclid's Elements, and The Nine Chapters on the Mathematical Art. .The geometric work of the Greeks, typified in the Elements, provided the framework for generalizing formulae beyond the solution of particular problems into more general systems of stating and solving equations, though this would not be realized until the medieval Muslim mathematicians.^ Consider the problem of solving a system of linear equations, .
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ Although formula (7) is much more accurate than (6) for this example, it would be nice to know how well (7) performs in general.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ Many problems, such as numerical integration and the numerical solution of differential equations involve computing sums with many terms.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

The Hellenistic mathematicians Hero of Alexandria and Diophantus [5] as well as Indian mathematicians such as Brahmagupta continued the traditions of Egypt and Babylon, though Diophantus' Arithmetica and Brahmagupta's Brahmasphutasiddhanta are on a higher level.[6] For example, the first complete arithmetic solution (including zero and negative solutions) to quadratic equations was described by Brahmagupta in his book Brahmasphutasiddhanta. Later, Arabic and Muslim mathematicians developed algebraic methods to a much higher degree of sophistication. .Although Diophantus and the Babylonians used mostly special ad hoc methods to solve equations, Al-Khwarizmi was the first to solve equations using general methods.^ Additional argument is needed for the special case where adding w does generate carry out.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

.He solved the linear indeterminate equations, quadratic equations, second order indeterminate equations and equations with multiple variable.^ Consider the problem of solving a system of linear equations, .
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

The word "algebra" is named after the Arabic word "al-jabr , الجبر" from the title of the book al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala , الكتاب المختصر في حساب الجبر والمقابلة, meaning The book of Summary Concerning Calculating by Transposition and Reduction, a book written by the Islamic Persian mathematician, Muhammad ibn Mūsā al-Khwārizmī, in 820. The word Al-Jabr means "reunion"[7]. .The Hellenistic mathematician Diophantus has traditionally been known as the "father of algebra" but in more recent times there is much debate over whether al-Khwarizmi, who founded the discipline of al-jabr, deserves that title instead.^ However, there is a much more efficient method which dramatically improves the accuracy of sums, namely .
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

[8] Those who support Diophantus point to the fact that the algebra found in Al-Jabr is slightly more elementary than the algebra found in Arithmetica and that Arithmetica is syncopated while Al-Jabr is fully rhetorical.[9] Those who support Al-Khwarizmi point to the fact that he introduced the methods of "reduction" and "balancing" (the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation) which the term al-jabr originally referred to,[10] and that he gave an exhaustive explanation of solving quadratic equations,[11] supported by geometric proofs, while treating algebra as an independent discipline in its own right.[12] His algebra was also no longer concerned "with a series of problems to be resolved, but an exposition which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study." He also studied an equation for its own sake and "in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems."[13]
The Persian mathematician Omar Khayyam is credited with identifying the foundations of algebraic geometry and found the general geometric solution of the cubic equation. .Another Persian mathematician, Sharaf al-Dīn al-Tūsī, found algebraic and numerical solutions to various cases of cubic equations.^ Many problems, such as numerical integration and the numerical solution of differential equations involve computing sums with many terms.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

[14] He also developed the concept of a function.[15] The Indian mathematicians Mahavira and Bhaskara II, the Persian mathematician Al-Karaji,[16] and the Chinese mathematician Zhu Shijie, solved various cases of cubic, quartic, quintic and higher-order polynomial equations using numerical methods. In 1637 Rene Descartes published La Géométrie, inventing analytic geometry and introducing modern algebraic notation.
Another key event in the further development of algebra was the general algebraic solution of the cubic and quartic equations, developed in the mid-16th century. The idea of a determinant was developed by Japanese mathematician Kowa Seki in the 17th century, followed by Gottfried Leibniz ten years later, for the purpose of solving systems of simultaneous linear equations using matrices. Gabriel Cramer also did some work on matrices and determinants in the 18th century. Abstract algebra was developed in the 19th century, initially focusing on what is now called Galois theory, and on constructibility issues.

Classification

Algebra may be divided roughly into the following categories:
.
  • Elementary algebra, in which the properties of operations on the real number system are recorded using symbols as "place holders" to denote constants and variables, and the rules governing mathematical expressions and equations involving these symbols are studied.^ These examples can be summarized by saying that optimizers should be extremely cautious when applying algebraic identities that hold for the mathematical real numbers to expressions involving floating-point variables.
    • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

    ^ Several different representations of real numbers have been proposed, but by far the most widely used is the floating-point representation.
    • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

    ^ Similarly, if the real number .0314159 is represented as 3.14 × 10 -2 , then it is in error by .159 units in the last place.
    • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

    This is usually taught at school under the title algebra (or intermediate algebra and college algebra in subsequent years). .University-level courses in group theory may also be called elementary algebra.
  • Abstract algebra, sometimes also called modern algebra, in which algebraic structures such as groups, rings and fields are axiomatically defined and investigated.
  • Linear algebra, in which the specific properties of vector spaces are studied (including matrices);
  • Universal algebra, in which properties common to all algebraic structures are studied.
  • Algebraic number theory, in which the properties of numbers are studied through algebraic systems.^ First of all, there are algebraic identities that are valid for floating-point numbers.
    • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

    ^ An algorithm that involves thousands of operations (such as solving a linear system) will soon be operating on numbers with many significant bits, and be hopelessly slow.
    • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

    ^ Of course, it would still be possible to compute the square of this number in extended double precision, but the resulting algorithm would no longer be portable to single/double systems.
    • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

    Number theory inspired much of the original abstraction in algebra.
  • Algebraic geometry applies abstract algebra to the problems of geometry.
  • Algebraic combinatorics, in which abstract algebraic methods are used to study combinatorial questions.
In some directions of advanced study, axiomatic algebraic systems such as groups, rings, fields, and algebras over a field are investigated in the presence of a geometric structure (a metric or a topology) which is compatible with the algebraic structure. The list includes a number of areas of functional analysis:

Elementary algebra

Elementary algebra is the most basic form of algebra. It is taught to students who are presumed to have no knowledge of mathematics beyond the basic principles of arithmetic. In arithmetic, only numbers and their arithmetical operations (such as +, −, ×, ÷) occur. In algebra, numbers are often denoted by symbols (such as a, x, or y). This is useful because:
  • It allows the general formulation of arithmetical laws (such as a + b = b + a for all a and b), and thus is the first step to a systematic exploration of the properties of the real number system.
  • It allows the reference to "unknown" numbers, the formulation of equations and the study of how to solve these (for instance, "Find a number x such that 3x + 1 = 10" or going a bit further "Find a number x such that ax+b=c". Step which lets to the conclusion that is not the nature of the specific numbers the one that allows us to solve it but that of the operations involved).
  • It allows the formulation of functional relationships (such as "If you sell x tickets, then your profit will be 3x − 10 dollars, or f(x) = 3x − 10, where f is the function, and x is the number to which the function is applied.").

Polynomials

A polynomial (see the article on polynomials for more detail) is an expression that is constructed from one or more variables and constants, using only the operations of addition, subtraction, and multiplication (where repeated multiplication of the same variable is standardly denoted as exponentiation with a constant non-negative integer exponent). For example, x2 + 2x − 3 is a polynomial in the single variable x.
An important class of problems in algebra is factorization of polynomials, that is, expressing a given polynomial as a product of other polynomials. The example polynomial above can be factored as (x − 1)(x + 3). A related class of problems is finding algebraic expressions for the roots of a polynomial in a single variable.

Abstract algebra

Abstract algebra extends the familiar concepts found in elementary algebra and arithmetic of numbers to more general concepts.
Sets: Rather than just considering the different types of numbers, abstract algebra deals with the more general concept of sets: a collection of all objects (called elements) selected by property, specific for the set. All collections of the familiar types of numbers are sets. Other examples of sets include the set of all two-by-two matrices, the set of all second-degree polynomials (ax2 + bx + c), the set of all two dimensional vectors in the plane, and the various finite groups such as the cyclic groups which are the group of integers modulo n. Set theory is a branch of logic and not technically a branch of algebra.
Binary operations: The notion of addition (+) is abstracted to give a binary operation, ∗ say. The notion of binary operation is meaningless without the set on which the operation is defined. For two elements a and b in a set S, ab is another element in the set; this condition is called closure. Addition (+), subtraction (-), multiplication (×), and division (÷) can be binary operations when defined on different sets, as is addition and multiplication of matrices, vectors, and polynomials.
Identity elements: The numbers zero and one are abstracted to give the notion of an identity element for an operation. Zero is the identity element for addition and one is the identity element for multiplication. For a general binary operator ∗ the identity element e must satisfy ae = a and ea = a. This holds for addition as a + 0 = a and 0 + a = a and multiplication a × 1 = a and 1 × a = a. Not all set and operator combinations have an identity element; for example, the positive natural numbers (1, 2, 3, ...) have no identity element for addition.
Inverse elements: The negative numbers give rise to the concept of inverse elements. For addition, the inverse of a is −a, and for multiplication the inverse is 1/a. A general inverse element a−1 must satisfy the property that aa−1 = e and a−1a = e.
Associativity: Addition of integers has a property called associativity. That is, the grouping of the numbers to be added does not affect the sum. For example: (2 + 3) + 4 = 2 + (3 + 4). In general, this becomes (ab) ∗ c = a ∗ (bc). This property is shared by most binary operations, but not subtraction or division or octonion multiplication.
Commutativity: Addition of integers also has a property called commutativity. That is, the order of the numbers to be added does not affect the sum. For example: 2+3=3+2. In general, this becomes ab = ba. Only some binary operations have this property. It holds for the integers with addition and multiplication, but it does not hold for matrix multiplication or quaternion multiplication .

Groups – structures of a set with a single binary operation

Combining the above concepts gives one of the most important structures in mathematics: a group. A group is a combination of a set S and a single binary operation ∗, defined in any way you choose, but with the following properties:
  • An identity element e exists, such that for every member a of S, ea and ae are both identical to a.
  • Every element has an inverse: for every member a of S, there exists a member a−1 such that aa−1 and a−1a are both identical to the identity element.
  • The operation is associative: if a, b and c are members of S, then (ab) ∗ c is identical to a ∗ (bc).
If a group is also commutative—that is, for any two members a and b of S, ab is identical to ba—then the group is said to be Abelian.
For example, the set of integers under the operation of addition is a group. In this group, the identity element is 0 and the inverse of any element a is its negation, −a. The associativity requirement is met, because for any integers a, b and c, (a + b) + c = a + (b + c)
The nonzero rational numbers form a group under multiplication. Here, the identity element is 1, since 1 × a = a × 1 = a for any rational number a. The inverse of a is 1/a, since a × 1/a = 1.
The integers under the multiplication operation, however, do not form a group. This is because, in general, the multiplicative inverse of an integer is not an integer. For example, 4 is an integer, but its multiplicative inverse is ¼, which is not an integer.
The theory of groups is studied in group theory. A major result in this theory is the classification of finite simple groups, mostly published between about 1955 and 1983, which is thought to classify all of the finite simple groups into roughly 30 basic types.
Examples
Set: Natural numbers N Integers Z Rational numbers Q (also real R and complex C numbers) Integers modulo 3: Z3 = {0, 1, 2}
Operation + × (w/o zero) + × (w/o zero) + × (w/o zero) ÷ (w/o zero) + × (w/o zero)
Closed Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes
Identity 0 1 0 1 0 N/A 1 N/A 0 1
Inverse N/A N/A a N/A a N/A 1/a N/A 0, 2, 1, respectively N/A, 1, 2, respectively
Associative Yes Yes Yes Yes Yes No Yes No Yes Yes
Commutative Yes Yes Yes Yes Yes No Yes No Yes Yes
Structure monoid monoid Abelian group monoid Abelian group quasigroup Abelian group quasigroup Abelian group Abelian group (Z2)
Semigroups, quasigroups, and monoids are structures similar to groups, but more general. They comprise a set and a closed binary operation, but do not necessarily satisfy the other conditions. A semigroup has an associative binary operation, but might not have an identity element. A monoid is a semigroup which does have an identity but might not have an inverse for every element. A quasigroup satisfies a requirement that any element can be turned into any other by a unique pre- or post-operation; however the binary operation might not be associative.
All groups are monoids, and all monoids are semigroups.

Rings and fields—structures of a set with two particular binary operations, (+) and (×)

Groups just have one binary operation. To fully explain the behaviour of the different types of numbers, structures with two operators need to be studied. The most important of these are rings, and fields.
Distributivity generalised the distributive law for numbers, and specifies the order in which the operators should be applied, (called the precedence). For the integers (a + b) × c = a × c + b × c and c × (a + b) = c × a + c × b, and × is said to be distributive over +.
A ring has two binary operations (+) and (×), with × distributive over +. Under the first operator (+) it forms an Abelian group. Under the second operator (×) it is associative, but it does not need to have identity, or inverse, so division is not allowed. The additive (+) identity element is written as 0 and the additive inverse of a is written as −a.
The integers are an example of a ring. The integers have additional properties which make it an integral domain.
A field is a ring with the additional property that all the elements excluding 0 form an Abelian group under ×. The multiplicative (×) identity is written as 1 and the multiplicative inverse of a is written as a−1.
The rational numbers, the real numbers and the complex numbers are all examples of fields.

Objects called algebras

The word algebra is also used for various algebraic structures:

See also

Notes

  1. ^ http://www.brusselsjournal.com/node/4107/print
  2. ^ A History of Mathematics: An Introduction (2nd Edition) (Paperback) Victor J katz Addison Wesley; 2 edition (March 6, 1998)
  3. ^ Roshdi Rashed (November 2009), Al Khwarizmi: The Beginnings of Algebra, Saqi Books, ISBN 0863564305 
  4. ^ Struik, Dirk J. (1987). A Concise History of Mathematics. New York: Dover Publications.
  5. ^ Diophantus, Father of Algebra
  6. ^ History of Algebra
  7. ^ Or rather restoration, according to RH Webster's 2nd ed.
  8. ^ Carl B. Boyer, A History of Mathematics, Second Edition (Wiley, 1991), pages 178, 181
  9. ^ Carl B. Boyer, A History of Mathematics, Second Edition (Wiley, 1991), page 228
  10. ^ (Boyer 1991, "The Arabic Hegemony" p. 229) "It is not certain just what the terms al-jabr and muqabalah mean, but the usual interpretation is similar to that implied in the translation above. The word al-jabr presumably meant something like "restoration" or "completion" and seems to refer to the transposition of subtracted terms to the other side of an equation; the word muqabalah is said to refer to "reduction" or "balancing" - that is, the cancellation of like terms on opposite sides of the equation."
  11. ^ (Boyer 1991, "The Arabic Hegemony" p. 230) "The six cases of equations given above exhaust all possibilities for linear and quadratic equations having positive root. So systematic and exhaustive was al-Khwarizmi's exposition that his readers must have had little difficulty in mastering the solutions."
  12. ^ Gandz and Saloman (1936), The sources of al-Khwarizmi's algebra, Osiris i, p. 263–277: "In a sense, Khwarizmi is more entitled to be called "the father of algebra" than Diophantus because Khwarizmi is the first to teach algebra in an elementary form and for its own sake, Diophantus is primarily concerned with the theory of numbers".
  13. ^ Rashed, R.; Armstrong, Angela (1994), The Development of Arabic Mathematics, Springer, pp. 11–2, ISBN 0792325656, OCLC 29181926 
  14. ^ O'Connor, John J.; Robertson, Edmund F., "Sharaf al-Din al-Muzaffar al-Tusi", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Biographies/Al-Tusi_Sharaf.html .
  15. ^ Victor J. Katz, Bill Barton (October 2007), "Stages in the History of Algebra with Implications for Teaching", Educational Studies in Mathematics (Springer Netherlands) 66 (2): 185–201 [192], doi:10.1007/s10649-006-9023-7 
  16. ^ (Boyer 1991, "The Arabic Hegemony" p. 239) "Abu'l Wefa was a capable algebraist as well as a trigonometer. [...] His successor al-Karkhi evidently used this translation to become an Arabic disciple of Diophantus - but without Diophantine analysis! [...] In particular, to al-Karkhi is attributed the first numerical solution of equations of the form ax2n + bxn = c (only equations with positive roots were considered),"

References

  • Donald R. Hill, Islamic Science and Engineering (Edinburgh University Press, 1994).
  • Ziauddin Sardar, Jerry Ravetz, and Borin Van Loon, Introducing Mathematics (Totem Books, 1999).
  • George Gheverghese Joseph, The Crest of the Peacock: Non-European Roots of Mathematics (Penguin Books, 2000).
  • John J O'Connor and Edmund F Robertson, MacTutor History of Mathematics archive (University of St Andrews, 2005).
  • I.N. Herstein: Topics in Algebra. ISBN 0-471-02371-X
  • R.B.J.T. Allenby: Rings, Fields and Groups. ISBN 0-340-54440-6
  • L. Euler: Elements of Algebra, ISBN 978-1-89961-873-6
  • Isaac Asimov Realm of Algebra (Houghton Mifflin), 1961

External links


Study guide

Up to date as of January 14, 2010
(Redirected to Topic:Algebra article)

From Wikiversity

Contents

The Wikiversity Division of Algebra is a collection of the various algebraic studies, and is part of the School of Mathematics and the Division of Pure Mathematics. Algebra is an ancient form of mathematical analytical methodology and is one of the most fundamental in our modern practice of such.

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1911 encyclopedia

Up to date as of January 14, 2010

From LoveToKnow 1911

ALGEBRA (from the Arab. .al-jebr wa'l-mugabala, transposition and removal (of terms of an equation), the name of a treatise by Mahommed ben Musa al-Khwarizmi), a branch of mathematics which may be defined as the generalization and extension of arithmetic.^ As a branch of mathematics , arithmetic may be treated logically, psychologically, or historically.

^ One of the standard checks when defining the operation + on any system of mathematical objects is that 0 (as named in that system) is the identity element for +.
  • Arithmetic: A Crash Review 3 February 2010 14:24 UTC www.zaimoni.com [Source type: Reference]

^ In general, applicants should have a bachelor's or master's degree in mathematics, an engineering discipline, or a branch of the natural sciences.
  • NJIT - Graduate Programs: Mathematics 28 January 2010 0:26 UTC catalog.njit.edu [Source type: Academic]

.The subject-matter of algebra will be treated in the following article under three divisions: - A. Principles of ordinary algebra; B. Special kinds of algebra; C. History.^ Three of the following: Math 407 Abstract Algebra (3) Math 412 Complex Analysis (3) Math 425 Differential Geometry (3) Math 430 Number Theory (3) Math 471 Combinatorics (3) .

^ Analytic geometry and linear algebra are closely related subjects studied at an intermediate level, following college algebra.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ Algebra and geometry were more advanced subjects, involving theory as well as practical matters, and were taught in high school and college.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

.Special phases of the subject are treated under their own headings, e.g. Algebraic Forms; Binomial; Combinatorial Analysis; Determin Ants; Equation; Continued Fraction; Function; Theory of groups; Logarithm; Number; Probability; Series.^ Basic theory of Lie groups and Lie algebras.

^ Equations and inequalities, algebraic expressions, functions and sequences and series.

^ The Pure Mathematics track offers research opportunities in many fields of specialization, including representation theory, number theory, low-dimensional topology, Riemann surfaces and Kleinian groups, geometric group theory, and 4-manifolds.
  • NJIT - Graduate Programs: Mathematics 28 January 2010 0:26 UTC catalog.njit.edu [Source type: Academic]

.A. Principles Of Ordinary Algebra 1. The above definition gives only a partial view of the scope of algebra.^ Or, on the quinary- binary system , we need only give independent definitions to the numbers up to five; the numbers six, seven,..

^ This view is supported, not only by the intelligibility of percentages to ordinary persons, but also by the tendency, noted above (§ 19), to group years into centuries, and to avoid the use of thousands.

.It may be regarded as based on arithmetic, or as dealing in the first instance with formal results of the laws of arithmetical number; and in this sense Sir Isaac Newton gave the title Universal Arithmetic to a work on algebra.^ The forms seem to result from a general tendency to visualization as an aid to memory; the letter-forms may in the first instance be quite as frequent as the numberforms, but they vanish in early childhood, being of no practical value, while the number-forms continue as an aid to arithmetical work.

^ In simple terms, an overflow will occur if the result produced by a 64 COMPUTER ARITHMETIC given operation is outside the range of the representable numbers.
  • 3.Computer Arithmetic 3 February 2010 14:24 UTC www.slideshare.net [Source type: Reference]

^ What is called modern algebra works with symbols that may obey different rules of composition or operations than the familiar ones of real numbers that we have just presented.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

.Any definition, however, must have reference to the state of development of the subject at the time when the definition is given.^ An introduction to algebra must spend a lot of time on these matters, to develop the student's facility in working with symbolic expressions.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ However, Up-State College has the ball with time out.
  • Math Standards to use in 6th Grade classes to help them meet student performance skills 2 February 2010 15:42 UTC www.internet4classrooms.com [Source type: FILTERED WITH BAYES]
  • Math Standards to use in 7th Grade classes to help them meet student performance skills 2 February 2010 15:42 UTC www.internet4classrooms.com [Source type: Reference]

^ As this is written, I have given my talk on "Arithmetic, Population, and Energy" over 1260 times in 48 of the 50 States in the 28 years since 1969.
  • Arithmetic Presentation -Complete 3 February 2010 14:24 UTC www.mnforsustain.org [Source type: FILTERED WITH BAYES]
  • Presentation by Albert Bartlett - Arithmetic, Population, and Energy 3 February 2010 14:24 UTC hawaii.gov [Source type: FILTERED WITH BAYES]

.2. The earliest algebra consists in the solution of equations.^ Algebraic numbers are numbers that are the solution of an algebraic equation, as x = √2 is the solution of x 2 = 2.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ It will develop mathematical tools extending the techniques of introductory calculus, including some matrix algebra and solution techniques for first order differential equations.

^ Rational numbers are, of course, algebraic, since they are the solutions of linear equations.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

.The distinction between algebraical and arithmetical reasoning then lies mainly in the fact that the former is in a more condensed form than the latter; an unknown quantity being represented by a special symbol, and other symbols being used as a kind of shorthand for verbal expressions.^ When the unit is not determined, the reasoning is algebraical rather than arithmetical.

^ (Is algebra harder than arithmetic?
  • Algebra = ‘most failed’ college class « Joanne Jacobs 10 February 2010 11:011 UTC www.joannejacobs.com [Source type: Original source]

^ The essence of algebra, then, is the use of literal symbols to stand for general numbers or other quantities.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

.This form of algebra was extensively studied in ancient Egypt; but, in accordance with the practical tendency of the Egyptian mind, the study consisted largely in the treatment of particular cases, very few general rules being obtained.^ All evidence, from the laboratory and from extensive case studies of professionals, indicates that real competence only comes with extensive practice.
  • Illinois Loop: Mathematics 28 January 2010 0:26 UTC www.illinoisloop.org [Source type: FILTERED WITH BAYES]

^ Finally, many mathematicians study the areas they do for purely aesthetic reasons, viewing mathematics as an art form rather than as a practical or applied science .

^ Systems of linear equations is a particular case for which a large amount of algebra exists, including matrices and determinants, which is called linear algebra .
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

.3. For many centuries algebra was confined almost entirely to the solution of equations; one of the most important steps being the enunciation by Diophantus of Alexandria of the laws governing the use of the minus sign.^ Algebraic numbers are numbers that are the solution of an algebraic equation, as x = √2 is the solution of x 2 = 2.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ Solve one-step linear equations.
  • Math Standards to use in 7th Grade classes to help them meet student performance skills 2 February 2010 15:42 UTC www.internet4classrooms.com [Source type: Reference]

^ This step appears to have been taken by Diophantus in the 3rd century, and re-introduced to Europe via the Arabs about a thousand years later together with algorism, which is arithmetic using written symbols, the Hindu-Arabic digits.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

The knowledge of these laws, however, does not imply the existence of a conception of negative quantities. .The development of symbolic algebra by the use of general symbols to denote numbers is due to Franciscus Vieta (Francois Viete, 1540-1603).^ The essence of algebra, then, is the use of literal symbols to stand for general numbers or other quantities.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ It has a number of useful features in algebra 2.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

^ The symbols denoting a number are called its digits.

This led to the idea of algebra as generalized arithmetic.
4. The principal step in the modern development of algebra was the recognition of the meaning of negative quantities. .This appears to have been due in the first instance to Albert Girard (1595-1632), who extended Vieta's results in various branches of mathematics.^ First, one can expect results in the direction of a better understanding of the applicability of mathematics to the world.
  • Explanation in Mathematics (Stanford Encyclopedia of Philosophy) 28 January 2010 0:26 UTC plato.stanford.edu [Source type: Academic]

^ Look at the TIMSS international results and at the percentage of fourth- and eighth-grade students who reached the TIMSS advanced international benchmark in mathematics, by country in 2007.
  • Algebra = ‘most failed’ college class « Joanne Jacobs 10 February 2010 11:011 UTC www.joannejacobs.com [Source type: Original source]

His work, however, was little known at the time, and later was overshadowed by the greater work of Descartes (1596-1650).
.5. The main work of Descartes, so far as algebra was concerned, was the establishment of a relation between arithmetical and geometrical measurement.^ Intel Math examines the arithmetic, geometric and algebraic aspects… About.com: Math .
  • Alltop - Top Math News 2 February 2010 15:42 UTC math.alltop.com [Source type: General]

^ Topics include algebra, applied math, arithmetic, geometry, and measurement.

^ Thus, while arithmetical numbering refers to units, geometrical numbering does not refer to units but to the intervals between units.

.This involved not only the geometrical interpretation of negative quantities, but also the idea of continuity; this latter, which is the basis of modern analysis, leading to two separate but allied developments, viz.^ Two to five faculty members from various disciplines will describe in detail a project in which they are engaged that involves all ingredients of computational engineering and science: a scientific or engineering problem, a mathematical problem leading to mathematical questions, and the solution and interpretation of these questions obtained by the use of modern computing techniques.

^ These are aimed at 3D and video applications that involve streaming data; data that is used only once (for geometric rendering) and then discarded.
  • Textbook on Computer Arithmetic 3 February 2010 14:24 UTC www.ece.ucsb.edu [Source type: Academic]

^ The transition is similar to that which arises in the case of geometrical measurement (§ 26), and it is an essential feature of all reasoning with regard to continuous quantity, such as we have to deal with in real life.

the theory of the function and the theory of limits.
.6. The great development of all branches of mathematics in the two centuries following Descartes has led to the term algebra being used to cover a great variety of subjects, many of which are really only ramifications of arithmetic, dealt with by algebraical methods, while others, such as the theory of numbers and the general theory of series, are outgrowths of the application of algebra to arithmetic, which involve such special ideas that they must properly be regarded as distinct subjects.^ Use is subject to license terms .
  • Math (Java 2 Platform SE 5.0) 2 February 2010 15:42 UTC java.sun.com [Source type: Academic]

^ Given two numbers such as 1/2 and 1/3, describe the real numbers between them.
  • Maine Learning Results 28 January 2010 0:26 UTC www.state.me.us [Source type: Reference]

^ The term is also used in a number of other specialty ways in mathematics, the best known being the "congruence modulus".
  • Origins of some arithmetic terms 3 February 2010 14:24 UTC www.pballew.net [Source type: FILTERED WITH BAYES]

.Some writers have attempted unification by treating algebra as concerned with functions, and Comte accordingly defined algebra as the calculus of functions, arithmetic being regarded as the calculus of values. 7. These attempts at the unification of algebra, and its separation from other branches of mathematics, have usually been accompanied by an attempt to base it, as a deductive science, on certain fundamental laws or general rules; and this has tended to increase its difficulty.^ As a branch of mathematics , arithmetic may be treated logically, psychologically, or historically.

^ Arithmetic-based science .
  • Reading Instruction for Arithmetic Word Problems: 3 February 2010 14:24 UTC www.math.umd.edu [Source type: FILTERED WITH BAYES]

^ This is especially true in a multi-step approximation, where over- and underestimates at various steps tend to cancel each other out, usually resulting in something not too far off from the truth.
  • mathematics (kottke.org) 28 January 2010 0:26 UTC www.kottke.org [Source type: General]

In reality, the variety of algebra corresponds to the variety of phenomena. .Neither mathematics itself, nor any branch or set of branches of mathematics, can be regarded as an isolated science.^ Whereas the issues treated in section 3a affect the methodology of science, a different set of issues has emerged in connection to the nominalism-platonism debate in philosophy of mathematics.
  • Explanation in Mathematics (Stanford Encyclopedia of Philosophy) 28 January 2010 0:26 UTC plato.stanford.edu [Source type: Academic]

^ There are two major areas in which the discussion of whether mathematics can play an explanatory role in science makes itself felt.
  • Explanation in Mathematics (Stanford Encyclopedia of Philosophy) 28 January 2010 0:26 UTC plato.stanford.edu [Source type: Academic]

^ The author's own article in the Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences may be useful in this regard.
  • Arithmetic - Mathematics and the Liberal Arts 3 February 2010 14:24 UTC math.truman.edu [Source type: FILTERED WITH BAYES]

.While, therefore, the logical development of algebraic reasoning must depend on certain fundamental relations, it is important that in the early study of the subject these relations should be introduced gradually, and not until there is some empirical acquaintance with the phenomena with which they are concerned.^ There appears therefore to be a tendency to use some larger number than ten as a basis for grouping into new units or for subdivision into parts.

^ The physically important concept of vector , generalized to vector spaces and studied in linear algebra , belongs to the two branches of structure and space.

^ Consumers consume when they don’t worry about the future, and there are lots of reasons for them to worry – some defined above and ad nauseum elsewhere.
  • The difficult arithmetic of Chinese consumption 3 February 2010 14:24 UTC mpettis.com [Source type: FILTERED WITH BAYES]

.8. The extension of the range of subjects to which mathematical methods can be applied, accompanied as it is by an extension of the range of study which is useful to the ordinary worker, has led in the latter part of the 19th century to an important reaction against the specialization mentioned in the preceding paragraph.^ In Bolzano's case, the aim of providing a reconstruction of parts of analysis and geometry, so that the exposition would use only “explanatory” proofs, also led to major mathematical results, such as his purely analytic proof of the intermediate value theorem.
  • Explanation in Mathematics (Stanford Encyclopedia of Philosophy) 28 January 2010 0:26 UTC plato.stanford.edu [Source type: Academic]

^ Myth #2: Children develop a deeper understanding of mathematics and a greater sense of ownership when they are expected to invent and use their own methods for performing the basic arithmetical operations, rather than study, understand and practice the standard algorithms.
  • Illinois Loop: Mathematics 28 January 2010 0:26 UTC www.illinoisloop.org [Source type: FILTERED WITH BAYES]

^ Typically, two opportunities to take each component are provided each year: Analysis and Linear Algebra - Numerical Methods (August and January), Applied Mathematics (January and May).
  • NJIT - Graduate Programs: Mathematics 28 January 2010 0:26 UTC catalog.njit.edu [Source type: Academic]

This reaction has taken the form of a return to the alliance between algebra and geometry (�5), on which modern analytical geometry is based; the alliance, however, being concerned with the application of graphical methods to particular cases rather than to general expressions. .These applications are sometimes treated under arithmetic, sometimes under algebra; but it is more convenient to regard graphics as a separate subject, closely allied to arithmetic, algebra, mensuration and analytical geometry.^ Other authors of physics texts have written chapters or sections in their texts using these applications of exponential arithmetic.
  • Presentation by Albert Bartlett - Arithmetic, Population, and Energy 3 February 2010 14:24 UTC hawaii.gov [Source type: FILTERED WITH BAYES]

^ As regards the first of these two examples, however, it would be more correct to write 1,820 for the former of the two meanings (cf.

^ This notation is more than a convenience; it is a powerful method of "doing algebra" on such array quantities, and can be used to prove all of the properties of matrices.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

.9. The association of algebra with arithmetic on the one hand, and with geometry on the other, presents difficulties, in that geometrical measurement is based essentially on the idea of continuity, while arithmetical measurement is based essentially on the idea of discontinuity; both ideas being equally matters of intuition.^ Topics include algebra, applied math, arithmetic, geometry, and measurement.

^ The application of arithmetical methods to geometrical measurement presents some difficulty.

^ The course presents ideas from an intuitive perspective that prepares teachers to discuss geometry with children, and from a computational perspective to enable teachers to work with students to calculate distance, area and volume in both customary and metric units, measure angles, construct figures, and more.

.The difficulty first arises in elementary mensuration, where it is partly met by associating arithmetical and geometrical measurement with the cardinal and the ordinal aspects of number respectively (see Arithmetic).^ Elementary number theory, modular arithmetic.

^ The study of structure starts with numbers , firstly the familiar natural numbers and integers and their arithmetical operations, which are recorded in elementary algebra .

^ This book is somewhere between simple arithmetic and elementary number theory, but develops the subjects quite differently than we do today.
  • Arithmetic - Mathematics and the Liberal Arts 3 February 2010 14:24 UTC math.truman.edu [Source type: FILTERED WITH BAYES]

.Later, the difficulty recurs in an acute form in reference to the continuous variation of a function.^ Each chapter has 15 or more exercises, of varying difficulty and nature, and its own references.
  • Textbook on Computer Arithmetic 3 February 2010 14:24 UTC www.ece.ucsb.edu [Source type: Academic]

^ Rigorous reconsideration of the real-number system, infinte series and of continuity, differentiation and integration for functions of one variable.

^ Algebraic and topological properties of the real numbers; limits of sequences and functions; continuity, differentiation and integration of functions of one variable; infinite series.

.Reference to a geometrical interpretation seems at first sight to throw light on the meaning of a differential coefficient; but closer analysis reveals new difficulties, due to the geometrical interpretation itself.^ Offered: AWSpS. MATH 309 Linear Analysis (3) NW First order systems of linear differential equations, Fourier series and partial differential equations, and the phase plane.

^ A string of statements of the form A ⇒ B ⇒ C ⇒ D should mean that A by itself implies B, and B by itself implies C, and C by itself implies D; that is the coventional interpretation given by mathematicians.

^ The evolution of the word to mean disorder seems to come from reference to the time before God created the universe.
  • Origins of some arithmetic terms 3 February 2010 14:24 UTC www.pballew.net [Source type: FILTERED WITH BAYES]

.One of the most recent developments of algebra is the algebraic theory of number, which is devised with the view of removing these difficulties.^ One of the earliest notations to indicate multiplication was by juxtaposition, placing the numbers adjacent to each other as we do for algebraic characters today.
  • Origins of some arithmetic terms 3 February 2010 14:24 UTC www.pballew.net [Source type: FILTERED WITH BAYES]

^ I will argue, on the contrary, that these developments confirmed his view.
  • Peter Suber, "Geometry and Arithmetic are Synthetic" 3 February 2010 14:24 UTC www.earlham.edu [Source type: FILTERED WITH BAYES]

^ One great area of responsibility of our community of scientists and engineers is vigorous pursuit of research and development in all these areas.
  • Arithmetic Presentation -Complete 3 February 2010 14:24 UTC www.mnforsustain.org [Source type: FILTERED WITH BAYES]

.The harmony between arithmetical and geometrical measurement, which was disturbed by the Greek geometers on the discovery of irrational numbers, is restored by an unlimited supply of the causes of disturbance.^ Thus, while arithmetical numbering refers to units, geometrical numbering does not refer to units but to the intervals between units.

^ The application of arithmetical methods to geometrical measurement presents some difficulty.

^ This book is somewhere between simple arithmetic and elementary number theory, but develops the subjects quite differently than we do today.
  • Arithmetic - Mathematics and the Liberal Arts 3 February 2010 14:24 UTC math.truman.edu [Source type: FILTERED WITH BAYES]

ro. .Two other developments of algebra are of special importance.^ "Take two algebra students -- one is still a little shaky on the distributive property, whereas the other knows it cold.
  • Illinois Loop: Mathematics 28 January 2010 0:26 UTC www.illinoisloop.org [Source type: FILTERED WITH BAYES]

^ But at least two other readings are possible, both of which preserve the possibility that mathematical theories of non-Euclidean spaces may be developed without contradiction.
  • Peter Suber, "Geometry and Arithmetic are Synthetic" 3 February 2010 14:24 UTC www.earlham.edu [Source type: FILTERED WITH BAYES]

.The theory of sequences and series is sometimes treated as a part of elementary algebra; but it is more convenient to regard the simpler cases as isolated examples, leading up to the general theory.^ It will show up again in elementary algebra.
  • Arithmetic: A Crash Review 3 February 2010 14:24 UTC www.zaimoni.com [Source type: Reference]

^ The great mathematician C. F. Gauss once wrote “Mathematics is the queen of sciences and number theory is the queen of mathematics.” Number theory is that part of mathematics dealing with the integers and certain natural generalizations.

^ Topics include real numbers and cardinality of sets, sequences and series of real numbers, metric spaces, continuous functions, integration theory, sequences and series of functions, and polynomial approximation.

.The treatment of equations of the second and higher degrees introduces imaginary and complex numbers, the theory of which is a special subject.^ Equation Solver This is an equation solver for binomials and trinomail(yes, a quadratic is a trinomial) and also solves for complex solutions, (imaginary numbers) .
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

^ The residue number system has provided an invaluable tool for researchers interested in complexity theory and limits of fast arithmetic as well as to the designers of fault-tolerant systems.
  • Textbook on Computer Arithmetic 3 February 2010 14:24 UTC www.ece.ucsb.edu [Source type: Academic]

^ To get the complex number field, we adjoin the element i, which satisfies the algebraic equation x 2 + 1 = 0.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

r. .One of the most difficult questions for the teacher of algebra is the stage at which, and the extent to which, the ideas of a negative number and of continuity may be introduced.^ Teachers will find plans for the strands of Number and Number Sense , Computation and Estimation , Measurement , Geometry , Probability and Statistics , and Patterns, Functions, and Algebra .

^ Rigorous reconsideration of the real-number system, infinte series and of continuity, differentiation and integration for functions of one variable.

^ After introducing how to divide by numbers of one digit, and then larger primes, he develops a set of "Composition Rules" for numbers with more than one digit.
  • Origins of some arithmetic terms 3 February 2010 14:24 UTC www.pballew.net [Source type: FILTERED WITH BAYES]

.On the one hand, the modern developments of algebra began with these ideas, and particularly with the idea of a negative number.^ The names of numbers give an idea of the way in which the idea of number has developed.

^ One great area of responsibility of our community of scientists and engineers is vigorous pursuit of research and development in all these areas.
  • Arithmetic Presentation -Complete 3 February 2010 14:24 UTC www.mnforsustain.org [Source type: FILTERED WITH BAYES]
  • Presentation by Albert Bartlett - Arithmetic, Population, and Energy 3 February 2010 14:24 UTC hawaii.gov [Source type: FILTERED WITH BAYES]

^ After introducing how to divide by numbers of one digit, and then larger primes, he develops a set of "Composition Rules" for numbers with more than one digit.
  • Origins of some arithmetic terms 3 February 2010 14:24 UTC www.pballew.net [Source type: FILTERED WITH BAYES]

.On the other land, the lateness of occurrence of any particular mathematical idea is usually closely correlated with its intrinsic difficulty.^ Emphasizes problem solving, communication of mathematical ideas, and analysis of sources of difficulty in learning/teaching these concepts.

^ Open-ended, vague, and/or ill-posed problems do not lend themselves to any particular mathematical approach or solution, nor do they generalize to other, future problems.
  • Illinois Loop: Mathematics 28 January 2010 0:26 UTC www.illinoisloop.org [Source type: FILTERED WITH BAYES]

^ If students see mathematical ideas in other times [and in other cultures], they can appreciate the ideas better in our own.
  • Arithmetic - Mathematics and the Liberal Arts 3 February 2010 14:24 UTC math.truman.edu [Source type: FILTERED WITH BAYES]

.Moreover, the ideas which are usually formed on these points at an early stage are incomplete; and, if the incompleteness of an idea is not realized, operations in which it is implied are apt to be purely formal and mechanical.^ In pure-2 counting, there are separate words for one and two and these are used to form all other number words.
  • Arithmetic - Mathematics and the Liberal Arts 3 February 2010 14:24 UTC math.truman.edu [Source type: FILTERED WITH BAYES]

^ Rather than using all these digits, floating-point hardware normally operates on a fixed number of digits.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ These strange vapors, without shape or form, reminded him of the Greek idea of Chaos, so he called them by the Germanic (Flemish is a dialect of German) spelling of chaos, gas .
  • Origins of some arithmetic terms 3 February 2010 14:24 UTC www.pballew.net [Source type: FILTERED WITH BAYES]

What are called negative numbers in arithmetic, for instance, are not really negative numbers but negative quantities (� 27 (i.)); and the difficulties incident to the ideas of continuity have already been pointed out.
.12. In the present article, therefore, the main portions of elementary algebra are treated in one section, without reference to these ideas, which are considered generally in two separate sections.^ The letter, addressed to the two authors as well as the fact-checker on the article and CC'd to David Remnick and the New Yorker's general counsel, runs 12 pages, so you may want to have a look at the press release instead .
  • mathematics (kottke.org) 28 January 2010 0:26 UTC www.kottke.org [Source type: General]

^ While taking a few liberties as far as input and output in COBOL in the above lines, the general idea is that PAY-HOURS has one decimal place and PAY-RATE and TOTAL-PAY have two decimal places.
  • Arithmetic: Programming Language History 3 February 2010 14:24 UTC hhh.gavilan.edu [Source type: FILTERED WITH BAYES]

^ If four coins are laid on a table, close together, they can (by most adults) be seen to be four, without counting; but seven coins have to be separated mentally into two groups, the numbers of which are added, or one group has to be seen and the remaining objects counted, before the number is known to be seven.

.These three sections may therefore be regarded as to a certain extent concurrent.^ These may be divided into three classes, as follows: (i) The fraction of a concrete quantity may itself not exist as a concrete quantity, but be represented by a token.

^ It may also be aided, to a certain extent, by the tendency to find rhythms in sequences of sounds.

They are preceded by two sections dealing with the introduction to algebra from4the arithmetical and the graphical sides, and are followed by a section dealing briefly with the developments mentioned in �� 9 and 1 o above.
.I. Arithmetical Introduction to Algebra. 13. Order of Arithmetical Operations.^ The next area of difference among languages is the order of evaluation or precedence of arithmetic operators.
  • Arithmetic: Programming Language History 3 February 2010 14:24 UTC hhh.gavilan.edu [Source type: FILTERED WITH BAYES]

^ The study of structure starts with numbers , firstly the familiar natural numbers and integers and their arithmetical operations, which are recorded in elementary algebra .

- .It
is important, before beginning the study of algebra, to have a clear idea as to the meanings of the symbols used to denote arithmetical operations.^ The meaning of the × symbol should be clear from the context.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ The percent symbol (%) is often used to indicate a modulus or remainder operation.
  • Arithmetic: Programming Language History 3 February 2010 14:24 UTC hhh.gavilan.edu [Source type: FILTERED WITH BAYES]

^ X3 and 3X4 mean different things, but the operations which they denote produce the same result.

(i.) .Additions and subtractions are performed from left to right.^ It is desirable, wherever possible, to perform operations on numbers or numerical quantities from the left, rather than from the right.

^ Commutative Law for Additions and Subtractions, that additions and subtractions may be performed in any order; e.g.

^ The intermediate values obtained by the successive additions are different according as we work from the left or from the right, being £9, 5s.

.Thus 3 lb + 5 lb - 7 lb + 2 lb means that 5 lb is to be added to 3 lb, 7 lb subtracted from the result, and 2 lb added to the new result.^ Matrices of the same shape can be added or subtracted, element by element: C = A + B means c ij = a ij + b ij .
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ The subtraction of 4 from 9 may mean either " What has to be added to 4 in order to make up a total of 9," or " To what has 4 to be added in order to make up a total of 9."

^ The only way that the result can be larger than x + y is if we right-shift y by more than g digits, thus losing some of its digits and, hence, subtracting a smaller magnitude from x .
  • Textbook on Computer Arithmetic 3 February 2010 14:24 UTC www.ece.ucsb.edu [Source type: Academic]

(ii.) .The above operation is performed with i lb as the unit of counting, and the process would be the same with any other unit; e.g. we should perform the same process to find 3s.+5s.^ It should be remembered that the counting is performed with something as unit.

^ If we count forwards we find that to convert £3, 5s.

^ Students should be able to evaluate the same author's statement about coal, "At least 220 billion tons of immediately recoverable coal - awaits mining in the United States."
  • Arithmetic Presentation -Complete 3 February 2010 14:24 UTC www.mnforsustain.org [Source type: FILTERED WITH BAYES]
  • Presentation by Albert Bartlett - Arithmetic, Population, and Energy 3 February 2010 14:24 UTC hawaii.gov [Source type: FILTERED WITH BAYES]

-7s.+2s. .Hence we can separate the numbers from the common unit, and replace 3 lb - I - 5 lb - 7 lb +2 lb by (3+5-7+2) lb, the additions and subtractions being then performed by means of an addition-table.^ Simply specify the type of addition, subtraction, multiplication, or division, problem for either the counting numbers, the negative numbers, decimals, or fractions and the maximum and minimum numbers to be used in the problems.

^ If the number to be subtracted from is less than the number being subtracted, reverse the order of subtraction -- and remember to apply a negative sign to the answer, at the end.
  • Arithmetic: A Crash Review 3 February 2010 14:24 UTC www.zaimoni.com [Source type: Reference]

^ Robert Bunge's WorksheetMaker generates online whole number practice worksheets in addition, subtraction, multiplication, and division for students in grades 2-7.

(iii.) .Multiplications, represented by X, are performed from right to left.^ It is desirable, wherever possible, to perform operations on numbers or numerical quantities from the left, rather than from the right.

^ Diagrams of Division.-Since we write from left to right or downwards, it may be convenient for division to interchange the rows or the columns of the multiplication-diagram.

^ If the Arabic numerals representing two numbers have the same length, we compare their digits going left to right.
  • Arithmetic: A Crash Review 3 February 2010 14:24 UTC www.zaimoni.com [Source type: Reference]

.Thus 5 X 3 X 7 X r lb means 5 times 3 times 7 times lb; i.e. it means that i lb is to be multiplied by 7, the result by 3, and the new result by 5. We may regard this as meaning the same as 5 X3 X7 lb, since 7 lb itself means 7 X 1 lb, and the lb is the unit in each case.^ In the case of numbers the X may be replaced by a dot; thus 4.3 means 4 times 3.

^ X3 and 3X4 mean different things, but the operations which they denote produce the same result.

^ Hence we must express I, which itself means $ times, as being 7 times something.

But it does not mean the same as 5 X 21 lb, though the two are equal, i.e. give the same result (see � 23).
This rule as to the meaning of X is important. .If it is intended that the first number is to be multiplied by the second, a special sign such as X should be used.^ The Hebrews had a notation containing separate signs (the letters of the alphabet) for numbers from t to to, then for multiplies of to up to zoo, and then for multiples of too up to 400, and later up to moo.

^ Suppose the second multiply raises an exception, and the trap handler wants to use the value of a .
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ Special cases: If either argument is NaN, then the result is NaN. If the first argument is positive zero and the second argument is positive, or the first argument is positive and finite and the second argument is positive infinity, then the result is positive zero.
  • Math (Java 2 Platform SE 5.0) 2 February 2010 15:42 UTC java.sun.com [Source type: Academic]

(iv.) .The sign means that the quantity or number preceding it is to be divided by the quantity or number following it.^ Only works for linear divisors which means the thing you are dividing by cannot excede x+/-a(any number).
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

^ The symbols - and = mean respectively that the first quantity mentioned is to be reduced or divided by the second; but there is some vagueness about + and X. In the present article a+b will mean that a is taken first, and b added to it; but a X b will mean that b is taken first, and is then multiplied by a.

^ What this rule really means is that if you take a medium-sized number and divide it by an enormous number, you get a number very close to 0.

(v.) .The use of the solidus / separating two numbers is for convenience of printing fractions or fractional numbers.^ Simply specify the type of addition, subtraction, multiplication, or division, problem for either the counting numbers, the negative numbers, decimals, or fractions and the maximum and minimum numbers to be used in the problems.

^ Generally, to find the sum or difference of two or more fractional numbers, we must replace them by other fractional numbers having the same denominator; it is usually most convenient to take as this denominator the L.C.M. of the original fractional numbers (cf.

^ To summarize, instructions that multiply two floating-point numbers and return a product with twice the precision of the operands make a useful addition to a floating-point instruction set.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

.Thus 16/4 does not mean 16+.^ Thus " ioo " on a page of a book does not mean that the page is ioo times the page numbered 1, but merely that it is the page after 99.

4, but .
(vi.) .Any compound operation not coming under the above descriptions is to have its meaning made clear by brackets, the use of a pair of brackets indicating that the expression between them is to be treated as a whole.^ The percent symbol (%) is often used to indicate a modulus or remainder operation.
  • Arithmetic: Programming Language History 3 February 2010 14:24 UTC hhh.gavilan.edu [Source type: FILTERED WITH BAYES]

^ We use a matching pair of parentheses ( ) or brackets [ ] to indicate explicitly that the operation inside the parentheses, or brackets, is to be done before using the result outside of the parentheses or brackets.
  • Arithmetic: A Crash Review 3 February 2010 14:24 UTC www.zaimoni.com [Source type: Reference]

^ Operators are used to indicate what type of arithmetic operation is needed, such as subtraction, multiplication, etc.
  • Arithmetic: Programming Language History 3 February 2010 14:24 UTC hhh.gavilan.edu [Source type: FILTERED WITH BAYES]

Thus we should not write 8X7+6, but (8 X7)+6, or 8X(7+6). The sign X coming immediately before, or immediately after, a bracket may be omitted; e.g. 8X(7+6) may be written 8(7+6).
.This rule as to using brackets is not always observed, the convention sometimes adopted being that multiplications or divisions are to be performed before additions or subtractions.^ Elementary arithmetic taught addition, subtraction, multiplication and division.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ The "My Dear Aunt Sally" interpretation (multiplication, division, addition, subtraction): Perform multiplication before division.

^ Here are some of the most widely used interpretations: The "BODMAS interpretation" (bracketed operations, division, multiplication, addition, subtraction): Perform division before multiplication.

.The convention is even pushed to such an extent as to make " 1 2 " " 2 ";, 42+33 of 7+5 mean 41+(3 3 of 7)+5 though it is not clear what " rind the value of 42+33 times 7+ 5" would then mean.^ This would mean that the number starving at the end of the doubling time would be twice the number that are starving today.
  • Arithmetic Presentation -Complete 3 February 2010 14:24 UTC www.mnforsustain.org [Source type: FILTERED WITH BAYES]
  • Presentation by Albert Bartlett - Arithmetic, Population, and Energy 3 February 2010 14:24 UTC hawaii.gov [Source type: FILTERED WITH BAYES]

^ China is a continent-size economy and it’s still mainly driven by domestic demand even though the export is very important and a bit too high for such a big economy.
  • The difficult arithmetic of Chinese consumption 3 February 2010 14:24 UTC mpettis.com [Source type: FILTERED WITH BAYES]

^ The length of this perpendicular would then be the mean proportional, since it is an altitude of a right triangle and makes two proportional right triangles, from which the result follows.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

There are grave objections to an arbitrary rule of this kind, the chief being the useless waste of mental energy in remembering it.
(vii.) .The only exception that may be made to the above rule is that an expression involving multiplication-dots only, or a simple fraction written with the solidus, may have the brackets omitted for additions or subtractions, provided the figures are so spaced as to prevent misunderstanding.^ Solve real-life problems involving addition and subtraction of simple fractions.
  • Maine Learning Results 28 January 2010 0:26 UTC www.state.me.us [Source type: Reference]

^ Distributive Law, that multiplications and divisions may be distributed over additions and subtractions, e.g.

^ Additions and subtractions are simple.

Thus 8+ (7 X 6) +3 may be written 8+7.6+3, and 8+s+3 may be written 8+7/6+3. But 2.4 should be written (3.5)/(2.4), not 3.5/2.4.
14. Latent Equations. - .The equation exists, without being shown as an equation, in all those elementary arithmetical processes which come under the head of inverse operations; i.e. processes which consist in obtaining an answer to the question " Upon what has a given operation to be performed in order to produce a given result?"^ Carols own answer given in 1896: Because it can produce a few notes, tho they are very flat; and it is nevar put with the wrong end in front!
  • Mathematics in Movies 28 January 2010 0:26 UTC www.math.harvard.edu [Source type: FILTERED WITH BAYES]

^ Hence, so long as the denominator remains unaltered, we can deal with, exactly as if they were numbers, any operations being performed on the numerators.

^ X3 and 3X4 mean different things, but the operations which they denote produce the same result.

or to the question ." What operation of a given kind has to be performed on a given quantity or number in order to produce a given result?"^ Hence, so long as the denominator remains unaltered, we can deal with, exactly as if they were numbers, any operations being performed on the numerators.

^ It is desirable, wherever possible, to perform operations on numbers or numerical quantities from the left, rather than from the right.

^ In simple terms, an overflow will occur if the result produced by a 64 COMPUTER ARITHMETIC given operation is outside the range of the representable numbers.
  • 3.Computer Arithmetic 3 February 2010 14:24 UTC www.slideshare.net [Source type: Reference]

(i.) .In the case of subtraction the second of these two questions is perhaps the simpler.^ Discount may be allowed twice in succession off quoted prices; in such cases the second discount is off the reduced price, and therefore it is not correct to add the two rates of discount together.

^ To compare two fractions: Subtract the second fraction from the first fraction.
  • Arithmetic: A Crash Review 3 February 2010 14:24 UTC www.zaimoni.com [Source type: Reference]

^ Although multiplication may arise in either of these two ways, the actual process in each case is performed by commencing with the unit and taking it the necessary number of times.

.Suppose, for instance, that we wish to know how much will be left out of ios.^ Although formula (7) is much more accurate than (6) for this example, it would be nice to know how well (7) performs in general.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ Use this easy new car payment calculator to find out how much your monthly car payment is when you buy a new car .
  • Alltop - Top Math News 2 February 2010 15:42 UTC math.alltop.com [Source type: General]

^ Students don’t know how to do basic arithmetic because our schools don’t teach it very well nor do they expect very much from the students.
  • Algebra = ‘most failed’ college class « Joanne Jacobs 10 February 2010 11:011 UTC www.joannejacobs.com [Source type: Original source]

after spending .3s., or how much has been spent out of 10s.^ Use this easy new car payment calculator to find out how much your monthly car payment is when you buy a new car .
  • Alltop - Top Math News 2 February 2010 15:42 UTC math.alltop.com [Source type: General]

^ Once you have used this program, you will see how much thinking and working out it can really save you!!!
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

if 3s. is left. .In either case we may put the question in two ways: - (a) What must be added to 3s.^ To subtract, we may proceed in either of two ways.

^ The various stages in the study of arithmetic may be arranged in different ways, and the arrangement adopted must be influenced by the purpose in view.

^ Discount may be allowed twice in succession off quoted prices; in such cases the second discount is off the reduced price, and therefore it is not correct to add the two rates of discount together.

in order to produce 10s., or (b) To what must 3s. be added in order to produce ros. .If the answer to the question is X, we have either (a) I os.^ Questions and answers are also stored in a data base that you can examine either through a table of contents (arranged by level and topic), or through a search.
  • Mathematics Archives - K12 Internet Sites 2 February 2010 15:42 UTC archives.math.utk.edu [Source type: Academic]

^ Kernighan and Ritchie do not have a clear answer for this question either.
  • Arithmetic: Programming Language History 3 February 2010 14:24 UTC hhh.gavilan.edu [Source type: FILTERED WITH BAYES]

=3S. +X, .'. X =rOS. - 3S.
or (b) ros. = X+3S., ..X =ros. - 3s.
(ii.) .In the above case the two different kinds of statement lead to arithmetical formulae of the same kind.^ The formula is correct when a and b are positive real numbers, but it leads to errors when generalized indiscriminately to other kinds of numbers.

^ It is hard to classify the different kinds of mistakes they make, but in many cases their mistakes are related to this one: Everything is additive.

^ Some students may be puzzled by the differences between the two versions of the Integration by Parts formula (in boxes, in the last few paragraphs).

.In the case of division we get two kinds of arithmetical formula, which, however, may be regarded as requiring a single kind of numerical process in order to determine the final result.^ Two exceptions, however, may be noted.

^ Numbers whose Arabic numeral representation end in the digits 0, 2, 4, 6, or 8 are divisible by the number two.
  • Arithmetic: A Crash Review 3 February 2010 14:24 UTC www.zaimoni.com [Source type: Reference]

^ In this method a division was accomplished by breaking the divisor into its factors, and then dividing the dividend by one of the factors, and sequentially dividing the resulting quotient by each remaining factor in turn to get a final quotient.
  • Origins of some arithmetic terms 3 February 2010 14:24 UTC www.pballew.net [Source type: FILTERED WITH BAYES]

(a) If 24d. is divided into 4 equal portions, how much will each portion be ?
Let the answer be X; then 24d. =4 X X, .'. X=1 of 24d.
.(b) Into how many equal portions of 6d.^ The author begins with a discussion of the division of the scale into twelve equal semitones, and how this appears natural from the continued fraction representation of log 2 3.
  • Arithmetic - Mathematics and the Liberal Arts 3 February 2010 14:24 UTC math.truman.edu [Source type: FILTERED WITH BAYES]

each may 24d. be divided ?
Let the answer be x; then 24d. = x X 6d., .. x =24d. - 6d.
(iii.) .Where the direct operation is evolution, for which there is no commutative law, the two inverse operations are different in kind.^ There is no difference in principle between addition (or subtraction) of numbers and addition (or subtraction) of numerical quantities.

^ There is no essential difference, however, between this and the denary basis.

^ If there is a unit element for multiplication in the ring, multiplication is commutative, and--most importantly--a multiplication inverse for every element is in the set, the ring becomes a field .
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

.(a) What would be the dimensions of a cubical vessel which would exactly hold 125 litres; a litre being a cubic decimetre ?^ The litre is equal to a cubic decimetre.

Let the answer be X; then 125 c.dm. =X 3, .'. X = ' 125 c.dm. = 1 125 dm.
(b) To what power must 5 be raised to produce 125 ? Let the answer be x; then 125 =5 x, .. x = logs 125.
.15. With regard to the above, the following points should be noted.^ In line 16, "86-bit floating-point format" should be "80-bit floating-point format (sign, 15-bit exponent field, 64-bit significand)".
  • Textbook on Computer Arithmetic 3 February 2010 14:24 UTC www.ece.ucsb.edu [Source type: Academic]

^ Alpha should be inserted in the numerator of the fraction that follows "Note: h =".
  • Textbook on Computer Arithmetic 3 February 2010 14:24 UTC www.ece.ucsb.edu [Source type: Academic]

(I) When what we require to know is a quantity, it is simplest to deal with this quantity as a whole. In (i.), for instance, we want to find the amount by which ios. exceeds 3s., not the number of shillings in this amount. .It is true that we obtain this result by subtracting 3 from io by means of a subtractiontable (concrete or ideal); but this table merely gives the generalized results of a number of operations of addition or subtraction performed with concrete units.^ Simply specify the type of addition, subtraction, multiplication, or division, problem for either the counting numbers, the negative numbers, decimals, or fractions and the maximum and minimum numbers to be used in the problems.

^ Robert Bunge's WorksheetMaker generates online whole number practice worksheets in addition, subtraction, multiplication, and division for students in grades 2-7.

^ Since all languages agreed on the symbol and result for addition, subtraction, and multiplication we will start our discussion where things start to disagree.
  • Arithmetic: Programming Language History 3 February 2010 14:24 UTC hhh.gavilan.edu [Source type: FILTERED WITH BAYES]

.We must count with something; and the successive somethings obtained by the addition of successive units are in fact numerical quantities, not numbers.^ Adding two Arabic numerals representing negative numbers uses a similar procedure: do the addition for the Arabic numerals as if they were positive, then prefix the - to the result.
  • Arithmetic: A Crash Review 3 February 2010 14:24 UTC www.zaimoni.com [Source type: Reference]

^ The base of a numeral system is defined when said system uses digits (including the digit representing the number zero) to use finite tables to define arbitrary-precision addition and multiplication.
  • Arithmetic: A Crash Review 3 February 2010 14:24 UTC www.zaimoni.com [Source type: Reference]

^ The fact that no numbers are given made this passage particularly hard to decipher, and it was not properly understood for many years; hence we can see the advantages of numerical examples.
  • Arithmetic - Mathematics and the Liberal Arts 3 February 2010 14:24 UTC math.truman.edu [Source type: FILTERED WITH BAYES]

.Whether this principle may legitimately be extended to the notation adopted in (iii.^ We might extend this principle to cases in which the terms of two series, whether of numbers or £1 A £ 1 53, 7 s.

) (a) of � 14 is a moot point. .But the present tendency is to regard the early association of arithmetic with linear measurement as important; and it seems to follow that we may properly (at any rate at an early stage of the subject) multiply a length by a length, and the product again by another length, the practice being dropped when it becomes necessary to give a strict definition of multiplication.^ The forms seem to result from a general tendency to visualization as an aid to memory; the letter-forms may in the first instance be quite as frequent as the numberforms, but they vanish in early childhood, being of no practical value, while the number-forms continue as an aid to arithmetical work.

^ If it can be simplified, that may reduce the length of the numerals in the following calculation.
  • Arithmetic: A Crash Review 3 February 2010 14:24 UTC www.zaimoni.com [Source type: Reference]

^ The negative sign - is used to indicate the following (among other possibilities): Measurement of a length, or angle, against the usual orientation.
  • Arithmetic: A Crash Review 3 February 2010 14:24 UTC www.zaimoni.com [Source type: Reference]

.(2) The results may be stated briefly as follows, the more usual form being adopted under (iii.^ Then, the program will do one of the following: Give both equations in slope-intercept form and state their intersection; State that the lines are parrallel, show their slopes, and their y-intercepts; State that the lines are coinciding and show their equations in slope-intercept form.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

^ This may cost more money for students, so the math department is looking at doing an “e-text” where students can download homework in PDF form, but nothing has been determined yet.” .
  • Algebra = ‘most failed’ college class « Joanne Jacobs 10 February 2010 11:011 UTC www.joannejacobs.com [Source type: Original source]

^ If the number of decimal places to which a result is to be accurate is determined beforehand, it is usually not necessary in the actual working to go to more than two or three places beyond this.

) (a): (1.) If A=B+X, or=X+B, then X=A-B.
(ii.) (a) If A = m times X, then X= - In ' of A.
(b) If A=x times M, then x (iii.) (a) If n= P then x = ?i n.
(b) If n = a z, then x =tog a n.
.The important thing to notice is that where, in any of these five cases, one statement is followed by another, the second is not to be regarded as obtained from the first by logical reasoning involving such general axioms as that " if equals are taken from equals the remainders are equal "; the fact being that the two statements are merely different ways of expressing the same relation.^ These are, of course, not the same thing in general.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ The two statements are in fact merely different aspects of a single relation, considered in the next section.

^ Do the following two statements obtain the same result: .
  • Arithmetic: Programming Language History 3 February 2010 14:24 UTC hhh.gavilan.edu [Source type: FILTERED WITH BAYES]

.To say, for instance, that X is equal to A -B, is the same thing as to say that X is a quantity such that X and B, when added, make up A; and the above five statements of necessary connexion between two statements of equality are in fact nothing more than definitions of the symbols -, m of, =,, and loga.^ No more than two indices can be the same in a term.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ We often round the probability of such things up or down because they are so likely or unlikely to occur, that it's easier to recognise them as a probability of one or zero.

^ Or, on the quinary- binary system , we need only give independent definitions to the numbers up to five; the numbers six, seven,..

.An apparent difficulty is that we use a single symbol - to denote the result of the two different statements in (i.^ [Do not list results using two digits.
  • Arithmetic: A Crash Review 3 February 2010 14:24 UTC www.zaimoni.com [Source type: Reference]

^ You still have supplied no evidence to support that statement, especially as you have apparently used absolutely zero measures to control for any biases you may have.
  • Algebra = ‘most failed’ college class « Joanne Jacobs 10 February 2010 11:011 UTC www.joannejacobs.com [Source type: Original source]

^ Growth Functions This does two different things using one formula.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

) (a) and (i.) (b) of � 14. This is due to the fact that there are really two kinds of subtraction, respectively involving counting forwards (complementary addition) and counting backwards (ordinary subtraction); and it suggests that it may be wise not to use the one symbol - to represent the result of both operations until the commutative law for addition has been fully grasped.
.16. In the same way, a statement as to the result of an inverse operation is really, by the definition of the operation, a statement as to the result of a direct operation.^ Thinking about floating-point in this fuzzy way stands in sharp contrast to the IEEE model, where the result of each floating-point operation is precisely defined.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ X3 and 3X4 mean different things, but the operations which they denote produce the same result.

^ Thus, when a program is moved from one machine to another, the results of the basic operations will be the same in every bit if both machines support the IEEE standard.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

If, for instance, we state that A=X - B, this is really a statement that X=Ad-B. Thus, corresponding to the results under � 15 (2), we have the iollowing: (i) Where the inverse operation is performed on the unknown quantity or number: (i.) If A=X-B, then X=A+B.
(ii.) (a) If M =m of X, then X=m times M.
(b) If m = X = M, then X =m times M.
(iii.) .(a) If a= then x = aP. (b) If p = log a x, then x = aP. (2) Where the inverse operation is performed with the unknown quantity or number: (i.^ It is desirable, wherever possible, to perform operations on numbers or numerical quantities from the left, rather than from the right.

^ Hence, so long as the denominator remains unaltered, we can deal with, exactly as if they were numbers, any operations being performed on the numerators.

^ It would therefore be better, in some ways, to retain the unit throughout, and to describe - 4A as a negative quantity, in order to avoid confusion with the " negative numbers " with which operations are performed in formal algebra.

) If B=A-X, then A=B+X.
(ii.) (a) If m =A=X, then A =m times X.
(b) If M=x of A, then A=x times M.
(iii.) (a) If p = log i n, then n = (b) If a = n, then n =a'.
In each of these cases, however, the reasoning which enables us to replace one statement by another is of a different kind from the reasoning in the corresponding cases of � 15. There we proceeded from the direct to the inverse operations; i.e. so far as the nature of arithmetical operations is concerned, we launched out on the unknown. .In the present section, however, we return from the inverse operation to the direct; i.e. we rearrange our statement in its simplest form.^ In math, we generally prefer to write our answers in simplest form (and we sometimes insist on it).

^ However, proofs in this system cannot verify the algorithms of sections Cancellation and Exactly Rounded Operations , which require features not present on all hardware.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ The first occurance in the text, on page 36, without prior definition introduces students to a set of problems with the directions, "Reduce the fractions below to simplest forms".
  • Origins of some arithmetic terms 3 February 2010 14:24 UTC www.pballew.net [Source type: FILTERED WITH BAYES]

The statement, for instance, that 32 - x = 25, is really a statement that 32 is the sum of x and 25.
17. The five equalities which stand first in the five pairs of equalities in � 15 (2) may therefore be taken as the main types of a simple statement of equality. .When we are familiar with the treatment of quantities by equations, we may ignore the units and deal solely with numbers; and (ii.^ The pair of compartments on either side may, as here, contain numerical quantities, or may contain numbers.

^ S. The simplest case, in which the quantity can be expressed as an integral number of the largest units B involved, has already been considered (§§ 37, 42).

^ The £I is termed the unit, A numerical quantity, therefore, represents a certain unit, taken a certain number of times.

) (a) and (ii.) .(b) may then, by the commutative law for multiplication, be regarded as identical.^ Commutative Law for Multiplications and Divisions, that multiplications and divisions may be performed in any order: e.g.

^ Commutative Law for Additions and Subtractions, that additions and subtractions may be performed in any order; e.g.

^ Division.-From the commutative law for multiplication, which shows that 3 X 4d.

The five processes of deduction then reduce to four, which may be described as (i.) subtraction, (ii.) division, (iii.) (a) taking a root, (iii.) (b) taking logarithms. .It will be found that these (and particularly the first three) cover practically all the processes legitimately adopted in the elementary theory of the solution of equations; other processes being sometimes liable to introduce roots which do not satisfy the original equation.^ The program will also FACTORIZE all cubic equations with three rational roots.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

^ An approximation to a square root by comparing with other solutions to an equation x 2 + D = y 2 .
  • Arithmetic - Mathematics and the Liberal Arts 3 February 2010 14:24 UTC math.truman.edu [Source type: FILTERED WITH BAYES]

^ In this method, we use the first equation to eliminate x 1 in all the other equations.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

18. It should be noticed that we are still dealing with the elementary processes of arithmetic, and that all the numbers contemplated in �� 14-17 are supposed to be positive integers. .If, for instance, we are told that 15= 4 of (x- 2), what is meant is that (I) there is a number u such that x=u+2, (2) there is a number v such that u=4 times v, and (3) 15=3 times v.^ There exists a positive number b such that for each positive number a we have b less than a.

^ For instance, here is the definition of continuity of a real-valued function f: f is continuous if for each real number p and each positive number ε there exists a positive number δ (which may depend on p and ε) such that, for each real number q, if .

^ If there is an integral number to be taken as well as a percentage, this number is written in front of the point; thus 23.76 X A means 23 times A, with 76% of A. We might therefore denote 76% by 0.76.

From these statements, working backwards, we find successively that v= 5, u = 20, X = 22. The deductions follow directly from the definitions, and such mechanical processes as "clearing of fractions " find no place (� 21 (ii.)). The extension of the methods to fractional numbers is part of the establishment of the laws governing these numbers (� 27 (ii.)).
lg. .Expressed Equations.-The simplest forms of arithmetical equation arise out of abbreviated solutions of particular problems.^ Thomas Harriot had described negative roots as the solution to an alternate form of the equation with the signs of the odd powers changed.
  • Origins of some arithmetic terms 3 February 2010 14:24 UTC www.pballew.net [Source type: FILTERED WITH BAYES]

^ Matrix exponential, fundamental solution matrix, phase-space and portraits, stability, initial- and boundary-value problems, introduction to partial differential equations.

^ Numerical solution of initial and boundary value problems of ordinary and partial differential equations.

In accordance with � 15, it is desirable that our statements should be statements of equality of quantities rather than of numbers; and it is convenient in the early stages to have a distinctive notation, e.g. to represent the former by capital letters and the latter by small letters.
As an example, take the following. I buy 2 lb of tea, and have 6s. 8d. left out of Ios.; how much per lb did tea cost? (I) In ordinary language we should say: Since 6s. 8d. was left, the amount spent was Ios. - 6s. 8d., i.e. was 3s. 4d. Therefore 2 lb of tea cost 3s. 4d. Therefore 1 lb of tea cost is. 8d.
.(2) The first step towards arithmetical reasoning in such a case is the introduction of the sign of equality.^ For example sums are a special case of inner products, and the sum ((2 × 10 -30 + 10 30 ) - 10 30 ) - 10 -30 is exactly equal to 10 -30 , but on a machine with IEEE arithmetic the computed result will be -10 -30 .
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ This program also features other such equations where theres are unkowns on both sides of the equal sign.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

^ In the leftmost column, m should be inserted before the first equal sign in each box (three instances).
  • Textbook on Computer Arithmetic 3 February 2010 14:24 UTC www.ece.ucsb.edu [Source type: Academic]

Thus we say: Cost of 2 lb tea+6s. 8d.= Ios.
.'. Cost of 2 lb tea = ios. -6s. 8d. =3s. 4d.
Cost of i lb tea = is. .8d (3) The next step is to show more distinctly the unit we are dealing with (in addition to the money unit), viz.^ The next problem is a "two-step" variant on the preceding problem, it requires more thought, when it is not preceded by Problem 22.
  • Reading Instruction for Arithmetic Word Problems: 3 February 2010 14:24 UTC www.math.umd.edu [Source type: FILTERED WITH BAYES]

the cost of 1 lb tea. We write: (2 Xcost of 1 lb tea) +6s. 8d. = Ios.
.'. 2 Xcost of I lb tea =Ios. -6s. 8d. =3s. 4d.
.'. Cost of 1 lb tea= Is. 8d.
.(4) The stage which is introductory to algebra consists merely in replacing the unit " cost of 1 lb tea " by a symbol, which may be a letter or a mark such as the mark of interrogation, the asterisk, &c.^ What is called modern algebra works with symbols that may obey different rules of composition or operations than the familiar ones of real numbers that we have just presented.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ Anecdotal evidence consisting merely of two countries does not justify such an extreme statement as the top third of American students are the best in the world.
  • Algebra = ‘most failed’ college class « Joanne Jacobs 10 February 2010 11:011 UTC www.joannejacobs.com [Source type: Original source]

^ Indeed, the very principle of using symbols to represent general quantities in algebra may make a problem easier to solve.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

If we denote this unit by X, we have (2 XX) +6s. 8d. = Ios.
. 2 X X = Ios. - 6s. 8d. =3s. 4d.
.. X= i s. 8d.
20. Notation of Multiples.-The above is arithmetic. .The only thing which it is necessary to import from algebra is the notation by which we write 2X instead of 2 X X or 2. X. This is rendered possible by the fact that we can use a single letter to represent a single number or numerical quantity, however many digits are contained in the number.^ Do this as many times as the number represented by the given Arabic numeral.
  • Arithmetic: A Crash Review 3 February 2010 14:24 UTC www.zaimoni.com [Source type: Reference]

^ Only one of these numbers is necessary.
  • Arithmetic: A Crash Review 3 February 2010 14:24 UTC www.zaimoni.com [Source type: Reference]

^ However, this instruction has many other uses.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

.It must be remembered that, if a is a number, 3a means 3 times a, not a times 3; the latter must be represented by aX3 or a.^ This would mean that the number starving at the end of the doubling time would be twice the number that are starving today.
  • Arithmetic Presentation -Complete 3 February 2010 14:24 UTC www.mnforsustain.org [Source type: FILTERED WITH BAYES]
  • Presentation by Albert Bartlett - Arithmetic, Population, and Energy 3 February 2010 14:24 UTC hawaii.gov [Source type: FILTERED WITH BAYES]

^ Hence we must express I, which itself means $ times, as being 7 times something.

^ Do this as many times as the number represented by the given Arabic numeral.
  • Arithmetic: A Crash Review 3 February 2010 14:24 UTC www.zaimoni.com [Source type: Reference]

3.
The number by which an algebraical expression is to be multiplied is called its coefficient. Thus in 3a the coefficient of a is 3. But in 3.4a the coefficient of 4a is 3, while the coefficient of a is3.4.
.21. Equations with Fractional Coefficients.-As an example of a special form of equation we may take zx+ 3x = Io.^ The above example will factor as (X-1)(X+1)(9X+13)(13X-21), If the polynomial has irrational or imaginary/complex roots in addition to rational roots, the program will show the coefficients of the remaining polynomial.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

^ To form a polynomial with reciprocal roots, just take the coefficients in opposite order: 1 + 2x - 3x 2 .
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ (Note this is not necessarily what modern teachers would call in "simplest form", for example 8/4 is a simple fraction) "4.
  • Origins of some arithmetic terms 3 February 2010 14:24 UTC www.pballew.net [Source type: FILTERED WITH BAYES]

(i.) .There are two ways of proceeding.^ Myth #3: There are two separate and distinct ways to teach mathematics.
  • Illinois Loop: Mathematics 28 January 2010 0:26 UTC www.illinoisloop.org [Source type: FILTERED WITH BAYES]

^ Every nonzero complex number b has two square roots, but in general there is no natural way to say which one should be associated with the expression √b.

^ On the other hand, if there is no way of writing as the coproduct of two other root systems we say it’s “irreducible”.
  • Alltop - Top Math News 2 February 2010 15:42 UTC math.alltop.com [Source type: General]

.(a) The statement is that (I) there is a number u such that x= 2u,(2) there is a number v such that x= 3v, and (3) u+v= Io.^ There exists a positive number b such that for each positive number a we have b less than a.

^ For instance, here is the definition of continuity of a real-valued function f: f is continuous if for each real number p and each positive number ε there exists a positive number δ (which may depend on p and ε) such that, for each real number q, if .

^ Much of the discussion on number systems may be familiar, but here there is also a little that may be a little less familiar, such as the use of Etruscan letters in the early Roman numerals.
  • Arithmetic - Mathematics and the Liberal Arts 3 February 2010 14:24 UTC math.truman.edu [Source type: FILTERED WITH BAYES]

We may therefore conveniently take as our unit, in place of x, a number y such that x=6y. We then have 3Y+2y=10, whence 5y= 10, y= 2, x=6y= 12.
(b) We can collect coefficients, i.e. combine the separate quantities or numbers expressed in terms of x as unit into a single quantity or number so expressed, obtaining sx =10.
By successive stages we obtain (� 18) s5 = 2, X = 12; or we may write at once x = of io= 5 of 10 = 12. The latter is the more advanced process, implying some knowledge of the laws of fractional numbers, as well as an application of the associative law (� 26 (i.)).
(ii.) .Perhaps the worst thing we can do, from the point of view of intelligibility, is to " clear of fractions " by multiplying both sides by 6. It is no doubt true that, if 2x+3x=io, then 3x+ 2x= 60 (and similarly if 2x+3x+6x=lo, then 3x+ 2x+x= 60); but the fact, however interesting it may be, is of no importance for our present purpose.^ The operation "multiply both sides by x–4" is not reversible.

^ In fact, to reverse the operation, we just have to multiply both sides of an equation by 1/2.

^ Note: Students without the mathematical prerequisites can take this course as Mathematics 4: no student may take both Mathematics 4 and 27 for credit, and only Mathematics 27 is eligible to count towards the major in mathematics.

.In the method (a) above there is indeed a multiplication by 6; but it is a multiplication arising out of subdivision, not out of repetition (see Arithmetic), so that the total (viz.^ The applications of modular arithmetic to cryptography and fast methods of multiplication are more widely known, but will come as a pleasant surprise to the uninitiated.
  • Arithmetic - Mathematics and the Liberal Arts 3 February 2010 14:24 UTC math.truman.edu [Source type: FILTERED WITH BAYES]

^ Or we might say that, since multiplication is a form of addition, and division a form of subtraction, there are really only two fundamental processes, viz.

^ Thus we get successive multiplication; but it represents quite different operations according as it is due to repetition, in the sense of § 34, or to subdivision, and these operations will be exhibited by different diagrams.

Io) is unaltered.
.22. Arithmetical and Algebraical Treatment of Equations.-The following will illustrate the passage from arithmetical to algebraical reasoning.^ Equations, and systems of equations, can be handled by certain "rules" that simply follow from the properties of arithmetic.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

" Coal costs 3s. a ton more this year than last year. .If 4 tons last year cost 1045., how much does a ton cost this year ?^ How much do 5 white T- shirts cost?
  • Reading Instruction for Arithmetic Word Problems: 3 February 2010 14:24 UTC www.math.umd.edu [Source type: FILTERED WITH BAYES]

^ How much does a radio cost?
  • Reading Instruction for Arithmetic Word Problems: 3 February 2010 14:24 UTC www.math.umd.edu [Source type: FILTERED WITH BAYES]

^ How much was the cost of the food.
  • Reading Instruction for Arithmetic Word Problems: 3 February 2010 14:24 UTC www.math.umd.edu [Source type: FILTERED WITH BAYES]

." If we write X for the cost per ton this year, we have 4(X-3s.^ When a quantity such as the rate of consumption of a resource (measured in tons per year or in barrels per year) is growing at a fixed percent per year, the growth is said to be exponential.
  • Arithmetic Presentation -Complete 3 February 2010 14:24 UTC www.mnforsustain.org [Source type: FILTERED WITH BAYES]
  • Presentation by Albert Bartlett - Arithmetic, Population, and Energy 3 February 2010 14:24 UTC hawaii.gov [Source type: FILTERED WITH BAYES]

) =104s.
From this we can deduce successively X - 3s.= 26s., X= 29s. But, if we transform the equation into 4X-12s. = 104s., we make an essential alteration. The original statement was with regard to X-3s. as the unit; and from this, by the application of the distributive law (� 26 (i.)), we have passed to a statement with regard to X as the unit. This is an algebraical process.
.In the same way, the transition from (x 2 +4x+4) - 4= 21 to x 2 +4x+4 = 25, or from (5+2) 2 =25 to x+2= 1 /25, is arithmetical; but the transition from 5 2 + 45+4= 25 to (5+2) 2 = 25 is algebraical, since it involves a change of the number we are thinking about.^ Thinking about floating-point in this fuzzy way stands in sharp contrast to the IEEE model, where the result of each floating-point operation is precisely defined.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ Concerned with the idea the different cultures have different ways of thinking about mathematical concepts.
  • Arithmetic - Mathematics and the Liberal Arts 3 February 2010 14:24 UTC math.truman.edu [Source type: FILTERED WITH BAYES]

^ Pettis: And since we are talking about China’s low consumption (and high investment) why do so many of your examples of high consumption involve investment items, like railways and airplanes?
  • The difficult arithmetic of Chinese consumption 3 February 2010 14:24 UTC mpettis.com [Source type: FILTERED WITH BAYES]

.Generally, we may say that algebraic reasoning in reference to equations consists in the alteration of the form of a statement rather than in the deduction of a new statement; i.e. it cannot be said that " If A = B, then E=F " is arithmetic, while " If C = D, then E=F " is algebra.^ When the unit is not determined, the reasoning is algebraical rather than arithmetical.

^ Examples of American 19th-century "higher arithmetic" texts that included much that was later part of intermediate algebra, if it dealt with computation rather than theory.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ Professor, Fordham University: "I more than agree with your statement about the need for children to learn arithmetic; and the necessity of being able to do simple arithmetic without a calculator.
  • Illinois Loop: Mathematics 28 January 2010 0:26 UTC www.illinoisloop.org [Source type: FILTERED WITH BAYES]

.Algebraic treatment consists in replacing either of the terms A or B by an expression which we know from the laws of arithmetic to be equivalent to it.^ Additional work will consist either of the development of further algebraic structures or applications of the previously developed theory to areas such as coding theory or crystallography.

The subsequent reasoning is arithmetical.
.23. Sign of Equality.-The various meanings of the sign of equality (_) must be distinguished.^ In order to apply arithmetical processes to a quantity expressed in two or more denominations, we must first express it in terms of a single denomination by means of a varying scale of notation.

(i.) 4X31b=12lb.
.This states that the result of the operation of multiplying 3/b by 4 is 12 lb.^ If the product is to be 12.51, then this would be rounded to 12.5 as part of the single-extended multiply operation.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

(ii.) 4X3 l b =3X4 lb.
.This states that the two operations give the same result; i.e. that they are equivalent. (iii.^ It should, of course, give the same results.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ Are they the same with how they do the % operations?
  • Arithmetic: Programming Language History 3 February 2010 14:24 UTC hhh.gavilan.edu [Source type: FILTERED WITH BAYES]

^ X3 and 3X4 mean different things, but the operations which they denote produce the same result.

) A's share=5s., or 3 times A's share =15s.
.Either of these is a statement of fact with regard to a particular quantity; it is usually called an equation, but sometimes a conditional equation, the term " equation " being then extended to cover (i.^ This quantity is usually called the remainder, although residue sometimes is used.
  • Origins of some arithmetic terms 3 February 2010 14:24 UTC www.pballew.net [Source type: FILTERED WITH BAYES]

^ Systems of linear equations is a particular case for which a large amount of algebra exists, including matrices and determinants, which is called linear algebra .
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ Under these conditions the period of time necessary to consume the known reserves of a resource may be called the exponential expiration time (EET) of the resource.
  • Arithmetic Presentation -Complete 3 February 2010 14:24 UTC www.mnforsustain.org [Source type: FILTERED WITH BAYES]
  • Presentation by Albert Bartlett - Arithmetic, Population, and Energy 3 February 2010 14:24 UTC hawaii.gov [Source type: FILTERED WITH BAYES]

) and (ii.). (iv.) x 3 =5 X x X x.
This is a definition of x 3; the sign = is in such cases usually replaced by =. (v.) 24d. =2s.
This is usually regarded as being, like (ii.), a statement of equivalence. It is, however, only true if is. is equivalent to 12d., and the correct statement is then 12d. i?24d. =2s.
.If the operator 12d X is omitted, the statement is really an equation, giving Is.^ Cubic Equation Solver and Factorizer This program will give you the roots for a cubic equation (real and complex roots).
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

in terms of id. or vice versa. The following statements should be compared: X=A's share =2 of £IO=3X£5=£15.
X=A's share=2 of =--1 of £30=£15.
.In each case, the first sign of equality comes under (iv.^ In the leftmost column, m should be inserted before the first equal sign in each box (three instances).
  • Textbook on Computer Arithmetic 3 February 2010 14:24 UTC www.ece.ucsb.edu [Source type: Academic]

^ A simple class of cases is that which deals with equivalence of sums of money in different currencies; these cases really come under § 120.

) above, the second under (iii.), and the fourth under (i.); but the third sign comes under (i.) in the first case (the statement being that z of £io=£5) and under (ii.) in the second.
It will be seen from � 22 that the application of algebra to equations consists in the interchange of equivalent expressions, and therefore comes under (i.) and (ii.). .We replace 4(x3), for instance, by 4x4.3, because we know that, whatever the value of x may be, the result of subtracting 3 from it and multiplying the remainder by 4 is the same as the result of finding 4x and 4.3 separately and subtracting the latter from the former.^ If a and b are the two values, this is expressed as a ≡ b (mod 24), read "a is congruent to b mod 24," and meaning that a and b give the same remainder when multiplied by 24, or that a = b + km, where k is some positive or negative integer.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ A similar analysis of ( x x ) ( y y ) cannot result in a small value for the relative error, because when two nearby values of x and y are plugged into x 2  - y 2 , the relative error will usually be quite large.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ We write this down as the first part of the quotient, multiply the divisor to give x 3 + 2x 2 + x and subtract from the dividend, with the result -4x 2 - 5.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

A statement such as (i.) or (ii.) is sometimes called an .identity. The two expressions whose equality is stated by an equation or an identity are its members.^ For n = 3, this states that the scalar product of two vectors is less than or equal to the product of the lengths of the vectors, but it is true for any n.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ The first member of the series is 3; the second is the product of n 1 two numbers, each equal to 3; the third is the pron e duct of three numbers, each equal to 3; and so on.

^ Function This program will evaluate a function and will tell you if your selected value will make two equations equal to each other.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

24. Use of Letters in General Reasoning.-It
may be assumed that the use of letters to denote quantities or numbers will first arise in dealing with equations, so that the letter used will in each case represent a definite quantity or number; such general statements as those of �� 15 and 16 being deferred to a later stage. .In addition to these, there are cases in which letters can usefully be employed for general arithmetical reasoning.^ In general, only linear interpolation can be trusted in these cases unless elaborate means are used to handle the errors.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ In these systems, there are generally two operations, analogous to addition and multiplication, that obey certain rules.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ Additional argument is needed for the special case where adding w does generate carry out.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

(i.) .There are statements, such as A+B = B+A, which are particular cases of the laws of arithmetic, but need not be expressed as such.^ Professor, Fordham University: "I more than agree with your statement about the need for children to learn arithmetic; and the necessity of being able to do simple arithmetic without a calculator.
  • Illinois Loop: Mathematics 28 January 2010 0:26 UTC www.illinoisloop.org [Source type: FILTERED WITH BAYES]

^ There are certain special cases that have interesting properties, such as functions where the sum of the exponents is constant, such as ax 2 + bxy + cy 2 , which is a homogeneous function of order 2, so that f(kx,ky) = k 2 f(x,y).
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ A is greater than B then A+X is greater than B+X. Such statements, however, arc capableof logical proof, and are generalizations of results obtained empirically at an elementary stage; they therefore belong more properly to the laws of arithmetic (§ 58).

.For multiplication, for instance, we have the statement that, if P and Q are two quantities, containing respectively p and q of a particular unit, then p X Q = q X P; or the more abstract statement that p X q= q X p.^ If M and N are respectively m and n times a unit, and P and Q are respectively p and q times a unit, then the quantities are in proportion if mq = np; and conversely.

^ Multiple-Tables.-The diagram C or D of § 35 is part of a complete table giving the successive multiples of the particular unit.

^ In order to apply arithmetical processes to a quantity expressed in two or more denominations, we must first express it in terms of a single denomination by means of a varying scale of notation.

(ii.) .The general theory of ratio and proportion requires the use of general symbols.^ Indeed, the very principle of using symbols to represent general quantities in algebra may make a problem easier to solve.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ The essence of algebra, then, is the use of literal symbols to stand for general numbers or other quantities.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ I might point out that the colon, ":" seems to be the division symbol of choice if this sample can be generalized as it was used in Norway, Germany, Italy, and Denmark.
  • Origins of some arithmetic terms 3 February 2010 14:24 UTC www.pballew.net [Source type: FILTERED WITH BAYES]

(iii.) .The general statement of the laws of operation of fractions is perhaps best deferred until we come to fractional numbers, when letters can be used to express the laws of multiplication and division of such numbers.^ A Mixed Number is a number expressed by an integer and a fraction."
  • Origins of some arithmetic terms 3 February 2010 14:24 UTC www.pballew.net [Source type: FILTERED WITH BAYES]

^ Division of Fractional Numbers 6.4 67.

^ Multiplication of Fractional Numbers 6.3 66.

(iv.) .Variation is generally included in text-books on algebra, but apparently only because the reasoning is general.^ Examples of American 19th-century "higher arithmetic" texts that included much that was later part of intermediate algebra, if it dealt with computation rather than theory.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ Special Thanks To: Edwin Howard For His Help*** This is the only algebra II program you'll ever need because it has everything all in one big package.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

^ I generally deduct only one point for these errors, not because they are unimportant, but because deducting more would involve swimming against a tide that is just too strong for me.

It is part of the general theory of quantitative relation, and in its elementary stages is a suitable subject for graphical treatment (� 31).
25. Preparation for Algebra. - .The calculation of the values of simple algebraical expressions for particular values of letters involved is a useful exercise, but its tediousness is apt to make the subject repulsive.^ Examples include algebra students using calculators to solve 300/3 or 63/9.
  • Illinois Loop: Mathematics 28 January 2010 0:26 UTC www.illinoisloop.org [Source type: FILTERED WITH BAYES]

^ A pocket calculator of this quality is very useful for algebra, as is mentioned several places in the text.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ All the algebraic properties of determinants can be worked out by algebra using this expression.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

.What is more important is to verify particular examples of general formulae.^ Although formula (7) is much more accurate than (6) for this example, it would be nice to know how well (7) performs in general.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ We next present more interesting examples of formulas exhibiting catastrophic cancellation that can be rewritten to exhibit only benign cancellation.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ More generally, if we have obtained a as an approximate value for the pth root of N, the binomial theorem gives as an approximate formula p,IN =a+6, where N = a P + pap - 19.

.These formulae are of two kinds: - (a) the general properties, such as m(a+b) = ma+mb, on which algebra is based, and (b) particular formulae such as (x - a) (x+a) = x 2 - a 2 . Such verifications are of value for two reasons.^ The formula is correct when a and b are positive real numbers, but it leads to errors when generalized indiscriminately to other kinds of numbers.

^ "Take two algebra students -- one is still a little shaky on the distributive property, whereas the other knows it cold.
  • Illinois Loop: Mathematics 28 January 2010 0:26 UTC www.illinoisloop.org [Source type: FILTERED WITH BAYES]

^ If P(x) and Q(x) are two algebraic functions of x, the statement P(x) = Q(x) may be either an identity , if it is true for any value of x, or just an equation otherwise.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

In the first place, they lead to an understanding of what is meant by the use of brackets and by such a statement as 3(7+2) = 3.7+3 � 2. This does not mean (cf. � .23) that the algebraic result of performing the operation 3(7+2) is 3.7+3.2; it means that if we convert 7+2 into the single number 9 and then multiply by 3 we get the same result as if we converted 3.7 and 3.2 into 21 and 6 respectively and added the results.^ Each number is converted to a single integer to which it is congruent mod 9, and the operation is easily done with the single integers.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ They are able to manipulate fractions in order to gain a greater understanding of the meaning of whole number that is broken into parts.
  • TeachNet: Lesson Plans: Elementary, Middle & High School: Math 28 January 2010 0:26 UTC teachersnetwork.org [Source type: Academic]

^ The two numbers that are multiplied together are most often called factors and the result is called the Product .
  • Origins of some arithmetic terms 3 February 2010 14:24 UTC www.pballew.net [Source type: FILTERED WITH BAYES]

In the second place, particular cases lay the foundation for the general formula.
.Exercises in the collection of coefficients of various letters occurring in a complicated expression are usually performed mechanically, and are probably of very little value.^ A rule that covers both of the previous two examples is to compute an expression in the highest precision of any variable that occurs in that expression.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ Complex roots always occur in pairs with conjugate values if the coefficients of the polynomial are real.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ Collecting terms in an expression means combining terms differing only in the numerical coefficient.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

.26. General Arithmetical Theorems. (i.) The fundamental laws of arithmetic should be constantly borne in mind, though not necessarily stated.^ Perhaps they have in mind that floating-point numbers model real numbers and should obey the same laws that real numbers do.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

.The following are some special points.^ The Following Are Some Special Rules: (I) To Multiply By 5, Multiply By Io And Divide By 2.

.(a) The commutative law and the associative law are closely related, and it is best to establish each law for the case of two numbers before proceeding to the general case.^ The above definitions of logarithms, &c., relate to cases in which n and p are whole numbers, and are generalized later.

^ If four coins are laid on a table, close together, they can (by most adults) be seen to be four, without counting; but seven coins have to be separated mentally into two groups, the numbers of which are added, or one group has to be seen and the remaining objects counted, before the number is known to be seven.

^ Closely related topics: Number Words , Number Systems , and Abraham Seidenberg .
  • Arithmetic - Mathematics and the Liberal Arts 3 February 2010 14:24 UTC math.truman.edu [Source type: FILTERED WITH BAYES]

.In the case of addition, for instance, suppose that we are satisfied that in a+b+c+d+e we may take any two, as b and c, together (association) and interchange them (commutation).^ In this case, f(x) has no negative roots, since it has two pairs of complex roots in addition to the positive root.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ For instance, 3' 14159 2 7 = 3 (1 +1) ` 122 106) ` I+ 333 1 113) ' ' .; but the last two of these factors may be combined as (I -).

^ Discount may be allowed twice in succession off quoted prices; in such cases the second discount is off the reduced price, and therefore it is not correct to add the two rates of discount together.

.Then we have a+b+c+d+e=a+c+b+d+e. Thus any pair of adjoining numbers can be interchanged, so that the numbers can be arranged in any order.^ Thus to divide by a fractional number we must multiply by the number obtained by interchanging the numerator and the denominator, i.e.

(b) The important form of the distributive law is m(A+B) = mA+mB. The form (m+n)A=mA+nA follows at once from the fact that A is the unit with which we are dealing.
.(c) The fundamental properties of subtraction and of division are that A - B +B = A and m X m of A = A, since in each case the second operation restores the original quantity with which we started.^ Shift/subtract division algorithms 13.2 Programmed division 13.3 Restoring hardware dividers 13.4 Non-restoring and signed division 13.5 Division by constants 13.6 Preview of fast dividers .
  • Textbook on Computer Arithmetic 3 February 2010 14:24 UTC www.ece.ucsb.edu [Source type: Academic]

^ In the second class of cases the fraction of the unit quantity is a quantity of the same kind, but cannot be determined with absolute exactness.

^ Or we might say that, since multiplication is a form of addition, and division a form of subtraction, there are really only two fundamental processes, viz.

(ii.) .The elements of the theory of numbers belong to arithmetic.^ The residue number system has provided an invaluable tool for researchers interested in complexity theory and limits of fast arithmetic as well as to the designers of fault-tolerant systems.
  • Textbook on Computer Arithmetic 3 February 2010 14:24 UTC www.ece.ucsb.edu [Source type: Academic]

.In particular, the theorem that if n is a factor of a and of b it is also a factor of pa= qb, where p and q are any integers, is important in reference to the determination of greatest common divisor and to the elementary treatment of continued fractions.^ Greatest Common Divisor 3.4.3 47.

^ The process depends on (ii) of § 45, in the extended form that, if x is a factor of a and b, it is a factor of pa-qb, where p and q are any integers.

^ This was also long known as the greatest common measure, GCM. We can always write any integer as a product of prime factors, and if we know the factors, the GCD is easily determined.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

Graphic methods are useful here (� 34 (iv.)). The law of relation of successive convergents to a continued fraction involves more advanced methods (see � 42 (iii.) and Continued Fraction).
(iii.) There are important theorems as to the relative value of fractions; e.g. (a) If a = d, then each = p b qd. (b) V - =--1_27:i nearer to I than b is � and, gener a lly, if - b = - d' then pa+qc l i es between the two. .(All the numbers are, of course, pb +qd supposed to be positive.^ The numbers represented by a, b, c, x and m are all supposed to be positive.

) .27. Negative Quantities and Fractional Numbers.^ Negative Fractional Numbers 6.5 68.

^ Simply specify the type of addition, subtraction, multiplication, or division, problem for either the counting numbers, the negative numbers, decimals, or fractions and the maximum and minimum numbers to be used in the problems.

^ For multiplication by a proper fraction or a decimal, it is sometimes convenient, especially when we are dealing with mixed quantities, to convert the multiplier into the sum or difference of a number of fractions, each of which has i as its numerator.

- .(i.
) What are usually called " negative numbers " in arithmetic are in reality not negative numbers but negative quantities. If a person has to receive 7s.^ A number of this kind is called a surd; the surd which is the pth root of N is written ¦JN, but if the index is 2 it is usually omitted, so that the square root of N is written ,/N. .

^ What is called modern algebra works with symbols that may obey different rules of composition or operations than the familiar ones of real numbers that we have just presented.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ Arithmetic is usually divided into Abstract Arithmetic and Concrete Arithmetic, the former dealing with numbers and the latter with concrete objects.

and pay .5s., with a net result of +2s., the order of the operations is immaterial.^ In higher mathematics, we say that two operations commute if we can perform them in either order and get the same result.

If he pays first, he then has - 5s. .This is sometimes treated as a debt of 5s.; an alternative method is to recognize that our zero is really arbitrary, and that in fact we shift it with every operation of addition or subtraction.^ The alternative method is to retrace the steps of addition, i.e.

^ Or we might say that, since multiplication is a form of addition, and division a form of subtraction, there are really only two fundamental processes, viz.

^ A popular avoid-the-reading method of instruction for Arithmetic Word problems is the "Key-Word" method, in which student are given lists of "addition" and "subtraction" words and phrases.
  • Reading Instruction for Arithmetic Word Problems: 3 February 2010 14:24 UTC www.math.umd.edu [Source type: FILTERED WITH BAYES]

But " " " ".
when we say - ss. we mean - (5s.), not (-5)s. the idea of (-5) as a number with which we can perform such operations as multiplication comes later (� 49)� (ii.) .On the other hand, the conception of a fractional number follows directly from the use of fractions, involving the subdivision of a unit.^ Simply specify the type of addition, subtraction, multiplication, or division, problem for either the counting numbers, the negative numbers, decimals, or fractions and the maximum and minimum numbers to be used in the problems.

^ The IEEE standard uses denormalized 18 numbers, which guarantee (10) , as well as other useful relations.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ This involves using the formula for the EET in which the quotient ( R / r 0 ) is the number of years the quantity R of the resource would last at the present rate of consumption, r 0 .
  • Arithmetic Presentation -Complete 3 February 2010 14:24 UTC www.mnforsustain.org [Source type: FILTERED WITH BAYES]
  • Presentation by Albert Bartlett - Arithmetic, Population, and Energy 3 February 2010 14:24 UTC hawaii.gov [Source type: FILTERED WITH BAYES]

.We find that fractions follow certain laws corresponding exactly with those of integral multipliers, and we are therefore able to deal with the fractional numbers as if they were integers.^ They are able to manipulate fractions in order to gain a greater understanding of the meaning of whole number that is broken into parts.
  • TeachNet: Lesson Plans: Elementary, Middle & High School: Math 28 January 2010 0:26 UTC teachersnetwork.org [Source type: Academic]

^ He argues from these similarities that these number words, and therefore the corresponding number concepts, arose one place and spread throughout the world by a diffusion process.
  • Arithmetic - Mathematics and the Liberal Arts 3 February 2010 14:24 UTC math.truman.edu [Source type: FILTERED WITH BAYES]

^ We further determined that exponential functions have certain “rules” they must follow, they are b> 0, b can’t equal 1, and a can’t equal 0.
  • Alltop - Top Math News 2 February 2010 15:42 UTC math.alltop.com [Source type: General]

28. Miscellaneous Developments in Arithmetic. - .The following are matters which really belong to arithmetic; they are usually placed under algebra, since the general formulae involve the use of letters.^ If and , then computing r 1 using formula (4) will involve a cancellation.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ The formula is correct when a and b are positive real numbers, but it leads to errors when generalized indiscriminately to other kinds of numbers.

^ 'Kids get to use calculators as a substitute for practice, and they never really understand arithmetic,' says Sandra Stotsky, deputy education commissioner in Massachusetts, a state that has taken a back-to-basics approach."
  • Illinois Loop: Mathematics 28 January 2010 0:26 UTC www.illinoisloop.org [Source type: FILTERED WITH BAYES]

(i.) Arithmetical Progressions such as 2, 5, 8, ... - .The formula for the rth term is easily obtained.^ There are also methods for obtaining the roots of fourth-order equations in terms of radicals, but they are as inconvenient as Cartan's formulas.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ If, for instance, three terms of a proportion are given, the fourth can be obtained by the relation given at the end of § 57, this relation being then called the Rule of Three; but this is equivalent to the use of an algebraical formula.

.The problem of finding the sum of r terms is aided by graphic representation, which shows that the terms may be taken in pairs, working from the outside to the middle; the two cases of an odd number of terms and an even number of terms may be treated separately at first, and then combined by the ordinary method, viz.^ This gives a rule for finding the L.C.M. of two numbers.

^ Shows the work graphically as well.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

^ We can thus find fractional numbers equivalent to the sum or difference of any two fractional numbers.

writing the series backwards.
.In this, as in almost all other cases, particular examples should be worked before obtaining a general formula.^ Ideally, all topics in each chapter should be covered before moving to the next chapter.
  • Textbook on Computer Arithmetic 3 February 2010 14:24 UTC www.ece.ucsb.edu [Source type: Academic]

^ The formula is correct when a and b are positive real numbers, but it leads to errors when generalized indiscriminately to other kinds of numbers.

^ It almost seems as though the U.S. Department of Energy has not studied the works of Hubbert, Campbell & Laherrre, Ivanhoe, Edwards, Masters and other prominent petroleum geologists.
  • Arithmetic Presentation -Complete 3 February 2010 14:24 UTC www.mnforsustain.org [Source type: FILTERED WITH BAYES]

(ii.) .The law of indices (positive integral indices only) follows at once from the definition of a 2, a 3, a 4,..^ Hence we may insert in the power-series numbers with fractional indices, provided that the multiplication of these numbers follows the same law.

^ We therefore only require a definite law for the formation of the successive names or symbols.

^ The arithmetical processes which we have considered in reference to positive integral numbers are subject to the following laws: .

.
as abbreviations of a.a, a.a.a, a.a.a.a,.. ., or (by analogy with the definitions of 2, 3, 4,. .. themselves) of a.a, a.a 2, a.a 3,.. . successively. .The treatment of roots and of logarithms (all being positive integers) belongs to this subject; a= n and p= log a n being the inverses of n=a P (cf.^ This equation is valid for all positive values of k and for those negative values of k for which the argument of the logarithm is positive.
  • Arithmetic Presentation -Complete 3 February 2010 14:24 UTC www.mnforsustain.org [Source type: FILTERED WITH BAYES]
  • Presentation by Albert Bartlett - Arithmetic, Population, and Energy 3 February 2010 14:24 UTC hawaii.gov [Source type: FILTERED WITH BAYES]

^ Solving, converting exponents and logs, and it will show you all ten of the logarithmic properties.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

^ All Root Simplifier Quickly simplifies roots of any power for any integer.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

�� 15, 16). .The theory may be extended to the cases of p= i and p = o; so that a 3 means a.a.a.1, a 2 means a.a.i, a 1 means a.i, and means I (there being then none of the multipliers a). The terminology is sometimes confused.^ In the case of numbers the X may be replaced by a dot; thus 4.3 means 4 times 3.

^ In the case of partition we can express the complete operation if we extend the meaning of division so as to enable us to divide 20 apples by 5 boys.

^ The symbols - and = mean respectively that the first quantity mentioned is to be reduced or divided by the second; but there is some vagueness about + and X. In the present article a+b will mean that a is taken first, and b added to it; but a X b will mean that b is taken first, and is then multiplied by a.

.In n = a P, a is the root or base, p is the index or logarithm, and n is the power or antilogarithm. Thus a, a !^ Powers, Roots and Logarithms .

^ Thus multiplication and division in the power-series correspond to addition and subtraction in the index-series, and vice versa.

^ Thus, to find the logarithm of a number to base 2, the number being greater than i, we first divide repeatedly by 2 until we get a number between I and 2; then divide repeatedly by 10 12 until we get a number between I and 10 y2; then divide repeatedly by ioo v 2; and so on.

,. a 3,.. .
are the first, second, third,. .. powers of .a. But a P is sometimes incorrectly described as " a to the power p "; the power being thus confused with the index or logarithm.^ Thus multiplication and division in the power-series correspond to addition and subtraction in the index-series, and vice versa.

^ Thus, to find the logarithm of a number to base 2, the number being greater than i, we first divide repeatedly by 2 until we get a number between I and 2; then divide repeatedly by 10 12 until we get a number between I and 10 y2; then divide repeatedly by ioo v 2; and so on.

^ For practical purposes the number taken as base is so; the convenience of this being that the increase of the index by an integer means multiplication by the corresponding power of 10, i.e.

(iii.) .Scales of Notation lead, by considering, e.g., how to express in the scale of to a number whose expression in the scale of 8 is 2222222, to (iv.^ We may therefore consider that there is some number x whose cube is 2000, and we can find this number to any degree of accuracy that we please.

^ As an example of the difference between geometry and algebra, consider the algebraic expression (a + b)(a - b), representing the product of the sum and difference of two numbers a and b.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ S. The simplest case, in which the quantity can be expressed as an integral number of the largest units B involved, has already been considered (§§ 37, 42).

) Geometrical Progressions. - .It
should be observed that the radix of the scale is exactly the same thing as the root mentioned under (ii.^ As the section Languages and Compilers mentions, many programming languages don't specify that each occurrence of an expression like 10.0*x in the same context should evaluate to the same value.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ In reference to the use of the sign X with the converting factor, it should be observed that " lb X " symbolizes the replacing of so many times 4 lb by the same number of times 7 lb, while " 4 X " symbolizes the replacing of 4 times something by 7 times that something.

^ This relation is of exactly the same kind as the relation of the successive digits in numbers expressed in a scale of notation whose base is n.

) above; and it is better to use the term " root " throughout. Denoting the root by a, and the number 2222222 in this scale by N, we have N = 2222222.
aN = 2222-2220.
.Thus by adding 2 to aN we can subtract N from aN+2, obtaining 2000-0000, which is =2. a 7; and from this we easily pass to the general formula for the sum of a geometrical progression having a given number of terms.^ The formula is correct when a and b are positive real numbers, but it leads to errors when generalized indiscriminately to other kinds of numbers.

^ Sometimes a term must be added and subtracted (which does not change the value of the expression) in order to do this, as in completing the square.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ An early systematic study is in the late Medieval Latin poem De Vetula , which gives the number of ways you can obtain any given total from a throw of 3 dice.
  • Arithmetic - Mathematics and the Liberal Arts 3 February 2010 14:24 UTC math.truman.edu [Source type: FILTERED WITH BAYES]

(v.) Permutations and Combinations may be regarded as arithmetical recreations; they become important algebraically in reference to the binomial theroem (�� 41, 44)� (vi.) Surds and Approximate Logarithms. - .From the arithmetical point of view, surds present a greater difficulty than negative quantities and fractional numbers.^ Negative Fractional Numbers 6.5 68.

^ They are able to manipulate fractions in order to gain a greater understanding of the meaning of whole number that is broken into parts.
  • TeachNet: Lesson Plans: Elementary, Middle & High School: Math 28 January 2010 0:26 UTC teachersnetwork.org [Source type: Academic]

^ Simply specify the type of addition, subtraction, multiplication, or division, problem for either the counting numbers, the negative numbers, decimals, or fractions and the maximum and minimum numbers to be used in the problems.

.We cannot solve the equation 7s.+X=4s.; but we are accustomed to transactions of lending and borrowing, and we can therefore invent a negative quantity - 3s.^ A commn rule is a negative quantity cannot be raised to a real power.
  • Arithmetic: Programming Language History 3 February 2010 14:24 UTC hhh.gavilan.edu [Source type: FILTERED WITH BAYES]

such that - 3s.+3s. = o. .We cannot solve the equation 7X =4s.; but we are accustomed to subdivision of units, and we can therefore give a meaning to X by inventing a unit w s.^ Also give you the equation before solving it!!
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

^ There appears therefore to be a tendency to use some larger number than ten as a basis for grouping into new units or for subdivision into parts.

^ Subdivision of Submultiple.-By 7 of A we mean 5 times the unit, 7 times which is A. If we regard this unit as being 4 times a lesser unit, then Aris 7.4 times this lesser unit, and 7 of A is 5.4 times the lesser unit.

such that .7 X 1 s = is., and can thence pass to the idea of fractional numbers.^ The idea and properties of a fractional number having been explained, we may now call it, for brevity, a fraction.

.When, however, we come to the equation x 2 --- 5, where we are dealing with numbers, not with quantities, we have no concrete facts to assist us.^ The fact that no numbers are given made this passage particularly hard to decipher, and it was not properly understood for many years; hence we can see the advantages of numerical examples.
  • Arithmetic - Mathematics and the Liberal Arts 3 February 2010 14:24 UTC math.truman.edu [Source type: FILTERED WITH BAYES]

^ These include the fact that in building odd numbers, the word one comes at the end, and also the fact that there is on connective.
  • Arithmetic - Mathematics and the Liberal Arts 3 February 2010 14:24 UTC math.truman.edu [Source type: FILTERED WITH BAYES]

^ Thanks to signed zero, x will be negative, so log can return a NaN. However, if there were no signed zero, the log function could not distinguish an underflowed negative number from 0, and would therefore have to return - .
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

.We can, however, find a number whose square shall be as nearly equal to 5 as we please, and it is this number that we treat arithmetically as 1 15. We may take it to (say) 4 places of decimals; or we may suppose it to be taken to 1000 places.^ A number can be correct to so many places of decimals.

^ As a branch of mathematics , arithmetic may be treated logically, psychologically, or historically.

^ Find the sum of the squares of these three numbers.
  • Reading Instruction for Arithmetic Word Problems: 3 February 2010 14:24 UTC www.math.umd.edu [Source type: FILTERED WITH BAYES]

.In actual practice, surds mainly arise out of mensuration; and we can then give an exact definition by graphical methods.^ Also most fractions cannot be expressed exactly as decimals; and this is also the case for surds and logarithms, as well as for the numbers expressing certain ratios which arise out of geometrical relations.

^ In this segment, we give some explanation of how Benford’s Law actually arises in so many settings: why are so many kinds of data logarithmically distributed?
  • Alltop - Top Math News 2 February 2010 15:42 UTC math.alltop.com [Source type: General]

.When, by practice with logarithms, we become familiar with the correspondence between additions of length on the logarithmic scale (on a slide-rule) and multiplication of numbers in the natural scale (including fractional numbers), A /5 acquires a definite meaning as the number corresponding to the extremity of a length x, on the logarithmic scale, such that 5 corresponds to the extremity of 2X. Thus the concrete fact required to enable us to pass arithmetically from the conception of a fractional number to the conception of a surd is the fact of performing calculations by means of logarithms.^ Multiplication of Fractional Numbers 6.3 66.

^ Logarithmic scales are the basis for the slide rule.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ They are able to manipulate fractions in order to gain a greater understanding of the meaning of whole number that is broken into parts.
  • TeachNet: Lesson Plans: Elementary, Middle & High School: Math 28 January 2010 0:26 UTC teachersnetwork.org [Source type: Academic]

.In the same way we regard log l 02, not as a new kind of number, but as an approximation.^ The G.C.D. of three or more numbers is found in the same way.

^ II in the same way that the number III includes the number II in fig.

^ Another way to measure the difference between a floating-point number and the real number it is approximating is relative error , which is simply the difference between the two numbers divided by the real number.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

(vii.) The use of fractional indices follows directly from this parallelism. .We find that the product a m X a m X a m is equal to a im; and, by definition, the product ;/a X -la X Ala is equal to a, which is a'. This suggests that we should write ;la as a l l3; and we find that the use of fractional indices in this way satisfies the laws of integral indices.^ The end product should be a translation of this wordy problem into the really simple statement: "Find 44% of $1400".
  • Reading Instruction for Arithmetic Word Problems: 3 February 2010 14:24 UTC www.math.umd.edu [Source type: FILTERED WITH BAYES]

^ Precedence [ find definition ] is the order of operator used.
  • Arithmetic: Programming Language History 3 February 2010 14:24 UTC hhh.gavilan.edu [Source type: FILTERED WITH BAYES]

^ But if we try to use negative arguments or floating point values, the best way to find out what happens is write a short program and check the results.
  • Arithmetic: Programming Language History 3 February 2010 14:24 UTC hhh.gavilan.edu [Source type: FILTERED WITH BAYES]

.It should be observed that, by analogy with the definition of a fraction, a P l q mean (al/q)P, not (aP)llq. II. Graphical Introduction to Algebra. 29. The science of graphics is closely related to that of mensuration. While mensuration is concerned with the representation of geometrical magnitudes by numbers, graphics is concerned with the representation of numerical quantities by geometrical figures, and particularly by lengths.^ There is little detail, as the article is rather brief, but the author does mention the number concept and counting, fractions (very briefly), elementary geometric notions (e.g., that of a line), symmetry, string figures, and games of strategy.
  • Arithmetic - Mathematics and the Liberal Arts 3 February 2010 14:24 UTC math.truman.edu [Source type: FILTERED WITH BAYES]

^ Closely related topics: Number Words , Number Systems , and Abraham Seidenberg .
  • Arithmetic - Mathematics and the Liberal Arts 3 February 2010 14:24 UTC math.truman.edu [Source type: FILTERED WITH BAYES]

^ The ratio of two quantities is, in algebra, the quotient of their numerical values, written a/b or a:b, where a is called the antecedent , and b the consequent .
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

An important development, covering such diverse matters as the equilibrium of forces and the algebraic theory of complex numbers (� 66), has relation to cases where the numerical quantity has direction as well as magnitude. .There are also cases in which graphics and mensuration are used jointly; a variable numerical quantity is represented by a graph, and the principles of mensuration are then applied to determine related numerical quantities.^ The same principle can be used in other cases.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ If a/b = c/d, then certain other relations between quantities exist, which were given Latin names that are now rarely used.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ It will autamaticly detect any other variables used and prompt for there input.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

General aspects of the subject are considered under Mensuration; Vector Analysis; Infinitesimal Calculus.
.30. The elementary use of graphic methods is qualitative rather than quantitative; i.e. it is for purposes of illustration and suggestion rather than for purposes of deduction and exact calculation.^ Students, using the Key Word method (as in the wrong solution to Problem 7), will calculate that 89 gallons is enough.
  • Reading Instruction for Arithmetic Word Problems: 3 February 2010 14:24 UTC www.math.umd.edu [Source type: FILTERED WITH BAYES]

^ Myth #2: Children develop a deeper understanding of mathematics and a greater sense of ownership when they are expected to invent and use their own methods for performing the basic arithmetical operations, rather than study, understand and practice the standard algorithms.
  • Illinois Loop: Mathematics 28 January 2010 0:26 UTC www.illinoisloop.org [Source type: FILTERED WITH BAYES]

^ Rather than travel into the sticky abyss of statistics it is better to rely on a few data and on the pristine simplicity of elementary mathematics.
  • Presentation by Albert Bartlett - Arithmetic, Population, and Energy 3 February 2010 14:24 UTC hawaii.gov [Source type: FILTERED WITH BAYES]

We start with related facts, and adopt a particular method of visualizing the relation. .One of the relations most commonly illustrated in this way is the time-relation; the passage of time being associated with the passage of a point along a straight line, so that equal intervals of time are represented by equal lengths.^ The reader may derive formulas for the average speed when the intervals of time or space are not equal.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ If we divide the journey into n equal time intervals, with an average speed of v i in each interval, then it is easy to show that the average speed for the entire journey is v av = (1/n)Σ v i , so that the average speed is the arithmetic mean.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ Real numbers are the rational numbers and all limits of sequences of rational numbers (Dedekind cuts), and are in one-to-one correspondence with the points on a line.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

.31. It is important to begin the study of graphics with concrete cases rather than with tracing values of an algebraic function.^ Examples of American 19th-century "higher arithmetic" texts that included much that was later part of intermediate algebra, if it dealt with computation rather than theory.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ (To save face, just in case the error is your own, formulate it as a question rather than a statement.

^ Iteration Suite Displays numerical values and graphical representations of the iterations of a function.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

.Simple examples of the time-relation are - the number of scholars present in a class, the height of the barometer, and the reading of the thermometer, on successive days.^ How many pages will I need to read a day to finish the book in time?'
  • Illinois Loop: Mathematics 28 January 2010 0:26 UTC www.illinoisloop.org [Source type: FILTERED WITH BAYES]

^ The expressions 4 r are then fractional numbers, their relation to n' n' n' ' ' ordinary or integral numbers being that /2 n times n times is equal to p times.

^ Written in about 10 min of class time should try it its reall simple read the readme !
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

.Another useful set of graphs comprises those which give the relation between the expressions of a length, volume, &c., on different systems of measurement.^ If a/b = c/d, then certain other relations between quantities exist, which were given Latin names that are now rarely used.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ The authors also discuss an interesting feature of the Nahua language which was spoken by the Aztecs, where a system of classifiers was used; the language included classifiers for round objects, for objects where length is a primary factor, and for objects that can be stacked.
  • Arithmetic - Mathematics and the Liberal Arts 3 February 2010 14:24 UTC math.truman.edu [Source type: FILTERED WITH BAYES]

^ Divided Differences This program uses Newton's Divided Differences method to find a polynomial expression that will interpolate up to eight given points.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

.Mechanical, commercial, economic and statistical facts (the latter usually involving the time-relation) afford numerous examples.^ The fact that no numbers are given made this passage particularly hard to decipher, and it was not properly understood for many years; hence we can see the advantages of numerical examples.
  • Arithmetic - Mathematics and the Liberal Arts 3 February 2010 14:24 UTC math.truman.edu [Source type: FILTERED WITH BAYES]

^ The division into abstract and concrete, for instance, is logical, if the former is taken as relating to number and the latter to numerical quantity (§ I I).

32. The ordinary method of representation is as follows. .Let X and Y be the related quantities, their expressions in terms of selected units A and B being x and y, so that X=x.A, Y = y.^ Comparison, Addition and Subtraction of Fractions.-The quantities 4 of A and 7 of A are expressed in terms of different units.

^ M, N and P, Q respectively, M will be to N in the same ratio that P is to Q. This is expressed by saying M p that M is to N as P to Q, the relation being written M: N ::P: Q; the four quantities are then said to be in proportion or to be proportionals.

^ A B C of numerical quantities, merely correspond with each other, the correspondence being the result of some relation.

B. For graphical representation we select units of length L and M, not necessarily identical. We take a fixed line OX, usually drawn horizontally; for each value of X we measure a length or abscissa ON equal to x.L, and draw an ordinate NP at right angles to OX and equal to the corresponding value of y . M. The assemblage of ordinates NP is then the graph of Y.
The series of values of X will in general be discontinuous, and the graph will then be made up of a succession of parallel and (usually) equidistant ordinates. .When the series is theoretically continuous, the theoretical graph will be a continuous figure of which the lines actually drawn are ordinates.^ Graph Solver This program prompts you to enter 2 points on a line, and it will figure out: the slope of the line, the distance between the points, the midpoint of the line, and the point-slope of the line.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

.The upper boundary of this figure will be a line of some sort; it is this line, rather than the figure, that is sometimes called the " graph."^ Graph Solver This program prompts you to enter 2 points on a line, and it will figure out: the slope of the line, the distance between the points, the midpoint of the line, and the point-slope of the line.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

^ It sometimes wants the student to estimate an answer rather than find the right one.
  • Illinois Loop: Mathematics 28 January 2010 0:26 UTC www.illinoisloop.org [Source type: FILTERED WITH BAYES]

^ The VMS math libraries on the VAX use a weak form of in-line procedure substitution, in that they use the inexpensive jump to subroutine call rather than the slower CALLS and CALLG instructions.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

It is better, however, to treat this as a secondary meaning. .In particular, the equality or inequality of values of two functions is more readily grasped by comparison of the lengths of the ordinates of the graphs than by inspection of the relative positions of their bounding lines.^ It is essentially interpolation between known values of the function f(a) and f(b) at two locations a and b.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ A more general rational function is the ratio of two polynomials, a numerator P(x) of order n, and a denominator Q(x) of order m.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ This discount, of course, is not equal to the interest which the present value would produce at that rate of interest, but is rather greater, so that the present value as calculated in this way is less than the theoretical present value.

.33. The importance of the bounding line of the graph lies in the fact that we can keep it unaltered while we alter the graph as a whole by moving OX up or down.^ When adding two floating-point numbers, if their exponents are different, one of the significands will have to be shifted to make the radix points line up, slowing down the operation.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

We might, for instance, read temperature from 60° instead of from o°. Thus we form the conception, not only of a zero, but also of the arbitrariness of position of this zero (cf. � 2 7 (i.)); and we are assisted to the conception of negative quantities. On the other hand. the alteration in the direction of the bounding line, due to alteration in the unit of measurement of Y, is useful in relation to geometrical projection.
.This, however, applies mainly to the representation of values of Y. Y is represented by the length of the ordinate NP, so that the representation is cardinal; but this ordinate really corresponds to the point N, so that the representation of X is ordinal.^ Several different representations of real numbers have been proposed, but by far the most widely used is the floating-point representation.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ Real numbers are the rational numbers and all limits of sequences of rational numbers (Dedekind cuts), and are in one-to-one correspondence with the points on a line.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ Since every bit pattern represents a valid number, the return value of square root must be some floating-point number.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

.It is therefore only in certain special cases, such as those of simple time-relations (e.g.^ It has been suggested that as many as six objects can be seen at once; but this is probably only the case with few people, and with them only when the objects have a certain geometrical arrangement.

^ There are certain special cases that have interesting properties, such as functions where the sum of the exponents is constant, such as ax 2 + bxy + cy 2 , which is a homogeneous function of order 2, so that f(kx,ky) = k 2 f(x,y).
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ The £I is termed the unit, A numerical quantity, therefore, represents a certain unit, taken a certain number of times.

." J is aged 40, and K is aged 26; when will J be twice as old as K? "), that the graphic method leads without arithmetical reasoning to the properties of negative values.^ This is a more advanced method, which leads easily to the idea of negative quantities, if the subtraction is such that we have to go behind the o of the standard series.

^ Arithmetic, although very useful and the foundation of mathematics, is not suited to making general statements about numbers, nor to reasoning about their abstract properties.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

.In other cases the continuation of the graph may constitute a dangerous extrapolation.^ In most cases the quantity in the second column may be regarded as increasing or decreasing continuously as the number in the first column increases, and it has intermediate values corresponding to intermediate (i.e.

34. Graphic representation thus rests on the principle that equal numerical quantities may be represented by equal lengths, and that a quantity mA may be represented by a length mL, where A and L are the respective units; and the science of graphics rests on the converse property that the quantity represented by pL is pA, i.e. that pA is determined by finding the number of times that L is contained in pL. The graphic method may therefore be used in arithmetic for comparing two particular magnitudes of the same kind by comparing the corresponding lengths P and Q measured along a single line OX from the same point O.
(i.) .To divide P by Q, we cut off from P successive portions each equal to Q, till we have a piece R left which is less than Q. Thus P = kQ+R, where k is an integer.^ This discount, of course, is not equal to the interest which the present value would produce at that rate of interest, but is rather greater, so that the present value as calculated in this way is less than the theoretical present value.

^ For n = 3, this states that the scalar product of two vectors is less than or equal to the product of the lengths of the vectors, but it is true for any n.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ It is also important to note that if the increase or decrease of an amount A by a certain percentage produces B, it will require a different percentage to decrease or increase B to A. Thus, if B is 20% less than A, A is 25% greater than B. .

(ii.) .To continue the division we may take as our new unit a submultiple of Q, such as Q/r, where r is an integer, and repeat the process.^ If as our unit we take i of A = 1% of A, the above quantity might equally be written 2376 X = 21:367-0 s_.

^ Some operations are not reversible, and so we may get new solutions when we perform such an operation.

^ Similarly we may take the farthing as a unit, and invent smaller units, represented either by tokens or by no material objects at all.

.We thus get P=kQ+m.Q/r+S=(k+m/r)Q+S, where S is less than Q/r.^ It is also important to note that if the increase or decrease of an amount A by a certain percentage produces B, it will require a different percentage to decrease or increase B to A. Thus, if B is 20% less than A, A is 25% greater than B. .

^ The C language floor function and the BASIC int function returns the whole number that is less than or equal to the argument, and thus matches Rule 2.
  • Arithmetic: Programming Language History 3 February 2010 14:24 UTC hhh.gavilan.edu [Source type: FILTERED WITH BAYES]

.Proceeding in this way, we may be able to express P= Q as the sum of a finite number of terms k+m/r+n/r 2 +..^ The sum or difference of two percentages is expressed by the sum or difference of the numbers expressing the two percentages.

^ There are three principal ways of expressing the degree of accuracy of any number, i.e.

^ When a mixed quantity or a mixed number has to be multiplied by a large number, it is sometimes convenient to express the former in terms of one only of its denominations.

.;
or, if r is not suitably chosen, we may not. .If, e.g. r= io, we get the ordinary expression of P/Q as an integer and a decimal; but, if P/Q were equal to 1/3, we could not express it as a decimal with a finite number of figures.^ This is the usual method; but the relative accuracy of two numbers expressed to the same number of significant figures depends to a certain extent on the magnitude of the first figure.

^ The points of the compass might similarly be expressed by numbers in a binary scale; but the numbers would be ordinal, and the expressions would be analogous to those of decimals rather than to those of whole numbers.

^ The expressions 4 r are then fractional numbers, their relation to n' n' n' ' ' ordinary or integral numbers being that /2 n times n times is equal to p times.

(iii.) In the above method the choice of r is arbitrary. We can avoid this arbitrariness by a different procedure. .Having obtained R, which is less than Q, we now repeat with Q and R the process that we adopted with P and Q; i.e. we cut off from Q successive portions each equal to R. Suppose we find Q = sR+T, then we repeat the process with R and T; and so on.^ This discount, of course, is not equal to the interest which the present value would produce at that rate of interest, but is rather greater, so that the present value as calculated in this way is less than the theoretical present value.

^ For n = 3, this states that the scalar product of two vectors is less than or equal to the product of the lengths of the vectors, but it is true for any n.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ The C language floor function and the BASIC int function returns the whole number that is less than or equal to the argument, and thus matches Rule 2.
  • Arithmetic: Programming Language History 3 February 2010 14:24 UTC hhh.gavilan.edu [Source type: FILTERED WITH BAYES]

.We thus express P=Q in the form of a continued fraction, k+ + I, which is usually written,for conciseness, k+ s + t + &c., s t+&c.^ The Babylonians expressed numbers less than r by the numerator of a fraction with denominator 60; the numerator only being written.

^ Any exact fraction can be expressed as a continued fraction, and there are methods for expressing as continued fractions certain other numbers, e.g.

^ Any rational fractional function can be expressed in this canonical form.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

or k+ s - + - t+ &C.
(iv.) .If P and Q can be expressed in the forms pL and qL, where p and q are integers, R will be equal to (p-kq)L, which is both less than pL and less than qL. Hence the successive remainders are successively smaller multiples of L, but still integral multiples, so that the series of quotients k, s, t,.^ On the most recent trend NAEP, both age groups were less proficient at computing with fractions than in 1982, twenty years ago."
  • Illinois Loop: Mathematics 28 January 2010 0:26 UTC www.illinoisloop.org [Source type: FILTERED WITH BAYES]

^ The series is formed by successive multiplication, and any antilogarithm to a larger number of decimal places is formed from it in the same way by multiplication.

^ This discount, of course, is not equal to the interest which the present value would produce at that rate of interest, but is rather greater, so that the present value as calculated in this way is less than the theoretical present value.

..
will ultimately come to an end. Moreover, if the last divisor is uL, then it follows from the theory of numbers (� 26 (ii.)) that (a) u is a factor of p and of q, and (b) any number which is a factor of p and q is also a factor of u. Hence u is the greatest common measure of p and q. 35. In relation to algebra, the graphic method is mainly useful in connexion with the theory of limits (�� 58, 61) and the functional treatment of equations (� 60). .As regards the latter, there are two classes of cases.^ There are two kinds of practice, simple practice and compound practice, but the latter is the simpler of the two.

^ There would be great convenience in a general adoption of this latter method; the combination of the two methods in such an expression as £123, 16s.

^ But in the latter case it must always be understood that there is some unit concerned, and the results have no meaning until the unit is reintroduced.

.In the first class come equations in a single unknown; here the function which is equated to zero is the Y whose values for different values of X are traced, and the solution of the equation is the determination of the points where the ordinates of the graph are zero.^ The values verses the first differences is (value/first difference) and is constant.
  • Alltop - Top Math News 2 February 2010 15:42 UTC math.alltop.com [Source type: General]

^ Since 3/7 is a repeating binary fraction, its computed value in double precision is different from its stored value in single precision.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ This Is a simple program that uses Cramer's Rule To figure out the intersection point of two linear equations and provides you with the determinants and the intersecting point.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

.The second class of cases comprises equations involving two unknowns; here we have to deal with two graphs, and the solution of the equation is the determination of their common ordinates.^ Here is a second case where cancer is prescribed as the cure for cancer.
  • Arithmetic Presentation -Complete 3 February 2010 14:24 UTC www.mnforsustain.org [Source type: FILTERED WITH BAYES]

^ This Is a simple program that uses Cramer's Rule To figure out the intersection point of two linear equations and provides you with the determinants and the intersecting point.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

^ Systems of linear equations is a particular case for which a large amount of algebra exists, including matrices and determinants, which is called linear algebra .
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

Graphic methods also enter into the consideration of irrational numbers (� 65).
III. Elementary Algebra of Positive Numbers. 36. Monomials. - .(i.) An expression such as a.2.a.a.b.c.3.a.a.c, denoting that a series of multiplications is to be performed, is called a monomial; the numbers (arithmetical or algebraical) which are multiplied together being its factors. An expression denoting that two or more monomials are to be added or subtracted is a multinomial or polynomial, each of the monomials being a term of it.^ Revisions: Added cubic factoring and multiplying.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

^ It multiplies the polynomial by the exponent number of times.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

^ The factor multiplying P is called the capital recovery factor .
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

.A multinomial consisting of two or of three terms is a binomial or a trinomial. (ii.^ It can also multiply two binomials to get one trinomial.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

^ TRIFACTR TRIFACTR will factor any quadriatic trinomial into two binomials, with i in mind (That was just to italicise "i", even though we both know it won't work), if of course, it has rational factors.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

^ Trinomial-> Binomial Factorer Converts a standard quadratic equation into two binomials.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

) .By means of the commutative law we can collect like terms of a monomial, numbers being regarded as like terms.^ Subdivision of Submultiple.-By 7 of A we mean 5 times the unit, 7 times which is A. If we regard this unit as being 4 times a lesser unit, then Aris 7.4 times this lesser unit, and 7 of A is 5.4 times the lesser unit.

^ Collecting terms in an expression means combining terms differing only in the numerical coefficient.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ Instead of regarding the 153 in 27.153 as meaning o h, we may regard the different figures in the expression as denoting numbers in the successive orders of submultiples of i on a denary scale.

Thus the above expression is equal to 6a 5 bc 2 , which is, of course, equal to other expressions, such as 6ba 5 c 2. The numerical factor 6 is called the coefficient of a 5 bc 2 (� 20); and, generally, the coefficient of any factor or of the product of any factors is the product of the remaining factors.
(iii.) The multiplication and division of monomials is effected by means of the law of indices. Thus 6a5bc2=5a2bc=1-a3c, since I. It must, of course, be remembered (� 23) that this is a statement of arithmetical equality; we call the statement an " identity," but we do not mean that the expressions are the same, but that, whatever the numerical values of a, b and c may be, the expressions give the same numerical result. .In order that a monomial containing a m as a factor may be divisible by a monomial containing a p as a factor, it is necessary that p should be not greater than m. (iv.^ To compute m x from mx involves rounding off the low order k digits (the ones marked with b ) so (32) m x = mx - x mod( k ) + r k The value of r is 1 if .bb...b is greater than and 0 otherwise.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ With a single guard digit, the relative error of the result may be greater than , as in 110 - 8.59.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ When only the order of magnitude of rounding error is of interest, ulps and may be used interchangeably, since they differ by at most a factor of .
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

) .In algebra we have a theory of highest common factor and lowest common multiple, but it is different from the arithmetical theory of greatest common divisor and least common multiple.^ Each worksheet can have up to 50 problems with a variety of whole number, integer, equation, greatest common factor, least common multiple, or algebra problems.

^ We might also be interested in the least common multiple (LCM) m = ka = jb, where k and j are integers.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ This book is somewhere between simple arithmetic and elementary number theory, but develops the subjects quite differently than we do today.
  • Arithmetic - Mathematics and the Liberal Arts 3 February 2010 14:24 UTC math.truman.edu [Source type: FILTERED WITH BAYES]

.We disregard numerical coefficients, so that by the H.C.F. or L.C.M. of 6a 5 bc 2 and 12a 4 b 2 cd we mean the H.C.F. or L.C.M. of a 5 bc 2 and a 4 b 2 cd. The H.C.F. is then an expression of the form a p b e c r d s , where p, q, r, s have the greatest possible values consistent with the condition that each of the given expressions shall be divisible by a p b e c r d s . Similarly the L.C.M. is of the form a p b e c r d s , where p, q, r, s have the least possible values consistent with the condition that a P PC'd s shall be divisible by each of the given expressions.^ Extended-based systems run most efficiently when expressions are evaluated in extended precision registers whenever possible, yet values that must be stored are stored in the narrowest precision required.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ Do real programs depend on the assumption that a given expression always evaluates to the same value?
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ In the numerical example given above, the computed value of (7) is 2.35, compared with a true value of 2.34216 for a relative error of 0.7 , which is much less than 11 .
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

In the particular case it is clear that the H.C.F. is a 4 bc and the L.C.M. is a5b2c2d. The extension to multinomials forms part of the theory of factors (� 51).
37. Products of Multinomials. - .(i.) Special arithmetical results may often be used to lead up to algebraical formulae.^ Lessons include Calendar Fun, in which students use a simple algebraic formula to determine which four days add up to a given sum and The Hot Tub, where students are asked to interpret data from a graph to tell a story.

^ For example sums are a special case of inner products, and the sum ((2 × 10 -30 + 10 30 ) - 10 30 ) - 10 -30 is exactly equal to 10 -30 , but on a machine with IEEE arithmetic the computed result will be -10 -30 .
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ As Johnny tried to work algebraic equations, his arithmetic kept bringing up weird results.
  • Illinois Loop: Mathematics 28 January 2010 0:26 UTC www.illinoisloop.org [Source type: FILTERED WITH BAYES]

.Thus a comparison of numbers occurring in a table of squares 1 2 =1 112=121 2 2= 4 122=144 3 2 =9 132=169 suggests the formula (A+a)2=A2+2Aa+a2. Similarly the equalities 99 X I o I = 9999 = wow - I 98 X 102 = 999 6 = moo() - 4 97 X 10 3 =9991 =1 0000 - 9 lead up to (A - a) (A+a) = A 2 - a 2. These, with (A - a)2= A 2 -2Aa+a 2, are the most important in elementary work.^ Thus there are (2 10 - 10 3 )2 14 = 393,216 different binary numbers in that interval.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ Thus 1878 in the quinary-binary scale would be 1131213, and 1828 would be 1130213; the meaning of these is seen at once by comparison with MDCCCLXXVIII and MDCCCXXVIII. Similarly the number which in the denary scale is 215 would in the quaternary scale (base 4) be 3113, being equal to 3.4.4.4+ 1.4.4+1.4+3.

^ Calculation of Square Root.-The calculation of the square root of a number depends on the formula (iii) of § 60.

(ii.) .These algebraical formulae involve not only the distributive law and the law of signs, but also the commutative law.^ Such an expression is also called an entire rational algebraic function, since it involves only addition, multiplication and raising to a power.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ I generally deduct only one point for these errors, not because they are unimportant, but because deducting more would involve swimming against a tide that is just too strong for me.

^ All these relations constitute the "laws of exponents" of elementary algebra and are easily proved for integral exponents.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

.Thus (A +a) 2 = (A +a) (A+a) =A(A+a) +a(A+a)=AA+Aa+aA+aa; and the grouping of the second and third terms as 2Aa involves treating Aa and aA as identical.^ Polynomial Term Expander Expands terms (ax+b) to form a second, third, or fourth-order polynomial.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

This is important when we come to the binomial theorem (� 41, and cf. � S4 (i.)).
(iii.) .By writing (A+a) 2 = A 2 + 2Aa+a 2 in the form (A+a)2= A 2 +(2A+a)a, we obtain the rule for extracting the square root in arithmetic.^ The diagrams, A, B, C are simple forms of tables; A giving a sum-series, B a multiple-series, and C a series of square roots, calculated approximately.

^ Simplify This program simplifies the process of calculating what square roots are when you need it in the form like 7 times the square root of 2.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

^ Thus to find the cube root of 1728, we write it in the form 2.3, and find that its cube root is 2.3=12; or, to find the cube root of 1 728, we write it as 17 r - _ 21_ s_ _ 2'.33.

(iv.) .When the terms of a multinoniial contain various powers of x, and we are specially concerned with x, the terms are usually arranged in descending (or ascending) order of the indices; terms which contain the same power being grouped so as to give a single coefficient.^ No more than two indices can be the same in a term.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ However, computing with a single guard digit will not always give the same answer as computing the exact result and then rounding.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ In order to apply arithmetical processes to a quantity expressed in two or more denominations, we must first express it in terms of a single denomination by means of a varying scale of notation.

.Thus 2bx-4x 2 +6ab+3ax would be written - 4x 2 +(3a+2b)x+6ab. It is not necessary to regard - 4 here as a negative number; all that is meant is that 4x2 has to be subtracted.^ Simply specify the type of addition, subtraction, multiplication, or division, problem for either the counting numbers, the negative numbers, decimals, or fractions and the maximum and minimum numbers to be used in the problems.

^ When the result of any arithmetical operation or operations is represented approximately but not exactly by a number, the excess (positive or negative) of this number over the number which would express the result exactly is called the error.

^ Here is one example that he has mentioned: Every positive number has two square roots: one positive, the other negative.

(v.) .When we have to multiply two multinomials arranged according to powers of x, the method of detached coefficients enables us to omit the powers of x during the multiplication.^ To multiply the roots by k, multiply each coefficient by a power of k, starting with k 0 , then k 1 , and so on.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ The table of multiples will then be as in C. The next step is to arrange the multiplier and the multiplicand above the partial products.

^ The method E of § ioi being adopted, the multiplicand and the multiplier are written with a space after as many digits (of each) as will be required in the product (on the principle explained in § 1 01); and the multiplication is performed from the left, two extra figures being kept in.

If any power is absent, we treat it as present, but with coefficient o. .Thus, to multiply x 3 -2x+1 by 2x 2 +4, we write the process +I +0-2+I +2+0+4 +2+0-4+2 +0+0 - 0+0 +4+0-8+4 +2+0+0+2-8+4 giving 2x 5 + 2x 2 -8x+4 as the result.^ The divisor goes -4 times into this, making the quotient x - 4, while multiplying the divisor gives -4x 2 - 8x - 4.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ We write this down as the first part of the quotient, multiply the divisor to give x 3 + 2x 2 + x and subtract from the dividend, with the result -4x 2 - 5.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

38. Construction and Transformation of Equations. - .(i.) The statement of problems in equational form should precede the solution of equations.^ Each lesson includes a statement describing the problem, objectives, teaching sequence, specific learning outcomes, extension tips, the solution, and a printable student worksheet.

^ The old-fashioned problems about the amount of work done by particular numbers of men, women and boys, are of this kind, and really involve the solution of simultaneous equations.

^ A string of statements of the form A ⇒ B ⇒ C ⇒ D should mean that A by itself implies B, and B by itself implies C, and C by itself implies D; that is the coventional interpretation given by mathematicians.

(ii.) .The solution of equations is effected by transformation, which may be either arithmetical or algebraical.^ Algebraic numbers are numbers that are the solution of an algebraic equation, as x = √2 is the solution of x 2 = 2.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ As Johnny tried to work algebraic equations, his arithmetic kept bringing up weird results.
  • Illinois Loop: Mathematics 28 January 2010 0:26 UTC www.illinoisloop.org [Source type: FILTERED WITH BAYES]

^ Rational numbers are, of course, algebraic, since they are the solutions of linear equations.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

The principles of arithmetical transformation follow from those stated in �� 15-18 by replacing X, A, B, m, M, x, n, a and p by any expressions involving or not involving the unknown quantity or number and representing positive numbers or (in the case of X, A, B and M) positive quantities. The principle of algebraic transformation has been stated in � 22; it is that, if A=B is an equation (i.e. if either or both of the expressions A and B involves x, and A is arithmetically equal to B for the particular value of x which we require), and if B = C is an identity (i.e. if B and C are expressions involving x which are different in form but are arithmetically equal for all values of x), then the statement A = C is an equation which is true for the same value of x for which A = B is true.
(iii.) .A special rule of transformation is that any expression may be transposed from one side of an equation to the other, provided its sign is changed.^ Numbers of the form x + i (+0) have one sign and numbers of the form x + i (-0) on the other side of the branch cut have the other sign .
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ The sides of an equation may be divided by equal quantities, provided the quantities are not zero.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ A term can be moved from one side of an equation to the other, provided that is sign is changed.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

.This is the rule of transposition. Suppose, for instance, that P+Q - R+S = T. This may be written (P+ Q - R) +S = T; and this statement, by definition of the sign -, is the same as the statement that (P+Q - R) = T - S. Similarly the statements P+Q - R - S = T and P+ Q - R= T+ S are the same.^ Suppose we have n equations in n unknowns, written in the usual form with the constant terms on the right of the equal signs.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ The implication "A implies B" is not the same as the implication "B implies A." For instance, if I went swimming at the beach today, then I got wet today is a true statement.

^ For instance, here is the definition of continuity of a real-valued function f: f is continuous if for each real number p and each positive number ε there exists a positive number δ (which may depend on p and ε) such that, for each real number q, if .

These transpositions are purely arithmetical. .To transpose a term which is not the last term on either side we must first use the commutative law, which involves an algebraical transformation.^ The first term x 1 is perturbed by n , the last term x n by only .
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ Simple " practice involves an application of the commutative law.

^ More complicated forms of arithmetical reasoning involve the use of series, each term in which corresponds to particular terms in two or more series jointly; and cases of this kind are usually dealt with by special methods, or by means of algebraical formulae.

Thus from the equation P+Q - R+S=T and the identity P+Q - R+S= P - R +S+Q we have the equation P - R +S+Q=T, which is the same statement as P - R +S=T - Q.
(iv.) .The procedure is sometimes stated differently, the transposition being regarded as a corollary from a general theorem that the roots of an equation are not altered if the same expression is added to or subtracted from both members of the equation.^ Equal quantities can be added to, subtracted from, or multiplied with, both sides of an equation.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ The expressions ∫ u 2 du and ∫ u 2 dx have very different meanings, but you're likely to confuse them if you write them both as ∫ u 2 .

^ Sometimes a term must be added and subtracted (which does not change the value of the expression) in order to do this, as in completing the square.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

The objection to this (cf. � 21 (ii.)) is that we do not need the general theorem, and that it is unwise to cultivate the habit of laying down a general law as a justification for an isolated action.
(v.) An alternative method of obtaining the rule of transposition is to change the zero from which we measure. Thus from P+Q - R+S=T we deduce P+(Q - R+S)=P+(T - P). .If instead of measuring from zero we measure from P, we find Q - R+ S = T-- P. The difference between this and (iii.^ Another way to measure the difference between a floating-point number and the real number it is approximating is relative error , which is simply the difference between the two numbers divided by the real number.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ But in measuring a distance we may find that it is " between " two distances differing by a unit of the lowest denomination used, and a subdivision of this unit follows naturally.

^ The zero, called " nought," is of course a different thing from the letter 0 of the alphabet , but there may be a historical connexion between them (§ 79).

) is that we transpose the first term instead of the last; the two methods corresponding to the two cases under (i.) of � 1 5 (2). (vi.) .In the same way, we do not lay down a general rule that an equation is not altered by multiplying both members by the same number.^ Every nonzero complex number b has two square roots, but in general there is no natural way to say which one should be associated with the expression √b.

^ It is typical for denormalized numbers to guarantee error bounds for arguments all the way down to 1.0 x .
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ To use, enter the amount of variables and the number of equations, then enter all the coefficients in HORIZONTALLY. Make sure if you enter 1, 2, 3 for the coefficients of X, Y, and Z respectively, that you also enter the coefficients of the second equation IN THE SAME EXACT ORDER!!!
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

Suppose, for instance, that -(x+ i) =1-(x-2). .Here each member is a number, and the equation may, by the commutative law for multiplication, be written 2(x+I) - 4(x-2) This means that, whatever unit A we take, 2(x+ I) A Sand 5 4(x-2) A are equal.^ Here are some things worth noting: Multiplication is commutative -- that is, xy = yx.

^ The pair of compartments on either side may, as here, contain numerical quantities, or may contain numbers.

^ Division.-From the commutative law for multiplication, which shows that 3 X 4d.

.We therefore take A to be 15, and find 3 that 6(x+I)=20(x-2).^ It Is Therefore Convenient, In Finding The Product Of Two Numbers, To Take The Smaller As The Multiplier.

^ Therefore, we can find a 0 by dividing f(x) by (x - b) and taking the remainder.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

.Thus, if we have an equation P=Q, where P and Q are numbers involving fractions, we can clear of fractions, not by multiplying P and Q by a number m, but by applying the equal multiples P and Q to a number m as unit.^ Multiplication of Fractional Numbers 6.3 66.

^ Simply specify the type of addition, subtraction, multiplication, or division, problem for either the counting numbers, the negative numbers, decimals, or fractions and the maximum and minimum numbers to be used in the problems.

^ The rule for multiplying a fractional number by a fractional number is therefore the same as the rule for finding a fraction of a fraction.

If the P and Q of our equation were quantities expressed in terms of a unit A, we should restate the equation in terms of a unit Anna, as explained in �� 18 and 21 (i.) (a).
(vii.) .One result of the rule of transposition is that we can transpose all the terms in x to one side of equation, and all the terms not containing x to the other.^ A term can be moved from one side of an equation to the other, provided that is sign is changed.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ This program also features other such equations where theres are unkowns on both sides of the equal sign.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

^ The resulting equation will have for its solutions all of the solutions of the original equation plus the additional new solution x=4.

.An equation of the form ax=b, where a and b do not contain x, is the standard form of simple equation. (viii.^ Diophantine equations solves diophantine equations of the form ax+by=c .
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

^ Form Converter v1.1 Use this program to convert an equation from standard form to slope/intercept form .
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

^ First, you are prompted to enter the coefficients of two equations in standard form.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

) .The quadratic equation is the equation of two expressions, monomial or multinomial, none of the terms involving any power of x except x and x 2 . The standard form is usually taken to be ax2+bx+c =0, from which we find, by transformation, (2 ax+b) 2 =b 2 - 4ac, 4 (}b 2 -4ac} -b and thence x = 2a (ii.^ Just input A,B, and C for Ax^2+Bx+C and you will recieve a factored form in return.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

^ Equation of a Line Given any two points or a point with a slope it can find the equation.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

^ Also, because the scaled values of m and q satisfy m /2 < q < 2 m , the corresponding value of n must have one of two forms depending on which of m or q is larger: if q   <  m , then evidently 1 < n < 2, and since n is a sum of two powers of two, n = 1 + 2 - k for some k ; similarly, if q > m , then 1/2 < n < 1, so n = 1/2 + 2 -( k + 1) .
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

) In an equation of the form Q=V, the expressions P, Q, U, V are usually numerical. We then have -. QV. =-. Q V, or PV = UQ, as in � 38 (vi.). This is the rule of cross-multiplication. (iii.) The restriction in (i.) is important. Thus 2 x 2 - I - x +x-2 (x _ I) (x+2) is equal to x + 2 q, except when x=1. For this latter value it becomes -°o, which has no direct meaning, and requires interpretation (� 61).
40. Powers of a Binomial. - .We know that (A +02= A2+ 2Aa+a 2. Continuing to develop the successive powers of A+a into multinomials, we find that (A+a)3=A3+3A2a+3Aa2+a3, &c.; each power containing one more term than the preceding power, and the coefficients, when the terms are arranged in descending powers of A, being given by the following table I I ' 'I 2 I 1 3 3 I 4 6 4 I 5 IO to 5 I I x 6 15 20 15 6 &c., where the first line stands for (A+a)°=1. A°a°, and the successive numbers in the (n+i)th line are the coefficients of A n a O An-lal ..^ No more than two indices can be the same in a term.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ Although formula (7) is much more accurate than (6) for this example, it would be nice to know how well (7) performs in general.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ From these we can find general rational functions of more than one variable.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

A°a n in the n+ I terms of the multinomial equivalent to (A+a)".
.In the same way we have (A-a) 2 =A 2 -2Aa+a 2, (A-a)3 = A 3 -3A 2 a+3Aa 2 -a 3, ..., so that the multinomial equivalent to (A-a)" has the same coefficients as the multinomial equivalent to (A+a)", but with signs alternately + and -.^ The most obvious symbol for this purpose is ≡, which means "is equivalent to," but that symbol has the disadvantage of looking too much like an equals sign, and thus possibly leading to the same confusion.

^ The next coefficient can be found in the same way by dividing q(x) by (x - b) and taking the remainder, and so on.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

.The multinomial which is equivalent to (A= a)", and has its terms arranged in ascending powers of a, is called the expansion of (A= a) n.^ If, for instance, three terms of a proportion are given, the fourth can be obtained by the relation given at the end of § 57, this relation being then called the Rule of Three; but this is equivalent to the use of an algebraical formula.

^ Findterm This program uses the Generla Formula for a Specific Term of an Expansion formula to find the coefficients and powers for a binomial raised to a power.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

.41. The binomial theorem gives a formula for writing down the coefficient of any stated term in the expansion of any stated power of a given binomial.^ Findterm This program uses the Generla Formula for a Specific Term of an Expansion formula to find the coefficients and powers for a binomial raised to a power.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

^ That's because most of the formulas involving trigonometric functions come out much simpler with radians than with degrees -- the formulas for the derivatives, for the power series expansions, etc.

^ Just plug in the given x, the coefficients, and the constant term to find the result.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

(i.) For the general formula, we need only consider (A+a)". It is clear that, since the numerical coefficients of A and of a are each 1, the coefficients in the expansions arise from the grouping and addition of like terms (� 37 (ii.)). We therefore determine the coefficients by counting the grouped terms individually, instead of adding them. To individualize the terms, we replace (A+a) (A+a) (A+a) ... by .(A+a) (B+b) (C+c) ..., so that no two terms are the same; the " like " -ness which determines the placing of two terms in one group being the fact that they become equal (by the commutative law) when B, C,.^ If four coins are laid on a table, close together, they can (by most adults) be seen to be four, without counting; but seven coins have to be separated mentally into two groups, the numbers of which are added, or one group has to be seen and the remaining objects counted, before the number is known to be seven.

^ There are no single symbols for two, three, &c.; but numbers are represented by combinations of symbols for one, five, ten, fifty, one hundred, five hundred, &c., the numbers which have single symbols, viz.

^ And in the same ad they noted that: "Conservation does no harm."
  • Arithmetic Presentation -Complete 3 February 2010 14:24 UTC www.mnforsustain.org [Source type: FILTERED WITH BAYES]
  • Presentation by Albert Bartlett - Arithmetic, Population, and Energy 3 February 2010 14:24 UTC hawaii.gov [Source type: FILTERED WITH BAYES]

.. and b, c, .. are each replaced by A and a respectively.
.Suppose, for instance, that n=5, so that we take five factors (A+a) (B+b) (C+c) (D+d) (E+e) and find their product.^ It Is Therefore Convenient, In Finding The Product Of Two Numbers, To Take The Smaller As The Multiplier.

^ Polynomial (up to five factors) Multiplier Polynomial Multiplier lets you enter up to FIVE polynomial factors and find out what the resulting polynomial will be.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

.The coefficient of A 2 a 3 in the expansion of (A+a) 5 is then the number of terms such as ABcde, AbcDe, AbCde, ...^ Binomial Expander This program will list the resulting terms of any binomial expansion of a binomial in [A(F)^X + B(S)^Y]^E format, where "F" and "S" are the variables, "A" and "B" are the variables' coefficients, "X" and "Y" are the variables' exponents, and "E" is the exponent of the binomial.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

^ Findterm This program uses the Generla Formula for a Specific Term of an Expansion formula to find the coefficients and powers for a binomial raised to a power.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

^ A number can often be expressed by a series of terms, such that by taking successive terms we obtain successively closer approximations.

,
in each of which there are two large and three small letters. .The first term is Abcde, in which all the letters are large; and the coefficient of A 2 a 3 is therefore the number of terms which can be obtained from Abcde by changing three, and three only, of the large letters into small ones.^ The first term x 1 is perturbed by n , the last term x n by only .
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ First of all, there are algebraic identities that are valid for floating-point numbers.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ If, however, we count these three as one, two, three, then the number of times we count is an abstract number.

.We can begin with any one of the 5 letters, so that the first change can be made in 5 ways.^ It therefore arises in one or other of two ways, according as the unit or the number exists first in consciousness.

^ Division.-In the same way, in performing approximate division, we can at a certain stage begin to abbreviate the divisor, taking off one figure (but with correction of the final figure of the partial product) at each stage.

There are then 4 letters left, and we can change any one of these. Then 3 letters are left, and we can change any one of these. Hence the change can be made in 3.4.5 ways.
.If, however, the 3.4.5 results of making changes like this are written down, it will be seen that any one term in the required product is written down several times.^ A common method is to reverse the digits in one of the numbers; but this is only appropriate to the old-fashioned method of writing down products from the right.

^ The ethanol production is seen to be approximately 1% of the annual consumption of gasoline by vehicles in the U.S. So one would have to multiply corn production by a factor of about 100 just to make the numbers match.
  • Arithmetic Presentation -Complete 3 February 2010 14:24 UTC www.mnforsustain.org [Source type: FILTERED WITH BAYES]

^ A term can be moved from one side of an equation to the other, provided that is sign is changed.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

.Consider, for instance, the term AbcDe, in which the small letters are bce. Any one of these 3 might have appeared first, any one of the remaining 2 second, and the remaining I last.^ [Rosenbach and Whitman] If, in the preceding problem, Achilles halves the distance in the first minute, then halves the remaining distance in the second minute, and so on, how long will it take him to overtake the tortoise?
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ For those of you that have not had your second cup of coffee today, I will explain the last one.
  • Arithmetic: Programming Language History 3 February 2010 14:24 UTC hhh.gavilan.edu [Source type: FILTERED WITH BAYES]

^ The scalar product of these vectors is x' i y' i = q ij q ik x j y k (note that the second dummy variable has been changed from j to k to avoid confusion with the first).
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

.The term therefore occurs 1.2. 3 times.^ The £I is termed the unit, A numerical quantity, therefore, represents a certain unit, taken a certain number of times.

.This applies to each of the terms in which there are two large and three small letters.^ There are no single symbols for two, three, &c.; but numbers are represented by combinations of symbols for one, five, ten, fifty, one hundred, five hundred, &c., the numbers which have single symbols, viz.

^ A similar analysis of ( x x ) ( y y ) cannot result in a small value for the relative error, because when two nearby values of x and y are plugged into x 2  - y 2 , the relative error will usually be quite large.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ As it becomes necessary to use higher but still small numbers, they are formed by combinations of one and two, or perhaps of three with one or two.

The total number of such terms in the multinomial equivalent to (A+a) (B+b) (C+c) (D+d) (E+e) is therefore (3.4. 5)- (1.2. 3); and this is therefore the coefficient of A 2 a 3 in the expansion of (A+a)5.
.The reasoning is quite general; and, in the same way, the coefficient of A n - r a r in the expansion of (A+a) n is { (n-r+ I) (n-r+2).^ The next coefficient can be found in the same way by dividing q(x) by (x - b) and taking the remainder, and so on.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

.. (n-I)n}= 11.2.3 ... r}. .It is usual to write this as a fraction, inverting the order of the factors in the numerator.^ Fraction in its Lowest Terms.-A fraction is said to be in its lowest terms when its numerator and denominator have no common the more correct method is to write it a: b.

^ The pth root of a number (§43) may, if the number is an integer, be found by expressing it in terms of its prime factors; or, if it is not an integer, by expressing it as a fraction in its lowest terms, and finding the pth roots of the numerator and of the denominator separately.

^ If we write 74 in the form 47 we may say that the value of a fraction is not altered by multiplying or dividing the numerator and denominator by any number.

Then, if we denote it by no), so that - n (n - I)...(n - r+I) (1), 72(r) - I. 2.3... r we have (A+a) n =n(o)A"+n(l)A"-la-{-... +n(r)An-rar+�.� +n(n)a n (2), where n(0), introduced for consistency of notation, is defined by n (o) EI (3). This is the binomial theorem for a positive integral index.
(ii.) To verify this, let us denote the true coefficient of An-rar by (,), so that we have to prove that (;`.) = n(r), where n(r ) is defined by (I); and let us inspect the actual process of multiplying the expansion of (A+ a) n -' by A+a in order to obtain that of (A+a)". Using detached coefficients (� 37 (v.)), the multiplication is represented by the following: - I+nI I) -} in-I) 2 n+...+(nrI)-F...+ I -{- (n r I) �...+ (r - I) �...+ (n -2) + I (111) (7) -} ... -} (n n I) (n r + (r - I). Now suppose that the formula (2) has been established for every power of A+a up to the (n-i)th inclusive, so that (ii_ I) = (n- I) (r), ( y I) = (n -1) (r _ l). Then (y) ,the coefficient - of A n - r a r in the expansion of (A+a)", is equal to (n-i)(0+ (r_l). .But it may be shown that (r being > o) This only gives one root.^ We'll only mention ones here that can be used for any kind of equation, not just for finding roots of polynomials; for others, see a book on the theory of equations.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ This relation is not one of proportion; but it may nevertheless be expressed by tabulation, as shown at D. .

^ You can use only one variable if you wish, but you still have to give the other variable a value.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

As to the other root, see � 47 (iii.). 39. Fractional Expressions. - .An equation may involve a fraction of the form Q, where Q involves x. (i.) If P and Q can (algebraically) be written in the forms RA and SA respectively, where A may or may not involve x, then Q = RA R, provided A is not o. SA so that n (r)_ (n - I) (r) -{- (n - I) (r-1) and therefore (n) = no.^ Hence we may insert in the power-series numbers with fractional indices, provided that the multiplication of these numbers follows the same law.

^ Suppose we have n equations in n unknowns, written in the usual form with the constant terms on the right of the equal signs.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ Financial calculations may involve progressions, and are usually included in intermediate algebra.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

).
Hence the formula (2) is also true for the nth power of A-I-a. But it is true for the 1st and the 2nd powers; therefore it is true for the 3rd; therefore for the 4th; and so on. Hence it is true for all positive integral powers of n. (iii.) The product 1.2. 3. .. .r is denoted by I?' or r!, and is called factorial r. The form r! is better for printing, but the form I r is more convenient for ordinary use.^ If mankind had had six fingers on each hand and six toes on each foot, we should be using a duodenary scale (base twelve), which would have been far more convenient.

^ This notation is more than a convenience; it is a powerful method of "doing algebra" on such array quantities, and can be used to prove all of the properties of matrices.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

If we denote n(n -i) .. . (n-r+ I) (r factors) by n (r), then n(r)-n(r)/r!. (iv.) We can write n (;) in the more symmetrical form n(r)- (n - r) ! r ! which shows that n (r) = n (n-r) (6). We should have arrived at this form in (i.) by considering the selection of terms in which there are to be two large and three small letters, the large letters being written down first. The terms can be built up in 5! ways; but each will appear 2! 3! times.
(v.) Since (r) is an integer, (r) is divisible by r!; i.e. the product of any r consecutive integers is divisible by r! (see � 42 (ii.)).
(vi.) The product r! arose in (i.) by the successive multiplication of r, 'I, r' - 2,. .. I. In practice the successive factorials I !, 2 !, 3!. .. are supposed to be obtained successively by introduction of new factors, so that r!=r. (r-I)! (7)� Thus in defining r! as I. 2.3. .. .r we regard the multiplications as taking place from left to right; and similarly in r A product in which multiplications are taken in this order is called a continued product. (vii.^ The result of the multiplication is called the product of the unit by the number of times it is taken.

^ Diagrams of Division.-Since we write from left to right or downwards, it may be convenient for division to interchange the rows or the columns of the multiplication-diagram.

^ In short division the divisor and the quotient are placed respectively on the left of and below the dividend, and the partial products and remainders are not shown at all.

) .In
order to make the formula (5) hold for the extreme values n (o) and n (n) we must adopt the convention that o !^ Sometimes a term must be added and subtracted (which does not change the value of the expression) in order to do this, as in completing the square.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ Higher-order interpolation formulas, such as Langrange's, are very good with accurate function values.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ This set of numbers can tell you about the central tendency of your data, the spread, the extreme values, and provide low order information about the shape of the distribution.
  • Alltop - Top Math News 2 February 2010 15:42 UTC math.alltop.com [Source type: General]

= I (8). This is consistent with (7), which gives I!= i.o!. .It should be observed that, for r=o, (4) is replaced by n (o) _ (n - I)(o) (9), and similarly, for the final terms, we should note that P0)= 0 if 4> p (io). (viii.^ In reference to the use of the sign X with the converting factor, it should be observed that " lb X " symbolizes the replacing of so many times 4 lb by the same number of times 7 lb, while " 4 X " symbolizes the replacing of 4 times something by 7 times that something.

) .If u, denotes the term involving a r in the expansion of (A-{-a) n, then ur/ur - 1 = {(n-r+ i)/r}.a/A. This decreases as r increases; its value ranging from na/A to al(nA). If na<A, the terms will decrease from the beginning; if n A< a, the terms will increase up to the end; if na > A and nA > a, the terms will first increase up to a greatest term (or two consecutive equal greatest terms) and then decrease.^ In order to apply arithmetical processes to a quantity expressed in two or more denominations, we must first express it in terms of a single denomination by means of a varying scale of notation.

^ In most cases the quantity in the second column may be regarded as increasing or decreasing continuously as the number in the first column increases, and it has intermediate values corresponding to intermediate (i.e.

^ The first is increased exponent range.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

(ix.) .The position of the greatest term will depend on the relative values of A and a; if a/A is small, it will be near the beginning.^ A similar analysis of ( x x ) ( y y ) cannot result in a small value for the relative error, because when two nearby values of x and y are plugged into x 2  - y 2 , the relative error will usually be quite large.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ In that case, if x is small but not quite small enough that 1.0   +   x rounds to 1.0 in single precision, then the value returned by log1p(x) can exceed the correct value by nearly as much as x , and again the relative error can approach one.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ Thus, if x is not so small that 1.0   +   x rounds to 1.0 in extended precision but small enough that 1.0   +   x rounds to 1.0 in single precision, then the value returned by log1p(x) will be zero instead of x , and the relative error will be one--rather larger than 5 .
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

Advantage can be taken of this, when n is large, to make approximate calculations, by omitting terms that are negligible.
(a) Let S r denote the sum u o+ u 1+. .. .+u,, this sum being taken so as to include the greatest term (or terms); and let u r+1 /u r = 0, so that 0< 1. Then the sum of the remaining terms ur+1+ u r+2+.^ Instead of being expressed as the sum of a series of terms, a number may be expressed as the product of a series of factors, which become successively more and more nearly equal to i.

� .+ u n is less than (I which is less than u r+1 /(i -0); and therefore (A+a) n lies between S r and Sr+ur+1/(i-8). We can therefore stop as soon as u r+i / (I -0) becomes negligible.
(b) In the same way, for the expansion of (Aa)", let o, denote uo-u 1 +. .. u r. Then, provided a r includes the greatest term, it will be found that (A - a)" lies between 0' r and ar+1� For actual calculation it is most convenient to write the theorem in the form methods of procedure. .We know that (A+a)" is equal to a multinomial of n+I terms with unknown coefficients, and we require to find these coefficients.^ Just plug in the given x, the coefficients, and the constant term to find the result.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

^ Suppose we have n equations in n unknowns, written in the usual form with the constant terms on the right of the equal signs.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ Logarithms With this program, you can find the base, term, or equal of a logarithm given the other two.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

.We therefore represent them by separate symbols, in the same way that we represent the unknown quantity in an equation by a symbol.^ A set of written symbols is sometimes read in more than one way, while on the other hand two different sets of symbols (at any rate if denoting numerical quantities) may be read in the same way.

^ The £I is termed the unit, A numerical quantity, therefore, represents a certain unit, taken a certain number of times.

^ We need therefore consider numerical quantities only, our results being applicable to numbers by regarding the digits as representing multiples of units in different denominations.

.This is the method of undetermined coefficients. We then obtain a set of equations, and by means of these equations we establish the required result by a process known as mathematical induction. This process consists in proving that a property involving p is true when p is any positive integer by proving (I) that it is true when p= 1, and (2) that if it is true when p=n, where n is any positive integer, then it is true when p = n+ I. The following are some further examples of mathematical induction.^ The following are some examples.

^ This method involves the derivative, which is outside of algebra, but for polynomials is very easy to obtain.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ Invalid operation covers the situations listed in TABLE D-3 , and any comparison that involves a NaN. The default result of an operation that causes an invalid exception is to return a NaN, but the converse is not true.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

(i.) By adding successively I, 3, 5.. . we obtain I, 4, 9, � � � This suggests that, if u n is the sum of the first n odd numbers, then u n = n 2 . Assume this true for u 1, u 2, ., u,,. Then u,,+1=un+(2n-+-I)=n2+(2n+I)=(n+I)2, so that it is true for u n+1. But it is true for u 1 . Therefore it is true generally.
(ii.) We can prove the theorem of � 41 (v.) by a double application of the method.
.(a) It is clear that every integer is divisible by I!.^ Instead of using synthetic division with every integer, it uses the rational zeros theorem to make the program up to 10x faster!!!!!!!!
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.(b) Let us assume that the product of every set of p consecutive integers is divisible by p!, and let us try to prove that the product of every set of p+ I consecutive integers is divisible by (p+i)!.^ Let us first assume the function f(t) is tabulated for t at equally-spaced values differing by h.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ Instead of using synthetic division with every integer, it uses the rational zeros theorem to make the program up to 10x faster!!!!!!!!
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

^ Let us examine the situation in regard to production of domestic crude oil in the U.S. Table IV gives the relevant data.
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  • Presentation by Albert Bartlett - Arithmetic, Population, and Energy 3 February 2010 14:24 UTC hawaii.gov [Source type: FILTERED WITH BAYES]

Denote the product n(n +I) .. . .(n+rI) by nlr1. Then the assumption is that, whatever positive integral value n may have, n [P] is divisible by p!. (I) n1P+11-(n- I)[P+11=n(n-l-I)...^ But in algebra a X b = b X a is called an identity, in the sense that it is true whatever a and b may be; while n X X = A is called an equation , as being true, when n and A are given, for one value only of X. Similarly the numbers represented by and a are not identical, but are equal.

(n+p - I)(n+p) - (n - I)n. .. (n+pi) _ (p+1). 'n 1P1. But, by hypothesis, n 1P1 is divisible by p!. Therefore n 1P+11 -(n-I) (P+11 is divisible by p!. Therefore, if (n-i) 1P+11 is divisible by (p+I)!, 'n 1P+11 is divisible b y ( p +i) !. (2) But 11P+11=(p+i)!, which is divisible by (p+I)!.
.(3) Therefore n1P+11 is divisible by (p+ I) !, whatever positive integral value n may have.^ But in algebra a X b = b X a is called an identity, in the sense that it is true whatever a and b may be; while n X X = A is called an equation , as being true, when n and A are given, for one value only of X. Similarly the numbers represented by and a are not identical, but are equal.

(c) Thus, if the theorem of � 41 (v.) is true for r= p, it is true for r= p+1. But it is true for r= I. Therefore it is true generally.
(iii.) .Another application of the method is to proving the law of formation of consecutive convergents to a continued fraction (see Continued Fractions).^ The slide-rule (see Calculating Machines ) is a simple apparatus for the mechanical application of the methods of logarithms.

^ Any exact fraction can be expressed as a continued fraction, and there are methods for expressing as continued fractions certain other numbers, e.g.

^ The theory of continued fractions gives a method of expressing a number, in certain cases, as a continued product.

43� Binomial Coefficients. - The numbers denoted by n (r) in � 41 are the binomial coefficients shown in the table in � 40; n (r) being the (r+ i) th number in the (n+ i) th row. They have arisen as the coefficients in the expansion of (A+a)"; but they may be considered independently as a system of numbers defined by (I) of � 41. The individual numbers are connected by various relations, some of which are considered in this section.
(i.) From (4) of � 41 we have n(r)- Changing n into n - i, 22-2,.. ., and adding the results, n(r)-(n-s)(r) =(n-I)(r-1)-+-(n-2)(r-1)+...+(n-s)(r-1) (12). .In particular, n (r) = (n - I) (r-1) +(n (r- (r _ i) (13). Similarly, by writing (4) in the form n (r) - (n- I) (r-1) _ (nI)(r) (14), changing n and r into ni and r-1, repeating the process, and adding, we find, taking account of (9), n(r)=(nI)(r)+ (n-2((r_1)+...+(n - r-I)0 (15). (ii.^ It also provides a progam that changes to points into slope intercept form.
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^ Similarly we cannot subtract 8 from 15, if 15 means 1 ten + 5 ones; we must either write 15-815-8=(10+5)-8= (I o - 8)+5 = 2+5 = 7, or else resolve the 15 into an inexpressible number of ones, and then subtract 8 of them, leaving 7.

^ Thus to find the cube root of 1728, we write it in the form 2.3, and find that its cube root is 2.3=12; or, to find the cube root of 1 728, we write it as 17 r - _ 21_ s_ _ 2'.33.

) It is therefore more convenient to rearrange the table of � 40 as shown below, on the left; the table on the right giving the key to the arrangement.
I I 2 I 1 3 I I 3644 Io I IO 15 &c.
5 (A =a)" =An(I?x)n =A n ?:C.An+ n 2 'I x.' n .Ix. An b... where x=a/A; thus the successive terms are obtained by successive multiplication.^ Thus we get successive multiplication; but it represents quite different operations according as it is due to repetition, in the sense of § 34, or to subdivision, and these operations will be exhibited by different diagrams.

^ In the case of multiplication we commence with the conception of the number " 5 " and the unit " boy "; and we then convert this unit into 4 apples, and thus obtain the result, 20 apples.

^ A number can often be expressed by a series of terms, such that by taking successive terms we obtain successively closer approximations.

.To apply the method to the calculation of N n, it is necessary that we should be able to express N in the form A+a or Aa, where a is small in comparison with A, A" is easy to calculate and a/A is convenient as a multiplier.^ Very small (less than 640 bytes + RAM for variables) so should work on any calculator without RAM worries.
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4 2. The reasoning adopted in � 41 (ii.) illustrates two general 0(0) I(1) 1 (0) 2(2) 2 (1) 3(3) 2 (0) 3(2) 4(4) 3(1) 4(3) 5(5) I 3(0) 4(2) 5(4) 6(6) 6 I 4(1) 5(3) 6(5) 7(7) 7 I 4(0) 5(2) 6(4) 7(6) 8(8> &c. +e+0 2 +. .. +Bnr -1)ur +1, Here we have introduced a number o (0) given by o (o) = 1 (16), which is consistent with the relations in (i.). .In this table any number is equal to the sum of the numbers which lie horizontally above it in the preceding Column, And The Difference Of Any Two Numbers In A Column Is Equal To The Sum Of The Numbers Horizontally Between Them In The Preceding Column.^ The remainder when a number is divided by 9 is equal to the remainder when the sum of its digits is divided by 9.

^ The example of the chessboard (Table I) shows us another important aspect of exponential growth; the increase in any doubling is approximately equal to the sum of all the preceding growth!
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^ The sum or difference of two percentages is expressed by the sum or difference of the numbers expressing the two percentages.

.The Coefficients In The Expansion Of (A A) N For Any Particular Value Of N Are Obtained By Reading Diagonally Upwards From Left To Right From The (N 1)Th Number In The First Column.^ It is desirable, wherever possible, to perform operations on numbers or numerical quantities from the left, rather than from the right.

^ Diagrams of Division.-Since we write from left to right or downwards, it may be convenient for division to interchange the rows or the columns of the multiplication-diagram.

^ In writing down any particular number, the successive powers of ten are written from right to left, e.g.

coefficients expressed in the form p [s1.
written
IN] ]
I[i]
2 [0] I [2]
2 [1] I[3]
3[o] 2 [2] I[4]
3[1] 2 [3] I
4[03
4[1] 3[3] 2[5]
5[0] 4[2] &c. 3[ 4 ]
The table in (ii.) may be
1[61
I[7]
2[6] I[8]
(iii.) .The table might be regarded as constructed by successive applications of (9) and (4); the initial data being (r6) and (Io).^ Let us examine the situation in regard to production of domestic crude oil in the U.S. Table IV gives the relevant data.
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^ It Is Usually Convenient To Make Out A Preliminary Table Of Multiples Up To Io Times; The Table Being Checked At 5 Times (§ Ioo) And At 10 Times.

^ We first construct the multiple-table C, and then subtract successively zoo times, 30 times and I times; these numbers being the partial quotients.

.Alternatively, we might consider that we start with the first diagonal row (downwards from the left) and construct the remaining diagonal rows by successive applications of (15).^ Diagrams of Division.-Since we write from left to right or downwards, it may be convenient for division to interchange the rows or the columns of the multiplication-diagram.

^ There is another method (Sarrus's formula), in which the first two columns are written to the right, and diagonal lines of 3 are multiplied, + when from lower left to upper right, and - when from lower right to upper left.
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^ The first object can be any of n, the second any of the n - 1 remaining, and so on, so the number of distinct arrangments in a row is n(n - 1)(n - 2)...2 1 = n!.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

.Constructed in this way, the successive diagonal rows, commencing with the first, give the figurate numbers of the first, second, third,.^ The first polynomial will then be divided by the second, and the quotient will be divided by the third, and so on.
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^ The names of numbers give an idea of the way in which the idea of number has developed.

^ As a reward for the invention the mathematician asked for the amount of wheat that would be determined by the following process: He asked the king to place 1 grain of wheat on the first square of the chess board, double this and put 2 grains on the second square, and continue this way, putting on each square twice the number of grains that were on the preceding square.
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.. order. .The (r+1)th figurate number of the nth order, i.e. the (r+ i) th number in the nth diagonal row, is n(n-}-1).^ Instead of regarding the 153 in 27.153 as meaning o h, we may regard the different figures in the expression as denoting numbers in the successive orders of submultiples of i on a denary scale.

.. (n+--r-1)lr!=n[r]lr!; this may, by analogy with the notation of �41, be denoted by n [ r 7. We then have (n+ 1)[r] _ (r + I)[n] = (n + r)!l(n ! r!) = (n + r)(r) = (n+r)(n) (17)� (iv.) .By means of (17) the relations between the binomial coefficients in the form p (4) may be replaced by others with the The most important relations are n[r] = n[r-i]+(n - I)(r) (r8); O[r] = 0 (19); n[r]-(n-s)[r] =n[r-i]+(n- I) [r-1]+...+ (n-s+I)[r-1] (20); n[r] =n[r-1]+(n-I)[r-1]+...+I[r-1] (21).^ If a/b = c/d, then certain other relations between quantities exist, which were given Latin names that are now rarely used.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ These numbers may represent the coefficients in a system of linear equations, a transformation of coordinates, or a quadratic form, with many applications in physics.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ Simplest, Fastest, Compact BINOMIAL EXPANDER Expands every Binomial in the form: (AX+B)^N I am aware that other Binomial Theorem expanders by other authors are present in the archive, but upon comparing them, this version proves to be the fastest, most compact and simple-to-use expander thus far.
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(v.) It should be mentioned that the notation of the binomial 'coefficients, and of the continued products such as n(n -1). .. (n-r+ r), is not settled. .Some writers, for instance, use the symbol n, in place, in some cases, of n (r) , and, in other cases, of n (r).^ This is the case, for instance, in the Celtic languages ; and the Breton or Gaulish names have affected the Latin system, so that the French names for some numbers are on the vigesimal system.

^ Algebra II Formulas I am currently taking Algebra II, and this program contains the quadratic and Compound Int and some other useful formulas .
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

^ Some compiler writers view restrictions which prohibit converting ( x + y ) + z to x + ( y + z ) as irrelevant, of interest only to programmers who use unportable tricks.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

It is convenient to retain x, to denote x r /r!, so that we have the consistent notation xr =x r /r!, n (r) =n(r)/r!, n[r] =n[r]/r!.
.The binomial theorem for positive integral index may then be written ( x + y) n = -iyi +.^ The binomial theorem can be proved by Mathematical Induction for integral values of n.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ The binomial theorem with integral exponents was usually the final topic in a school algebra course.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

� �
x0 yn � This must not be confused with the use of suffixes to denote particular terms of a series or a progression (as in � 41 (viii.) and (ix.)).
44. Permutations and Combinations. - The discussion, in � of the number of terms of a particular kind in a particular product, forms part of the theory of combinatorial analysis, which deals with the grouping and arrangement of individuals taken from a defined stock. .The following are some particular cases; the proof usually follows the lines already indicated.^ Thus the uncompleted diagram for partition is F or G, while for measuring it is usually H; the vacant compartment being for the unit in F or G, and for the number in H. In some cases it may be convenient in measuring to show both the units, as in K. .

^ The possibility of replacing them by a standard form, which could be utilized for performing arithmetical operations, is worthy of consideration; some of the difficulties in the way of standardization have already been indicated (§ 14).

.Certain of the individuals may be distinguishable from the remainder of the stock, but not from each other; these may be called a type. (i.^ These are still called in text-books the " four simple rules "; but this name ignores certain essential differences.

^ In that case, it may be true for certain roots of the equation, or may never be true, in which case the equation is called inconsistent .
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^ The introduction of these other symbols produces a compound scale, which may be called a quinarybinary, or, less correctly, a quinary-denary scale.

) A permutation is a linear arrangement, read in a definite direction of the line. .The number (n P r) of permutations of r individuals out of a stock of n, all being distinguishable, is n`r).^ So all you have to do is enter the the numbers for value A, B, and C. Then, it will compute the answers for you following the rules, such as dividing out all common factors.
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^ You enter the base and number being log'd and it spits out the log!
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

.In particular, the number of permutations of the whole stock is n!. If a of the stock are of one type, b of another, c of another,.^ One number is less or greater than another, according as the symbol (or ordinal) of the former comes earlier or later than that of the latter in the number-series.

^ The systems adopted for numeration and for notation do not always agree with one another; nor do they always correspond with the idea which the numbers subjectively present.

^ It finds two numbers that multiply together to make one number and that add together to make another.
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.. the number of distinguishable permutations of the whole stock is n!= (a?b!cI (ii.) .A combination is a group of individuals without regard to arrangement.^ If we sort objects into groups of ten, and find that there are five groups of ten with three over, we regard the five and the three as names for the actual sets of groups or of individuals.

The number (n C r) of combinations of r indi viduals out of a stock of n has in effect been proved in � 41 (i.) to be .n (r) . This property enables us to establish, by simple reasoning, certain relations between binomial coefficients.^ If a/b = c/d, then certain other relations between quantities exist, which were given Latin names that are now rarely used.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ But accurate operations are useful even in the face of inexact data, because they enable us to establish exact relationships like those discussed in Theorems 6 and 7.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

Thus (4) of � 41 (ii.) follows from the fact that, if .A is any one of the n individuals, the nCr groups of r consist of n _ 1 C r _ 1 which contain A and n_1C, which do not contain A. Similarly, considering the various ways in which a group of r may be obtained from two stocks, one containing m and the other containing n, we find that m+nCr mCr'nC0 +mCO'nCTp which gives (m -1--n) (,) =m(r).n(o)+m(r-i).n(1)+...^ Correspondence of Numerical Quantities.-Again, in § 89, we have considered cases of multiple-tables of numerical quantities, where each quantity in one series is equivalent to the corresponding quantity in the other series.

^ Also, because the scaled values of m and q satisfy m /2 < q < 2 m , the corresponding value of n must have one of two forms depending on which of m or q is larger: if q   <  m , then evidently 1 < n < 2, and since n is a sum of two powers of two, n = 1 + 2 - k for some k ; similarly, if q > m , then 1/2 < n < 1, so n = 1/2 + 2 -( k + 1) .
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^ A set of written symbols is sometimes read in more than one way, while on the other hand two different sets of symbols (at any rate if denoting numerical quantities) may be read in the same way.

+m(o).n(r) (22).
.This may also be written } n)(r)=m(r).n(a)-I-ro) m('-').n(')+...---r(r).m(o). n(r) (23). If r is greater than m or n (though of course not greater than m+n), some of the terms in (22) and (23) will be zero.^ This discount, of course, is not equal to the interest which the present value would produce at that rate of interest, but is rather greater, so that the present value as calculated in this way is less than the theoretical present value.

^ The element a 23 in the index notation would be expressed as a 23 j k in dyadics, and all terms had to be explicitly written.
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^ With a single guard digit, the relative error of the result may be greater than , as in 110 - 8.59.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

(iii.) .If there are n types, the number of individuals in each type being unlimited (or at any rate not less than r), the number (n H r) of distinguishable groups of r individuals out of the total stock is n[r].^ There is a small snag when = 2 and a hidden bit is being used, since a number with an exponent of e min will always have a significand greater than or equal to 1.0 because of the implicit leading bit.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ There is more than one way to split a number.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ There appears therefore to be a tendency to use some larger number than ten as a basis for grouping into new units or for subdivision into parts.

.This is sometimes called the number of homogeneous products of r dimensions formed out of n letters; i.e. the number of products such as x r , xr-3y3, x r-2 z 2,.^ If for some value of the number λ we can find a vector x such that Ax = λx, then x is called an eigenvector of A belonging to the eigenvalue λ.
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^ The system which is now almost universally in use amongst civilized nations for representing cardinal numbers is the Hindu, sometimes incorrectly called the Arabic, system.

^ If none of the partial products are out of range, the trap handler is never called and the computation incurs no extra cost.
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.. that can be formed with positive integral indices out of n letters x, y, z,.. ., the sum of the indices in each product being r. (iv.) Other developments of the theory deal with distributions, partitions, &c. (see Combinatorial Analysis).
(v.) The theory ofprobability also comes under this head. .Suppose that there are a number of arrangements of r terms or elements, the first of which a is always either A or not-A, the second b is B or not-B, the third c is C or not-C, and so on.^ First of all, there are algebraic identities that are valid for floating-point numbers.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ The first polynomial will then be divided by the second, and the quotient will be divided by the third, and so on.
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^ The second approach represents higher precision floating-point numbers as an array of ordinary floating-point numbers, where adding the elements of the array in infinite precision recovers the high precision floating-point number.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

.If, out of every N cases, where N may be a very large number, a is A in pN cases and not-A in (I - p) N cases, where p is a fraction such that pN is an integer, then p is the probability or frequency of occurrence of A. We may consider that we are dealing always with a single arrangement abc..^ This is not very helpful if the interval turns out to be large (as it often does), since the correct answer could be anywhere in that interval.
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^ In a large number of cases, however, the direction is steadily upwards from t to 12, then changing.

^ A matrix may be represented by a single capital letter, as A. The elements may be thought of as real numbers, though more general elements are possible.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

.,
and that the number of times that a is made A bears to the number of times that a is made not-A the ratio of p to r--p; or we may consider that there are N individuals, for pN of which the attribute a is A, while for (1-p)N it is not-A. If, in this latter case, the proportion of cases in which b is B to cases in which b is not-B is the same for the group of pN individuals in which a is A as for the group of (I-p)N in which a is not-A, then the frequencies of A and of B are said to be independent; if this is not the case they are said to be correlated. The possibilities of a, instead of being A and not-A, may be A 1, A2, � � ., each of these having its own frequency; and similarly for b, c,.. . If the frequency of each A is independent of the frequency of each B, then the attributes a and b are independent; otherwise they are correlated.
.45. Application of Binomial Theorem to Rational Integral Functions.^ The binomial theorem can be proved by Mathematical Induction for integral values of n.
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^ Newton's Binomial Theorem, a remarkable result with many useful applications, dates from 1665 or 1666.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ The binomial theorem with integral exponents was usually the final topic in a school algebra course.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

- An
expression of the form cox --c i x'+ ... --cn, where co, c 1,. .. do not involve .x, and the indices of the powers of x are all positive integers, is called a rational integral function of x of degree n. If we represent this expression by f (x), the expression obtained by changing x into x-+-h is f(x+h); and each term of this may be expanded by the binomial theorem.^ We are quite familiar with the expression in terms of indices.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ The numbers represented by a, b, c, x and m are all supposed to be positive.

^ Sometimes a term must be added and subtracted (which does not change the value of the expression) in order to do this, as in completing the square.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

Thus we have 2 f(x+)h =cox � +ncox n - i i i+n(n - 1) coxn- ? +... 2 +cix�-1+(n- I) c i x n I I +(n- I) (n2) c i x n - 3 - I+...
2 n-2 -}- (n - 2) c2xn-31 I ? + (n - 2) (n - 3) c 2 x n - 4 +... -? &c.
C2xn-2+...
+ j ncox�-' + (n - I) clxn-2 + (n - 2) c 2 x n - 3 +... I (l(}) h2 + j n(n- I)coxn-2+ (n- I) (n cixn-3+... 21 }- &c. } .It will be seen that the expression in curled brackets in each line after the first is obtained from the corresponding expression in the preceding line by a definite process; viz.^ The theory of the process is shown fully in F. Treating x as the unknown quotient corresponding to the original dividend, we obtain successive dividends corresponding to quotients x-200, x- 230 and x-231.

^ In the first place, an operation then corresponds more closely, at an elementary stage, with the concrete process which it represents.

x r is replaced by 1, except for r =o, when is replaced by o. .The expressions obtained in this way are called the first, second,.^ P are called the first power, second power,.

^ When a first approximation has been obtained in this way, further approximations can be obtained in various ways.

^ If n < 0, then a more accurate way to compute x n is not to call PositivePower(1/x, -n) but rather 1/PositivePower(x, -n) , because the first expression multiplies n quantities each of which have a rounding error from the division (i.e., 1/ x ).
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

.. derived functions
of f(x). If we denote these by f i (x), f 2 (x),.. ., so that fs(x) is obtained from f s _ 1 (x) by the above process, we have f(x +h) =.f(fi(x). h+f2(x)h212!+... +fr(x)hr/r!+... This is a particular case of Taylor's theorem (see Infinitesimal Calculus).
.46. Relation of Binomial Coefficients to Summation of Series. (i.) The sum of the first n terms of an ordinary arithmetical progression (a+b), (a+2b), ...^ Binomial Expander This program will list the resulting terms of any binomial expansion of a binomial in [A(F)^X + B(S)^Y]^E format, where "F" and "S" are the variables, "A" and "B" are the variables' coefficients, "X" and "Y" are the variables' exponents, and "E" is the exponent of the binomial.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

^ Sum of Arithmetic Series .
  • Mathematics: Algebra 2 - Educator.com 10 February 2010 11:011 UTC www.educator.com [Source type: Reference]

^ Findterm This program uses the Generla Formula for a Specific Term of an Expansion formula to find the coefficients and powers for a binomial raised to a power.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

(a+nb
) is (� 28 (i.)) Zn t(a+b)+ (a+nb)} = na+zn(n+1)b = nr1 1 .a+nr2 1. b. Comparing this with the table in �43 (iv.), and with formula .(21), we see that the series expressing the sum may be regarded as consisting of two, viz.^ The coefficients may be written in the symmetrical Pascal's triangle where each coefficient is the sum of the two on either side in the row above.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ Theorem 6 gives a way to express the product of two working precision numbers exactly as a sum.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

a+a+.. . and b+2b+3b+.. .; for the first series we multiply the table (i.e. each number in the table) by a, and for the second series we multiply it by b, and the terms and their successive sums are given for the first series by the first and the second columns, and for the second series by the second and the third columns.
(ii.) .In the same way, if we multiply the table by c, the sum of the first n numbers in any column is equal to the nth number in the next following column.^ One can prove that the sum, difference, product, or quotient of two p -bit numbers, or the square root of a p -bit number, rounded first to q bits and then to p bits gives the same value as if the result were rounded just once to p bits provided q 2 p + 2.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ The first equality of (31) shows that the computed value of is the same as if an exact summation was performed on perturbed values of x j .
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ A partition with an equal number of rows and columns has a determinant.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

.Thus we get a formula for the sum of n terms of a series such as 2.4.6+4.6.8+..., or 6.8.10.12+8.10.12.14+...^ Thus there are (2 10 - 10 3 )2 14 = 393,216 different binary numbers in that interval.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ The Taylor series of µ '( x ) is also alternating, and if x has decreasing terms, so - µ '( x ) - + 2 x /3, or - µ'( x ) 0, thus .
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ The second term in this expression is just the sum of the series d[1 + 2 + ...
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

(iii.) Suppose we have such a series as 2.5+5.8+8.11+ ... .This cannot be summed directly by the above method.^ We will give below a method of summing the series directly.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

But the nth term is (3n-I)(3n+2)=18nr 21 -6nr 11 - 2. The sum of n terms is therefore (� 43 (iv.)) 18nr 31 -6n [21 -2nr 11 =3n3+6n2+n.
(iv.) Generally, let N be any rational integral function of n of degree r. Then, since nr rl is also a rational integral function of n of degree r, we can find a coefficient c r , not containing n, and such as to make N-c r nr ri contain no power of n higher than n r - 1. Proceeding in this way, we can express N in the form cr.nrr l + cr_I.nrr_1 1 + �. ., where c r, c r _ l , c r _ 2, ... do not contain n; and thence we can obtain the sum of the numbers found by putting n= 1, 2, 3,. .. .n successively in N. These numbers constitute an arithmetical progression of the rth order. (v.^ Instead of regarding the 153 in 27.153 as meaning o h, we may regard the different figures in the expression as denoting numbers in the successive orders of submultiples of i on a denary scale.

^ We first construct the multiple-table C, and then subtract successively zoo times, 30 times and I times; these numbers being the partial quotients.

) A particular case is that of the sum I'd2r +3 r + ... +nr, where r is a positive integer. .It can be shown by the above reasoning that this can be expressed as a series of terms containing descending powers of n, the first term being nr+1/(r+I).^ A number can often be expressed by a series of terms, such that by taking successive terms we obtain successively closer approximations.

^ In order to apply arithmetical processes to a quantity expressed in two or more denominations, we must first express it in terms of a single denomination by means of a varying scale of notation.

^ If the denominator of the fraction, when it is in its lowest terms, contains any other prime factors than 2 and 5, it cannot be expressed exactly as a decimal; but after a certain point a definite series of figures will constantly recur.

The most important cases are I +2 +3 +...+n = 2n(n +I), 12+22+32+...+n2 = sn(n+I) (2n+1), I 3 +2 3 +3 3 +... +n 3 = 4n 2 (n+ I) 2 = (I +2 +... +n)2.
.The general formula (which is established by more advanced methods) is 4. Or+Ir+2r+...^ Although formula (7) is much more accurate than (6) for this example, it would be nice to know how well (7) performs in general.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

+(n - I)r+2nr I 'n' r+1 +B1(r+I)(2)n r-1 -B2(r+I)(4)n r-3 + � � � j -r+ I where B 1, B 2, ... are certain numbers known as .Bernoulli's numbers, and the terms within the bracket, after the first, have signs alternately + and -.^ Note that we must prefix each term with a sign, that is ±1 alternating as we count from the 0,0 element at the upper left-hand corner.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ In either of the above cases, and generally in any case where a number is known to be within a certain limit on each side of the stated value, the limit of error is expressed by the sign =.

^ The first step is to find a value near the root, or better, to bracket the root by two values at which the signs of the function are opposite.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

.The values of the first ten of Bernoulli's numbers are B1= t, B2= 1, B3 =412, B4 =30, B5 =6 = 6 9 1 B7 = l, B =3 =4 4 fl, IV. Negative Numbers and Formal Algebra. 47. Negative quantities will have arisen in various ways, e.g. (i.^ First of all, there are algebraic identities that are valid for floating-point numbers.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ The ratio of two quantities is, in algebra, the quotient of their numerical values, written a/b or a:b, where a is called the antecedent , and b the consequent .
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ In order to apply arithmetical processes to a quantity expressed in two or more denominations, we must first express it in terms of a single denomination by means of a varying scale of notation.

) .The logical result of the commutative law, applied to a succession of additions and subtractions, is to produce a negative quantity-3s.^ Commutative Law for Additions and Subtractions, that additions and subtractions may be performed in any order; e.g.

^ There is no difference in principle between addition (or subtraction) of numbers and addition (or subtraction) of numerical quantities.

^ Comparison, Addition and Subtraction of Fractions.-The quantities 4 of A and 7 of A are expressed in terms of different units.

such that -3s.+3s. = o(� 28 (vi.)).
(ii.) Simple equations, especially equations in which the unknown quantity is an interval of time, can often only be satisfied by a negative solution (� 33).
(iii.) In solving a quadratic equation by the method of � 38 (viii.) we may be led to a result which is apparently absurd. .If, for instance, we inquire as to the time taken to reach a given height by a body thrown upwards with a given velocity, we find that the time increases as the height decreases.^ The determination of a submultiple is therefore equivalent to completion of the diagram E or E' of § 35 by entry of the unit, when the number of times it is taken, and the product, are given.

^ This File Contains: Increase & Decrease Of Numbers; The Percent Of Increase Or Decrease; Proportion (Shadow & Height); Rates; Pythag; Precent Equations; Proportions; And Lots More!!
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

.Graphical representation shows that there are two solutions, and that an equation X2= 9a2 may be taken to be satisfied not only by X=3a but also by X= -3a.^ Or we might say that, since multiplication is a form of addition, and division a form of subtraction, there are really only two fundamental processes, viz.

^ The proof also shows that extending our reasoning to include the possibility of double-rounding can be challenging even for a program with only two floating-point operations.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ These programs do not merely provide you with a solution, but also show each and every step required to get there.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

.48. The occurrence of negative quantities does not, however, involve the conception of negative numbers.^ S. The simplest case, in which the quantity can be expressed as an integral number of the largest units B involved, has already been considered (§§ 37, 42).

^ To express the quantity in terms of £, it ought (20) (12) to be written £254 13 v 6; this would mean £254 1202 or £(254+20 20 + 6 12), and therefore would involve a fractional number.

^ The positive quantity or number obtained from a negative quantity or number by omitting the " - " is called its numerical value.

In (iii.) of � .47, for instance, " -3a" does not mean that a is to be taken (-3.) times, but that a is to be taken 3 times, and the result treated as subtractive; i.e.-3a means -(3a), not (-3)a (cf.^ Multiplication occurs when a certain number or numerical quantity is treated as a unit (§ II), and is taken a certain number of times.

^ If there is an integral number to be taken as well as a percentage, this number is written in front of the point; thus 23.76 X A means 23 times A, with 76% of A. We might therefore denote 76% by 0.76.

^ The result of the multiplication is called the product of the unit by the number of times it is taken.

� 27 (i.)). .In the graphic method of representation the sign - may be taken as denoting a reversal of direction, so that, if + 3 represents a length of 3 units measured in one direction,-3 represents a length of 3 units measured in the other direction.^ Examples of other representations are floating slash and signed logarithm [Matula and Kornerup 1985; Swartzlander and Alexopoulos 1975].
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ The scalar product is represented as a · b = ab cos θ, where a and b are the lengths of the vectors and θ is the angle between their positive directions.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ The components of a unit vector are the direction cosines of the direction represented, and the sum of their squares, the length of the unit vector, is unity.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

.But even so there are two distinct operations concerned in the-3, viz.^ The proof also shows that extending our reasoning to include the possibility of double-rounding can be challenging even for a program with only two floating-point operations.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ In these systems, there are generally two operations, analogous to addition and multiplication, that obey certain rules.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

the multiplication by 3 and the reversal of direction. .The graphic method, therefore, does not give any direct assistance towards the conception of negative numbers as operators, though it is useful for interpreting negative quantities as results.^ It is desirable, wherever possible, to perform operations on numbers or numerical quantities from the left, rather than from the right.

^ It is also useful to look out for pairs of numbers or quantities which make I of the next denomination, e.g.

^ The rounding mode affects overflow, because when round toward 0 or round toward - is in effect, an overflow of positive magnitude causes the default result to be the largest representable number, not + .
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

.49. In algebraical transformations, however, such as (x-a)2 = x 2 - 2ax+a 2 , the arithmetical rule of signs enables us to combine the sign-with a number and to treat the result as a whole, subject to its own laws of operation.^ It is doubtful, however, whether such a rule, giving a test which is necessarily incomplete, is of much educational value.

^ If the numbers which are the subject of operations are not all exact, the accuracy of the result requires special investigation in each case.

^ Sir O. Lodge, Easy Mathematics, chiefly Arithmetic (1905), treats the subject broadly in its practical aspects.

.We see first that any operation with 4a-3b can be regarded as an operation with (+)4a+(-)3b, subject to the conditions (I) that the signs (+) and (-) obey the laws (+)=(+),(+)(-)=(-)(+)= (-), (-) (-)=(+), and (2) that, when processes of multiplication are completed, a quantity is to be added or subtracted according as it has the sign (+) or (-) prefixed.^ Or we might say that, since multiplication is a form of addition, and division a form of subtraction, there are really only two fundamental processes, viz.

^ This law includes the rule of signs, that a - (b - c) = a - b+c; and it states that, subject to this, successive operations of addition or subtraction may be grouped in sets in any way; e.g.a - b+c+d+e - f =a - (b - c)+(d+e - f).

^ To find the square root of N, we first find some number a whose square is less than N, and subtract a 2 from N. If the complete square root is a+b, the remainder after subtracting a 2 is ( 2a+b)b.

.We are then able to combine any number with the + or the - sign inside the bracket, and to deal with this constructed symbol according to special laws; i.e. we can replace pr or -pr by (+p)r or (- p)r, subject to the conditions that (+p) (+q) = (-p) (- q) = (+pq), (+p) (-q) = (- p) (+ q) = (-pq), and that + (-s) means that s is to be subtracted.^ Similarly we cannot subtract 8 from 15, if 15 means 1 ten + 5 ones; we must either write 15-815-8=(10+5)-8= (I o - 8)+5 = 2+5 = 7, or else resolve the 15 into an inexpressible number of ones, and then subtract 8 of them, leaving 7.

^ There are no single symbols for two, three, &c.; but numbers are represented by combinations of symbols for one, five, ten, fifty, one hundred, five hundred, &c., the numbers which have single symbols, viz.

^ One number is less or greater than another, according as the symbol (or ordinal) of the former comes earlier or later than that of the latter in the number-series.

.These constructed symbols may be called positive and negative coefficients; or a symbol such as (- p) may be called a negative number, in the same way that we call 3 a fractional number.^ Negative Fractional Numbers 6.5 68.

^ In the same way, by dividing by powers of so we may get negative indices.

^ The symbols denoting a number are called its digits.

.This increases the extent of the numbers with which we have to deal; but it enables us to reduce the number of formulae.^ The answer is that it does matter, because accurate basic operations enable us to prove that formulas are "correct" in the sense they have a small relative error.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

.The binomial theorem may, for instance, be stated for (x+a)n alone; the formula for (x-a)" being obtained by writing it as {x+(-)a} n or Ix+(- a) } n, so that (x-a) n =x"- 1)xn-laF...+(-)rn(r)xn-rar+..., where + (-) r means - or + according as r is odd or even.^ A general 3 x 3 determinant may be expressed as D = ε ijk a 1i a 2j a 3k , where ε ijk = +1 if i,j,k is an even permutation of 1,2,3, -1 if an odd permutation, and 0 otherwise.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

.The result of the extension is that the number or quantity represented by any symbol, such as P, may be either positive or negative.^ The numbers represented by a, b, c, x and m are all supposed to be positive.

^ The pair of compartments on either side may, as here, contain numerical quantities, or may contain numbers.

^ What is called modern algebra works with symbols that may obey different rules of composition or operations than the familiar ones of real numbers that we have just presented.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

.The numerical value is then represented by (P (; thus " (x (< i " means that xis between -1 and +1.1 50. The use of negative coefficients leads to a difference between arithmetical division and algebraical division (by a multinomial), in that the latter may give rise to a quotient containing subtractive terms.^ Collecting terms in an expression means combining terms differing only in the numerical coefficient.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ There is no difference in principle between addition (or subtraction) of numbers and addition (or subtraction) of numerical quantities.

^ In polar plots, the radius r may be negative, and the angle may be multiple-valued, multiples of 2π giving the same information.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

The most important case is division by a binomial, as illustrated by the following examples: - 2.10+1) 6.100+5.10+ 1(3.10+I 2.10+I) 6.100+I.10 - I (3.10 - I 6.100+3.10 6.100+3.10 2.10+ I - 2.10 - I 2.10 +I - 2.10 - I In (1) the division is both arithmetical and algebraical, while in (2) it is algebraical, the quotient for arithmetical division being 2.10+9.
.It may be necessary to introduce terms with zero coefficients.^ To prevent confusion the zero or " nought " is introduced, so that the successive figures, beginning from the right, may represent ones, tens, hundreds,.

^ Quadric surfaces in three dimensions may be classified on the basis of the principal values of the coefficient matrix (linear terms are supposed to have been eliminated).
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

.Thus, to divide i by i +x algebraically, we may write it in the form I+o.x+o.x 2 +o.x 3 +o.x 4, and we then obtain I I +0.x+0.x2+0.x3 '+0.x4 = I' x+x2 - x 3 + x4 I+x I+x' where the successive terms of the quotient are obtained by a process which is purely formal.^ If we take several different units, and write down their successive multiples in parallel columns, preceded by the numberseries, we obtain a multiple-table such as the following: It is to be considered that each column may extend downwards indefinitely.

^ If we write 74 in the form 47 we may say that the value of a fraction is not altered by multiplying or dividing the numerator and denominator by any number.

^ The theory of the process is shown fully in F. Treating x as the unknown quotient corresponding to the original dividend, we obtain successive dividends corresponding to quotients x-200, x- 230 and x-231.

.51. If we divide the sum of x 2 and a 2 by the sum of x and a, we get a quotient x - a and remainder 2a 2, or a quotient a - x and remainder 2x 2, according to the order in which we work.^ The remainder when a number is divided by 9 is equal to the remainder when the sum of its digits is divided by 9.

^ The quotient will be the desired rational expression, and the remainder R(x) will give us a function R(x)/Q(x) that can be expressed as the sum of partial fractions.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ We therefore guess b by dividing the remainder by 2a, and form the product ( 2a+b) b.

Algebraical division therefore has no definite meaning unless dividend and divisor are rational integral functions of some expression such as x which we regard as the root of the notation (� 28 (iv.)), and are arranged in descending or ascending powers of x. If P and M are rational integral functions of x, arranged in descending powers of x, the division of P by M is complete when we obtain a remainder R whose degree (� 45) is less than that of M. If R= o, then M is said to be a factor of P.
.The highest common factor (or common factor of highest degree) of two rational integral functions of x is therefore found in the same way as the G.C.M. in arithmetic; numerical coefficients of the factor as a whole being ignored (cf.^ FACTOR1 This program is the same as FACTOR ( factor.zip ) in this directory, except that it can also find the rational roots in a polynomial of one variable with rational coefficients.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

^ The principle of subtraction from a higher number, which appeared in notation, also appeared in numeration, but not for exactly the same numbers or in exactly the same way; thus XVIII was two-from-twenty, and the next number was onefrom-twenty, but it was written XIX, not IXX. .

^ A set of written symbols is sometimes read in more than one way, while on the other hand two different sets of symbols (at any rate if denoting numerical quantities) may be read in the same way.

� 36 (iv.)).
52. Relation between Roots and Factors. - (i.) If we divide the multinomial P=p0xn+plxn-1+... -Fp n by x-a, according to algebraical division, the remainder is R= po an +pla? -1+...+pn. This is the remainder-theorem; it may be proved by induction.
(ii.) If x = a satisfies the equation P = o, then poan+plan-1+. ... + p n = o; and therefore the remainder when P is divided by x-a is o, i.e. x-a is a factor of P.
(iii.) Conversely, if x-a is a factor of P, then poand plan - 1 +... + p n = o; i.e. x= a satisfies the equation P = o. (iv.) .Thus the problems of determining the roots of an equation P = o and of finding the factors of P, when P is a rational integral function of x, are the same.^ The program will also FACTORIZE all cubic equations with three rational roots.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

^ It can also factorize any cubic equation if it has three real and rational roots.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

^ FACTOR1 This program is the same as FACTOR ( factor.zip ) in this directory, except that it can also find the rational roots in a polynomial of one variable with rational coefficients.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

(v.) In particular, the equation P = o, where P has the value in (i.), cannot have more than n different roots.
.The consideration of cases where two roots are equal belongs to the theory of equations (see Equation).^ Secant Method This program uses the Secant Method to find a root of an equation between two initial guesses.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

^ Function This program will evaluate a function and will tell you if your selected value will make two equations equal to each other.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

(vi.) .It follows that, if two multinomials of the nth degree in x have equal values for more than n values of x, the corresponding coefficients are equal, so that the multinomials are equal for all values of x.^ Using the values of a , b , and c above gives a computed area of 2.35, which is 1 ulp in error and much more accurate than the first formula.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ The IEEE standard specifies the following special values (see TABLE D-2 ): ± 0, denormalized numbers, ± and NaNs (there is more than one NaN, as explained in the next section).
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ The combination of these two roundings can yield a value that is different than what would have been obtained by rounding the first result correctly to double precision.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

53. Negative Indices and Logarithms. - (i.
) Applying the general principles of �� 47-49 to indices, we find that we can interpret Xm as being such that X m .Xm =X 0 =I; i.e. 'X-m = I/Xm. .' In the same way we interpret X-P!4 as meaning I/XP14. (ii.^ II in the same way that the number III includes the number II in fig.

^ We can obtain negative fractional numbers in the same way that we obtain negative integral numbers; thus - 4 or - 7A means that 4 or 7A is taken negatively.

) This leads to negative logarithms (see Logarithm).
54. Laws of Algebraic Form. - (i.) The results of the addition, subtraction and multiplication of multinomials (including monomials as a particular case) are subject to certain laws which correspond with the laws of arithmetic (� 26 (i.)) but differ from them in relating, not to arithmetical value, but to algebraic form. The commutative law in arithmetic, for instance, states that adb and b+a, or ab and ba, are equal. .The corresponding law of form regards a+b and bda, or ab and ba, as being not only equal but identical (cf.^ Hence we can only regard £153 as being equal to 3060s.

^ We need therefore consider numerical quantities only, our results being applicable to numbers by regarding the digits as representing multiples of units in different denominations.

^ The result of taking 13 sixths of A is then seen to be the same as the result of taking twice A and one-sixth of A, so that we may regard as being equal to 21.

� 37 (ii.)), and then says that A+B and 134-A, or AB and BA, are identical, where A and B are any multinomials. .Thus a(b+c) and (b+c)a give the same result, though it may be written in various ways, such as abdac, ca+ab, &c.^ However, computing with a single guard digit will not always give the same answer as computing the exact result and then rounding.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ Thus, when a program is moved from one machine to another, the results of the basic operations will be the same in every bit if both machines support the IEEE standard.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ The result of 3-7 is the same as that of o-4; and we may write it " - 4," and call it a negative number, if by this we mean something possessing the property that - 4+4 = o.

.In the same way the associative law is that A(BC) and (AB)C give the same formal result.^ However, computing with a single guard digit will not always give the same answer as computing the exact result and then rounding.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ You can choose numerical input, or graphical input, and either way it will give you the numeric result and the graphic result.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

^ That this gives the same result as adding 4 to 5 may be seen by reckoning the series backwards.

These laws can be established either by tracing the individual terms in a sum or a product or by means of the general theorem in � 52 (vi.).
(ii.) One result of these laws is that, when we have obtained any formula involving a letter a, we can replace a by a multinomial. For instance, having found that (x+a)2=x2+2axda2, we can deduce that (x+b+c) 2 = }xd(b+c)}2=x2+2(b+c)x+ (b+c)2. (iii.) .Another result is that we can equate coefficients of like powers of x in two multinomials obtained from the same expression by different methods of expansion.^ However, these two expressions do not have the same semantics on a binary machine, because 0.1 cannot be represented exactly in binary.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ One can prove that the sum, difference, product, or quotient of two p -bit numbers, or the square root of a p -bit number, rounded first to q bits and then to p bits gives the same value as if the result were rounded just once to p bits provided q 2 p + 2.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ Thus, when a program is moved from one machine to another, the results of the basic operations will be the same in every bit if both machines support the IEEE standard.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

For instance, by equating coefficients of or in the expansions of (I +x) m+n and of (I dx) m . (I dx) n we obtain (22) of � 44 (ii.).
(iv.) .On the other hand, the method of equating coefficients often applies without the assumption of these laws.^ System So A program to solve Systems of equation in an easy prompt coefficient method.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

In � 41 (ii.), for instance, the coefficient of A n - r a r in the expansion of (Ada) (A+a) n - 1 has been called (n ,.); and it has then been shown that (ii) = (n _ I) d (i). .This does ot involve any assumption of the identity of results obtained in different ways; for the expansions of (A+a) 2, (A+a) 3, ...^ In fact, the authors of the standard intended to allow different implementations to obtain different results.
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^ One way of obtaining this 50% behavior to require that the rounded result have its least significant digit be even.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ One way to restore the identity 1/(1/ x ) = x is to only have one kind of infinity, however that would result in the disastrous consequence of losing the sign of an overflowed quantity.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

are there supposed to be obtained in one way only, viz. by successive multiplications by Ada.
55� Algebraical Division. - .In order to extend these laws so as to include division, we need a definition of algebraical division.^ This law includes a rule, similar to the rule of signs, to the effect that a= (b= c) = a= b x c; and it states that, subject to this, successive operations of multiplication or division may be grouped in sets in any way; e.g.

^ Commutative Law for Multiplications and Divisions, that multiplications and divisions may be performed in any order: e.g.

The divisions in �� 50-52 have been supposed to be performed by a process similar to the process of arithmetical division, viz. by a series of subtractions. This latter process, however, is itself based on a definition of division in terms of multiplication (�� 15, 16). If, moreover, we examine the process of algebraical division as illustrated in � 50, we shall find that, just as arithmetical division is really the solution of an equation (� 14), and involves the tacit use of a symbol to denote an unknown quantity or number, so algebraical division by a multinomial really implies the use of undetermined coefficients (� 42). .When, for instance, we find that the quotient, when 6+5x+7x2+13x1+5x4 is divided by 2+35+5 2, is made up of three terms+3, - 2x, and +5x 2 , we are really obtaining successively the values of co, c 1 , and c 2 which satisfy the identity 6+ 5x+ 7x 2 + 13x 3 + 5x4 = (co+c i x+c 2 x 2) (2+3x+5 2); and we could equally obtain the result by expanding the right-hand side of this identity and equating coefficients in the first three terms, the coefficients in the remaining terms being then compared to see that there is no remainder.^ First of all, there are algebraic identities that are valid for floating-point numbers.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ If = 2, then there is no further error committed when dividing by 4.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ There are no single symbols for two, three, &c.; but numbers are represented by combinations of symbols for one, five, ten, fifty, one hundred, five hundred, &c., the numbers which have single symbols, viz.

.We therefore define algebraical division by means of algebraical multiplication, and say that, if P and M are multinomials, the statement " P/M = Q " means that Q is a multinomial such that MQ (or QM) and P are identical.^ Synthetic Substitution This program can help any student with the synthetic multiplication and division in the Algebra 2 curriculum.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

^ Or we might say that, since multiplication is a form of addition, and division a form of subtraction, there are really only two fundamental processes, viz.

^ To do this, consider what the ordinary processes of multiplication and division mean in reference to concrete objects.

In this sense, the laws mentioned in � 54 apply also to algebraical division.

56. Extensions of the Binomial Theorem

It has been mentioned in � 41 (ix.) that the binomial theorem can be used for obtaining an approximate value for a power of a number; the most important terms only being taken into account. .There are extensions of the binomial theorem, by means of which approximate calculations can be made of fractions, surds, and powers of fractions and of surds; the main difference being that the number of terms which can be taken into account is unlimited, so that, although we may approach nearer and nearer to the true value, we never attain it exactly.^ Thus there are (2 10 - 10 3 )2 14 = 393,216 different binary numbers in that interval.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ Since 3/7 is a repeating binary fraction, its computed value in double precision is different from its stored value in single precision.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ Algebra Formulas Displays the formulas for arithmetic operations, exponents/ radicals, factoring special polynomials, binomial theorem, quadratic formula, and inequalities/absolute value separated into categories.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

.The argument involves the theorem that, if 0 is a positive quantity less than I, 0 t can be made as small as we please by taking t large enough; this follows from the fact that tlog 0 can be made as large (numerically) as we please.^ In the numerical example given above, the computed value of (7) is 2.35, compared with a true value of 2.34216 for a relative error of 0.7 , which is much less than 11 .
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ Very small (less than 640 bytes + RAM for variables) so should work on any calculator without RAM worries.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

^ It is based on the following fact, which is proven in the section Theorem 14 and Theorem 8 .
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

(i.) By algebraical division, I _ I +0.X+0.x2�...+0.xr+l I+ x - 1+x = I- x-f-x2-...--(-)rxrd-(-)'+1 + x (24).
.If, therefore, we take r i /(i+x) as equal to I -xdx2-...+ (-) r x r , there is an error whose numerical magnitude is I xr+1/ (I +x) I; and, if I x <1, this can be made as small as we please.^ We may therefore consider that there is some number x whose cube is 2000, and we can find this number to any degree of accuracy that we please.

^ When a fraction cannot be expressed by an integral percentage, it can be so expressed approximately, by taking the nearest integer to the numerator of an equal fraction having ioo for its denominator.

.This is the foundation of the use of recurring decimals; thus we can replace = s s = 1 o o /(' - 1 + 0 -)1 by .363636(=36/102 +36/ 104 +3 6 / 106), with an error (in defect) of only 36/(10 6.99).^ Thus the 2 error bound for the Kahan summation formula (Theorem 8) is not as good as using double precision, even though it is much better than single precision.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ If x  = 3  ×  10 70 and y = 4 × 10 70 , then x 2 will overflow, and be replaced by 9.99 × 10 98 .
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ However, when using exactly rounded operations, this formula is only true for = 2, and not for = 10 as the example x  = .99998, y  = .99997 shows.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

(ii.) Repeated divisions of (24) by x+x, r being replaced by rd I before each division, will give (I +xy 2 = I -25+3x2-4x3-F...+(- )r(r (I)xr + (-) r+l x r+1 1(r+ I) (I +5)- 1 + (1 + 5)-21, (I-Fx)-3=I - (3x-6x2 - IOx3+...+(-)r� 2l(r+I)(r+2)xr +(-) r+l x r+1 12 (r+I) (r+2) (' +x)-1+(r+I)(I - Fx) - 2 +(I +x)-3},&c.
Comparison with the table of binomial coefficients in � 43 suggests that, if m is any positive integer, (I +x)-m =Sr+Rr (25), where Sr=I -m[1]x+m[2]x2...+(-)rm[r]xr (26), Rr_(_)r+1xr+11m[r] (1Fx) - 1+(m - I[r](I+x) m) (27). This can be verified by induction. The same result would (�55) be obtained if we divided I -}-o. x+o. x 2 +... at once by the expansion of (I dx)m.
(iii.) From (21) of � 43 (iv.) we see that I R r l is less than m[r+l]xr+1 if x is positive, or than 1 m [r+1] x m+1 (I+x)- m I if x is negative; and it can hence be shown that, if 1 x I < 1, 1 R r I can be made as small as we please by taking r large enough, so that we can make S r approximate as closely as we please to (r +x)-1n. (iv.) To assimilate this to the binomial theorem, we extend the definition of n (r) in (I) of � 41 (i.) so as to cover negative integral values of n; and we then have (-m)(r)- iI m- = (-) rm [ T ] (28), so that, if n=--- -m, Sr1 +n(ox+n(2)x2+... +n (r) x r (29).
(v.) .The further extension to fractional values (positive or negative) of n depends in the first instance on the establishment of a method of algebraical evolution which bears the same relation to arithmetical evolution (calculation of a surd) that algebraical division bears to arithmetical division.^ Do real programs depend on the assumption that a given expression always evaluates to the same value?
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ The calculation of a logarithm can be performed by successive divisions; evolution requires special methods.

^ When the result of any arithmetical operation or operations is represented approximately but not exactly by a number, the excess (positive or negative) of this number over the number which would express the result exactly is called the error.

In calculating 1/2, for instance, we proceed as if 2.0000... were the exact square of some number of the form co+c 1 /io+c 2 /io 2 +.. .
In the same way, to find X I I 4, where X= i+aix+a2x2+. and q is a positive integer, we assume that i /4 = i+bix+b2x2..., and we then (cf. � 55) determine b 1, b 2, ... in succession so that (r+b i x+b 2 x 2 +.. .) 4 shall be identical with X.
.The application of the method to the calculation of (I +x) n, when n= p/q, q being a positive integer and p a positive or negative integer, involves, as in the case where n is a negative integer, the separate consideration of the form of the coefficients b 1 , b 2, ...^ The slide-rule (see Calculating Machines ) is a simple apparatus for the mechanical application of the methods of logarithms.

^ In Some Cases, However, E.G. In The Case Of Negative Numbers And Reciprocals, Only One Is Involved; And There Might Be Three Or More, As In The Case Of A Number Expressed By ( A B)".

^ More complicated forms of arithmetical reasoning involve the use of series, each term in which corresponds to particular terms in two or more series jointly; and cases of this kind are usually dealt with by special methods, or by means of algebraical formulae.

and of the numerical value of i+b i x+b 2 x 2 +... + bixr. (vi.) The definition of no.) , which has already been extended in (iv.) above, has to be further extended so as to cover fractional values of n, positive or negative. Certain relations still hold, the most important being (22) of � 44 (ii.), which holds whatever the values of m and of n may be; r, of course, being a positive integer. This may be proved either by induction or by the method of � 52 (vi.). .The relation, when written in the form (23), is known as Vandermonde's theorem. By means of this theorem it can be shown that, whatever the value of n may be, f 1 + (plq)(i)x+(p/q)(2)x2+...^ This is true, whatever the arrangement of the original objects may be, and wherever the new one is introduced; and therefore, if the theorem is true for 8, it is true for 9.

^ If there is an integral number to be taken as well as a percentage, this number is written in front of the point; thus 23.76 X A means 23 times A, with 76% of A. We might therefore denote 76% by 0.76.

^ If we write 74 in the form 47 we may say that the value of a fraction is not altered by multiplying or dividing the numerator and denominator by any number.

+(plq)(r)xr} 4 =1 +p(1)x+p(2)x2+...+p(r)xr
+terms in xr+1, xr+2, x4r. (vii.) The comparison of the numerical value of I-Fn(1)x +n(242+...+n(r)xr, when n is fractional, with that of (i+x)n, involves advanced methods (� 64). .It is found that this expression can be used for approximating to the value of (1+x)n, provided that IxI<1; the results are as follows, where u r denotes n (r) x r and S r denotes uo+ul+u2+...+ur. (a) If n> - i, then, provided r> n, (I) If i >x>o, (I+x) n lies between Sr and Sr+1; (2) If o>x> - I, (I+x) n lies between Sr and Sr+ur+1l(I +x). (b) If n<-1, the successive terms will either constantly decrease (numerically) from the beginning or else increase up to a greatest term (or two equal consecutive greatest terms) and then constantly decrease.^ This Is a simple program that uses Cramer's Rule To figure out the intersection point of two linear equations and provides you with the determinants and the intersecting point.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

^ One can prove that the sum, difference, product, or quotient of two p -bit numbers, or the square root of a p -bit number, rounded first to q bits and then to p bits gives the same value as if the result were rounded just once to p bits provided q 2 p + 2.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ Thus when = 2, the number 0.1 lies strictly between two floating-point numbers and is exactly representable by neither of them.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

If S r is taken so as to include the greatest term (or terms), then, (1) If 1>x >o, (1+x). lies between Sr and Sr+1; (2) Ifo>x> -I, (1+x) n lies between Sr and Sr+ur+1/(I -ur+i/ur). The results in (b) apply also if n is a negative integer.
(viii.) .In applying the theorem to concrete cases, conversion of a number into a continued fraction is often useful.^ This means that programs which wish to use IEEE rounding can't use the natural language primitives, and conversely the language primitives will be inefficient to implement on the ever increasing number of IEEE machines.
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^ We will verify the theorem when no guard digits are used; the general case is similar.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ In the case of 0 0 , plausibility arguments can be made, but the convincing argument is found in "Concrete Mathematics" by Graham, Knuth and Patashnik, and argues that 0 0 = 1 for the binomial theorem to work.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

Suppose, for instance, that we require to calculate (23/13). .We want to express (23/13) 3 in the form a l b, where b is nearly equal to i.^ Instead of being expressed as the sum of a series of terms, a number may be expressed as the product of a series of factors, which become successively more and more nearly equal to i.

We find that - loglo (23/13) = � 37 1 6767 =log i n (2.3533) =logio (40/17) nearly; and thence that (23/13) 3 = (40/17) (I+1063/351520o), which can be calculated without difficulty to a large number of significant figures.
(ix.) The extension of n (r) , and therefore of n [ r ] , to negative and fractional values of n, enables us to extend the applicability of the binomial coefficients to the summation of series (� 46 (ii.)). .Thus the nth term of the series 2.5+5.8+8.11+...^ The Taylor series of µ '( x ) is also alternating, and if x has decreasing terms, so - µ '( x ) - + 2 x /3, or - µ'( x ) 0, thus .
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

in � 46 (iii.) is 18(n3)[2]; formula (20) of � 43 (iv.) holds for the extended coefficients, and therefore the sum of n terms of this series is 18.(n-3) [3] -18. (-3)[3]=3n3+6n2+n. In this way we get the general rule that, to find the sum of n terms of a series, the rth term of which is (a+rb) (a+r+ i � b) ... (a+r+ p - i �b), we divide the product of the p+1 factors which occur either in the nth or in the (n+i)th term by p+ 1, and by the common difference of the factors, and add to a constant, whose value is found by putting n= o.

57. Generating Functions

The series i- m [1] x+m [2] x 2 - ... obtained by dividing i+o. x+o.x 2 +... by (i+x) n, or the series 1 +(p/q)a)x+(p/q)(2)x2+ � � . obtained by taking the qth root of I + p (1) x+ p (2) x 2 + ..., is an infinite series, i.e. a series whose successive terms correspond to the numbers I, 2, 3, ... It is often convenient, as in � 56 (ii.) and (vi.), to consider the mode of development of such a series, without regard to arithmetical calculation; .i.e. to consider the relations between the coefficients of powers of x, rather than the values of the terms themselves.^ This problem can be avoided by introducing a special value called NaN, and specifying that the computation of expressions like 0/0 and produce NaN, rather than halting.
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^ To show that the spacing between binary numbers is always greater than the spacing between decimal numbers, consider an interval [10 n , 10 n + 1 ].
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ Findterm This program uses the Generla Formula for a Specific Term of an Expansion formula to find the coefficients and powers for a binomial raised to a power.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

.From this point of view, the function which, by algebraical operations on i+o.x+o.x2+..., produces the series, is called its generating function. The generating functions of the two series, mentioned above, for example, are (I +x)-' n and (I+x) P / Q. In the same way, the generating function of the series I+2x+x 2 +0.x 3 +o.x 4 +...^ For example, on a calculator, if the internal representation of a displayed value is not rounded to the same precision as the display, then the result of further operations will depend on the hidden digits and appear unpredictable to the user.
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^ This operation, called a fused multiply-add , can cause the same program to produce different results across different single/double systems, and, like extended precision, it can even cause the same program to produce different results on the same system depending on whether and when it is used.
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^ In addition to the two examples just mentioned (guard digits and extended precision), the section Systems Aspects of this paper has examples ranging from instruction set design to compiler optimization illustrating how to better support floating-point.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

is (1+x)2.
.Considered in this way, the relations between the coefficients of the powers of x in a series may sometimes be expressed by a formal equality involving the series as a whole.^ S. The simplest case, in which the quantity can be expressed as an integral number of the largest units B involved, has already been considered (§§ 37, 42).

^ On the counting system we may consider that we have a series of objects (represented in the adjoining diagram by dots), and that we attach to these objects in succession the symbols 1, 2, 3, 4, 5, 6, 7, 8, 9, o, repeating this series indefinitely.

^ Hence we may insert in the power-series numbers with fractional indices, provided that the multiplication of these numbers follows the same law.

Thus (4) of � 41 (ii.) may be written in the form I +n(i)x+n(2)x2+...+n(r)xr-}-... f (I +x) tI+(n-1)(1))x+(n-1)(2)x2-{-...
(n- I)(r)xr+���}; the symbol " being used to indicate that the equality is only formal, not arithmetical.
.This accounts for the fact that the same table of binomial coefficients serves for the expansions of positive powers of i+x and of negative powers of i - x.^ In the same way, by dividing by powers of so we may get negative indices.

^ Findterm This program uses the Generla Formula for a Specific Term of an Expansion formula to find the coefficients and powers for a binomial raised to a power.
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^ Binomial Expander This program will list the resulting terms of any binomial expansion of a binomial in [A(F)^X + B(S)^Y]^E format, where "F" and "S" are the variables, "A" and "B" are the variables' coefficients, "X" and "Y" are the variables' exponents, and "E" is the exponent of the binomial.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

For (4) may (� 43 (iv.)) be written (n - i) [71 = n [ r] - n[r_1], and this leads to relations of the form 1 +2x+3x2+..� i (I -x) (I +3x+6x2+Iox3+...) .(30), each set of coefficients being the numbers in a downward diagonal of the table.^ On the continent of Europe the figures are taken in sets of three, but are merely spaced, the comma being used at the end of a number to denote the commencement of a decimal.

^ We first construct the multiple-table C, and then subtract successively zoo times, 30 times and I times; these numbers being the partial quotients.

In the same way (21) of � 43 (iv.) leads to such relations as x +3x+6x2+... f (1+x+x2+...)(1+2x+3x2+...) (31), the relation of which to (30) is obvious.
An application of the method is to the summation of a recurring series, i.e. a series co+c i x+c 2 x 2 +.. . whose coefficients are connected by a relation of the form pocr+plcr_1+...-i-pkcr-k= o, where po,pi, � ..pk are independent of x and of r. 58. Approach to a Limit. - .There are two kinds of approach to a limit, which may be illustrated by the series forming the expansion of (x+h) n, where n is a negative integer and 1> h/x> o.^ Or we might say that, since multiplication is a form of addition, and division a form of subtraction, there are really only two fundamental processes, viz.

^ Where there is a division into sixteen parts, a binary scale may be formed by dividing into groups of two, four or eight.

^ It is not stated, in most cases, whether all the numbers within the limits of the series have definite positions, or whether there are only certain numbers which form an essential part of the figure, while others only exist potentially.

(i.) Denote n (r) x n - r h r by u r, and uo+u i +... +u r by Sr. Then (� 56 (iii.)) (x+h)" lies between S r and Sr+i; and provided S r includes the numerically greatest term, I Sr+i-Sri constantly decreases as r increases, and can be made as small as we please by taking r large enough. Thus by taking r = o, 4, 2, ... we have a sequence So, Si, S2, ... (i.e. a succession of numbers corresponding to the numbers I, 2, 3, ...) which possesses the property that, by starting far enough in the sequence, the range of variation of all subsequent terms can be made as small as we please, but .(x+h) n always lies between the two values determining the range.^ Thus when = 2, the number 0.1 lies strictly between two floating-point numbers and is exactly representable by neither of them.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ This can happen when the result as rounded to extended double precision is a "halfway case", i.e., it lies exactly halfway between two double precision numbers, so the second rounding is determined by the round-ties-to-even rule.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

This is expressed by saying that the sequence converges to (x+h)" as its limit; it may be stated concisely in any of the three ways, (x+h) n =lim(x"+n(1)xn-lh+....+ n(T)xn-rhr+��.),(x+h)n =lim Sr, Sr. .(x+h)n. It will be noticed that, although the differences between successive terms of the sequence will ultimately become indefinitely small, there will always be intermediate numbers that do not occur in the sequence.^ Thus there are (2 10 - 10 3 )2 14 = 393,216 different binary numbers in that interval.
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^ There is a small snag when = 2 and a hidden bit is being used, since a number with an exponent of e min will always have a significand greater than or equal to 1.0 because of the implicit leading bit.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ There is no difference in principle between addition (or subtraction) of numbers and addition (or subtraction) of numerical quantities.

.The approach to the limit will therefore be by a series of jumps, each of which, however small, will be finite; i.e. the approach will be discontinuous.^ In practice, however, both a/3 and a certain portion of a'b are small in comparison with a'(3 and b' a, and we therefore replace a'b' + a(3 by an approximate value, and increase the limit of error so as to cover the further error thus introduced.

(ii.) Instead of examining what happens as r increases, let us examine what happens as h/x decreases, r remaining unaltered. .Denote h/x by 0, where i> e> o; and suppose further that 0< i/n I, so that the first term of the series uo+ul+u2+...^ Posite The Corresponding Term Of The First Series, Each Column Being Headed .

^ If we precede the series of convergents by i and - 1 6 -, then the numerator (or denominator) of each term of the series o i a, ab?-1 after the first two, is found by multiplying 1, o?

^ Later, a system similar to the Hebrew was adopted, and extended by reproducing the first nine symbols of the series, preceded by accents, to denote multiplication by moo.

is
the greatest (numerically). .Then {(x+h)n - S T } /h r + l lies between n (r+1>x n-r-1 and n(r+1}xn-r-1(i +O)n; and the difference between these can be made as small as we please by taking h small enough.^ We have focused on differences between extended-based systems and single/double systems, but there are further differences among systems within each of these families.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

.Thus we can say that the limit of {(x+h)n - Sr }lhr +1 is n (r+1) x n - 1 ` 1; but the approach to this limit is of a different kind from that considered in (1.), and its investigation involves the idea of continuity.^ A continued fraction, of the kind we are considering, is an expression of the form a+ b+ c + d+ &c.

^ The third involves also the idea of continuity and therefore of unlimited subdivision.

V. Continuity. 59. The idea of continuity must in the first instance be introduced from the graphical point of view; arithmetical continuity being impossible without a considerable extension of the idea of number (� 65). .The idea is utilized in the elementary consideration :of a differential coefficient; and its importation into the treatment of certain functions as continuous is therefore properly associated with the infinitesimal calculus.^ The third involves also the idea of continuity and therefore of unlimited subdivision.

60. The first step consists in the functional treatment of equations. Thus, to solve the equation ax e +bx+c = o, we consider, not merely the value of x for which ax 2 +bx+c is o, but the value of ax e +bx+c for every possible value of x. By graphical treatment we are able, not merely to see why the equation has usually two roots, and also to understand why there is in certain cases only one root (i.e. two equal roots) and in other cases no root, but also to see why there cannot be more than two roots.
.Simultaneous equations in two unknowns x and y may be treated in the same way, except that each equation gives a functional relation between x and y.^ Simultaneous Equation Solver Simultaneous Equation Solver is offered as a replacement for Texas Instruments' PolySmlt Application which serves the same purpose.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

^ A set of written symbols is sometimes read in more than one way, while on the other hand two different sets of symbols (at any rate if denoting numerical quantities) may be read in the same way.

^ That this gives the same result as adding 4 to 5 may be seen by reckoning the series backwards.

("
Indeterminate equations " belong properly to the theory of numbers.) 61. From treating an expression involving x as a function of x which may change continuously when x changes continuously, we are led to regard two functions x and y as changing together, so that (subject to certain qualifications) to any succession of values of x or of y there corresponds a succession of values of y or of x; and thence, if (x, y) and (x+h, y+k) are pairs of corresponding values, we are led to consider the limit (� 58 (ii.)) of the ratio k/h when h and k are made indefinitely small. Thus we arrive at the differential coefficient of f(x) as the limit of the ratio of f (x+8) - f (x) to 0 when 0 is made indefinitely small; and this gives an interpretation of nx n-1 as the derived function of xn (� 45)� This conception of a limit enables us to deal with algebraical expressions which assume such forms as -° o for particular values of the variable (� 39 (iii.)). .We cannot, for instance, say that the fraction C _2 I is arithmetically equal to x+I when x= I, as well as for other values of x; but we can say that the limit of the ratio of x 2 - I to x - I when x becomes indefinitely nearly equal to I is the same as the limit of x+ On the other hand, if f(y) has a definite and finite value for y = x, it must not be supposed that this is necessarily the same as the limit which f (y) approaches when y approaches the value x, though this is the case with the functions with which we are usually concerned.^ For example sums are a special case of inner products, and the sum ((2 × 10 -30 + 10 30 ) - 10 30 ) - 10 -30 is exactly equal to 10 -30 , but on a machine with IEEE arithmetic the computed result will be -10 -30 .
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^ Generally, to find the sum or difference of two or more fractional numbers, we must replace them by other fractional numbers having the same denominator; it is usually most convenient to take as this denominator the L.C.M. of the original fractional numbers (cf.

^ In the second class of cases the fraction of the unit quantity is a quantity of the same kind, but cannot be determined with absolute exactness.

.62. The elementary idea of a differential coefficient is useful in reference to the logarithmic and exponential series.^ Of the two kinds of division, although the idea of partition is perhaps the more elementary, the process of measuring is the easier to perform, since it is equivalent to a F series of subtractions.

.We know that log l oN(I+9) = log l oN+log 10 (I+0), and inspection of a table of logarithms shows that, when 0 is small, log 10 (I+B);s approximately equal to X0, where X is a certain constant, whose value is.^ In practice, however, both a/3 and a certain portion of a'b are small in comparison with a'(3 and b' a, and we therefore replace a'b' + a(3 by an approximate value, and increase the limit of error so as to cover the further error thus introduced.

^ We take out log 2 from the table, halve it, and then find from the table the number of which this is the logarithm.

^ The gramme was intended to be equal to the weight of a cubic centimetre of pure water at a certain temperature, but the equality is only approximate.

434. � � If we took logarithms to base a, we should have loga(I+B) =logoIOXXO, approximately. .If therefore we choose a quantity e such that log e I o X X= I, log i oe = X, which gives (by more accurate calculation) e=2.71828..., we shall have lim(loge(I+0))}/0=I, and conversely 'lim' {ex+0 - e x } 143= The deduction of the expansions log e (' +x) = x - Zx 2 + 3x 3 - ..., e x = I +.x+x2/2!+x3/3!-}-..., is then more simply obtained by the differential calculus than by ordinary algebraic methods.^ Even if there are over/underflows, the calculation is more accurate than if it had been computed with logarithms, because each p k was computed from p k - 1 using a full precision multiply.
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^ Furthermore, Brown's axioms are more complex than simply defining operations to be performed exactly and then rounded.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ The problem with this approach is that it is less accurate, and that it costs more than the simple expression , even if there is no overflow.
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63. The theory of inequalities is closely connected with that of maxima and minima, and therefore seems to come properly under this head. .The more simple properties, however, only require the use of elementary methods.^ In particular, the IEEE standard requires a careful implementation, and it is possible to write useful programs that work correctly and deliver accurate results only on systems that conform to the standard.
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^ A more useful zero finder would not require the user to input this extra information.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ However, there is a much more efficient method which dramatically improves the accuracy of sums, namely .
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

Thus to show that the arithmetic mean of n positive numbers is greater than their geometric mean (i.e. than the nth root of their product) we show that if any two are unequal their product may be increased, without altering their sum, by making them equal, and that if all the numbers are equal their arithmetic mean is equal to their geometric mean.
VI. Special Developments. 64. One case of convergence of a sequence has already been considered in � 58 (i.). .The successive terms of the sequence in that case were formed by successive additions of terms of a series; the series is then also said to converge to the limit which is the limit of the sequence.^ We might extend this principle to cases in which the terms of two series, whether of numbers or £1 A £ 1 53, 7 s.

^ It is not stated, in most cases, whether all the numbers within the limits of the series have definite positions, or whether there are only certain numbers which form an essential part of the figure, while others only exist potentially.

^ Instead of being expressed as the sum of a series of terms, a number may be expressed as the product of a series of factors, which become successively more and more nearly equal to i.

Another example of a sequence is afforded by the successive convergents to a continued fraction of the form ao+ I I, al+ a2+� where ao,a 1 ,a 2, ... are integers. Denoting these convergents by Po/Qo, P1/Q1, P2/Q2, ... they may be regarded as obtained from a series Po + (Qi - Qo) + (Q2 (P2 - 1:11 ) +.. �; the successive terms of this series, after the first, are alternately positive and negative, and consist of fractions with numerators I and denominators continually increasing.
.Another kind of sequence is that which is formed by introducing the successive factors of a continued product; e.g. the successive factors on the right-hand side of Wallis's theorem it 2.2 4.4 6.6 2 = 1.3. 3.5.5.7...^ The left hand factor can be computed exactly, but the right hand factor µ ( x ) = ln(1 +  x )/ x will suffer a large rounding error when adding 1 to x .
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^ Instead of being expressed as the sum of a series of terms, a number may be expressed as the product of a series of factors, which become successively more and more nearly equal to i.

^ It is straightforward to check that the right-hand sides of (6) and (7) are algebraically identical.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

A continued product of this kind can, by taking logarithms, be replaced by an infinite series.
In the particular case considered in � 58 (i.) we were able to.. examine the approach of the sequence So, Si, S2, ... to its limit X by direct examination of the value of X - S r. .In most cases this is not possible; and we have first to consider the convergence of the sequence or of the series which it represents, and then to determine its limit by indirect methods.^ Correspondence of Numerical Quantities.-Again, in § 89, we have considered cases of multiple-tables of numerical quantities, where each quantity in one series is equivalent to the corresponding quantity in the other series.

^ On the counting system we may consider that we have a series of objects (represented in the adjoining diagram by dots), and that we attach to these objects in succession the symbols 1, 2, 3, 4, 5, 6, 7, 8, 9, o, repeating this series indefinitely.

^ If a is zero, we may regard as the first convergent, and precede the series by and °-.

This constitutes the general theory of convergence of series (see Series).
The word " sequence," as defined in � 58 (i.), includes progressions such as the arithmetical and geometrical progressions, and, generally, the succession of terms of a series. It is usual, however, to confine it to those sequences (e.g. the sequence formed by taking successive sums of a series) which have to be considered in respect of their convergence or non-convergence.
.In order that numerical results obtained by summing the first few terms of a series may be of any value, it is usually necessary that the series should converge to a limit; but there are exceptions to this rule.^ If a is zero, we may regard as the first convergent, and precede the series by and °-.

^ One can prove that the sum, difference, product, or quotient of two p -bit numbers, or the square root of a p -bit number, rounded first to q bits and then to p bits gives the same value as if the result were rounded just once to p bits provided q 2 p + 2.
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^ A similar analysis of ( x x ) ( y y ) cannot result in a small value for the relative error, because when two nearby values of x and y are plugged into x 2  - y 2 , the relative error will usually be quite large.
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.For instance, when n is large, n! is approximately equal to -J(27rn).^ In Germany , for instance, the Pfund is kilogramme, and is approximately equal to iY-ol-b English.

(nle) n;
the approximation may be improved by Stirling's theorem log e 2 +log e 3 +... +log e (n - I) + Zlog e n = Zlog e (21r) +nlog e n - n { - B, B. -? (-)r1Br -1 ?... I.2.n 3.4.n3+ - (2r - I),2r.n2r where B1, B2, ... are Bernoulli's numbers (� 46 (v.)), although the series is not convergent.
65. Consideration of the binomial theorem for fractional index, or of the continued fraction representing a surd, or of theorems such as Wallis's theorem (� 64), shows that a sequence, every term of which is rational, may have as its limit an irrational number, i.e. a number which cannot be expressed as the ratio of two integers.
.These are isolated cases of irrational numbers.^ In a few cases the names of certain small numbers are the names of objects which present these numbers in some conspicuous way.

^ From 1 to 12 the numbers sometimes lie in the circumference of a circle, an arrangement obviously suggested by a clock-face; in these cases the series usually mounts upwards from 12.

^ Although multiplication may arise in either of these two ways, the actual process in each case is performed by commencing with the unit and taking it the necessary number of times.

.Other cases arise when we consider the continuity of a function.^ Correspondence of Numerical Quantities.-Again, in § 89, we have considered cases of multiple-tables of numerical quantities, where each quantity in one series is equivalent to the corresponding quantity in the other series.

^ The transition is similar to that which arises in the case of geometrical measurement (§ 26), and it is an essential feature of all reasoning with regard to continuous quantity, such as we have to deal with in real life.

Suppose, for instance, that y=x 2; then to every rational value of x there corresponds a rational value of y, but the converse does not hold. .Thus there appear to be discontinuities in the values of y. The difficulty is due to the fact that number is naturally not continuous, so that continuity can only be achieved by an artificial development.^ Thus there are (2 10 - 10 3 )2 14 = 393,216 different binary numbers in that interval.
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^ There appears therefore to be a tendency to use some larger number than ten as a basis for grouping into new units or for subdivision into parts.

^ The problem can be traced to the fact that square root is multi-valued, and there is no way to select the values so that it is continuous in the entire complex plane.
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.The development is based on the necessity of being able to represent geometrical magnitude by arithmetical magnitude; and it may be regarded as consisting of three stages.^ This tendency is common in adults as well as in children; the strokes of a clock may, for instance, be grouped into fours, and thus eleven is represented as two fours and three.

^ From the educational point of view, the value of arithmetic has usually been regarded as consisting in the stress it lays on accuracy.

^ The various stages in the study of arithmetic may be arranged in different ways, and the arrangement adopted must be influenced by the purpose in view.

.Taking any number n to be represented by a point on a line at distance nL from a fixed point 0, where L is a unit of length, we start with a series of points representing the integers I, 2, 3,.^ More precisely, x is rounded by taking the significand of x, imagining a radix point just left of the k least significant digits, and rounding to an integer.
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^ The second approach represents higher precision floating-point numbers as an array of ordinary floating-point numbers, where adding the elements of the array in infinite precision recovers the high precision floating-point number.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ Rather than using all these digits, floating-point hardware normally operates on a fixed number of digits.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

..
This series is of course discontinuous. .The next step is to suppose that fractional numbers are represented in the same way.^ The numbers represented by a, b, c, x and m are all supposed to be positive.

^ The principle of subtraction from a higher number, which appeared in notation, also appeared in numeration, but not for exactly the same numbers or in exactly the same way; thus XVIII was two-from-twenty, and the next number was onefrom-twenty, but it was written XIX, not IXX. .

^ The rule for multiplying a fractional number by a fractional number is therefore the same as the rule for finding a fraction of a fraction.

.This extension produces a change of character in the series of numbers.^ On machines that have an instruction that multiplies two single precision numbers to produce a double precision number, dx   =   x*y can get mapped to that instruction, rather than compiled to a series of instructions that convert the operands to double and then perform a double to double precision multiply.
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.In the original integral series each number had a definite number next to it, on each side, except 1, which began the series.^ This definition applies whether the original number is integral or fractional.

^ It will be seen that the definition includes integral numbers.

^ Quotients.-The converse of subdivision is the formation of units into groups, each constituting a larger unit; the number of the groups so formed out of a definite number of the original units is called a quotient.

.But in the new series there is no first number, and no number can be said to be next to any other number, since, whatever two numbers we take, others can be inserted between them.^ If two numbers have no factor in common (except 1) each is said to be prime to the other.

^ There is no difference in principle between addition (or subtraction) of numbers and addition (or subtraction) of numerical quantities.

^ First of all, there are algebraic identities that are valid for floating-point numbers.
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.On the other hand, this new series is not continuous; for we know that there are some points on the line which represent surds and other irrational numbers, and these numbers are not contained in our series.^ In the majority of cases the numbers lie on a continuous (but possibly zigzag) line.

^ In a few cases the names of certain small numbers are the names of objects which present these numbers in some conspicuous way.

^ Or the words signifying these numbers may have reference to the completion of some act of counting.

.We therefore take a third step, and obtain theoretical continuity by considering that every point on the line, if it does not represent a rational number, represents something which may be called an irrational number. This insertion of irrational numbers (with corresponding negative numbers) requires for its exact treatment certain special methods, which form part of the algebraic theory of number, and are dealt with under Number.^ Any exact fraction can be expressed as a continued fraction, and there are methods for expressing as continued fractions certain other numbers, e.g.

^ The theory of continued fractions gives a method of expressing a number, in certain cases, as a continued product.

^ More complicated forms of arithmetical reasoning involve the use of series, each term in which corresponds to particular terms in two or more series jointly; and cases of this kind are usually dealt with by special methods, or by means of algebraical formulae.

.66. The development of the theory of equations leads to the amplification of real numbers, rational and irrational, positive and negative, by imaginary and complex numbers. The quadratic equation x 2 +b 2 =o, for instance, has no real root; but we may treat the roots as being +b-' - I, and - b 1 ,1 - 1, if -J - i is treated as something which obeys the laws of arithmetic and emerges into reality under the condition 1 1 - I. -!^ It can also factorize any cubic equation if it has three real and rational roots.
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^ I am working on a version that will display irrational factors, if there are no rational factors, using tha Quadriatic Formula.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

^ Equation Solver This is an equation solver for binomials and trinomail(yes, a quadratic is a trinomial) and also solves for complex solutions, (imaginary numbers) .
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

- I = - i.
.Expressions of the form b1,1 - I and a-}-b A l - I, where a and b are real numbers, are then described as imaginary and complex numbers respectively; the former being a particular case of the latter.^ Equation Solver This is an equation solver for binomials and trinomail(yes, a quadratic is a trinomial) and also solves for complex solutions, (imaginary numbers) .
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

^ The prefixes for multiplication by io, io 2, io 3 and Io 4 are deca-, hecto-, kilo- and myria-, and those for division by io, io 2 and io 3 are deci-, centi- and milli-; the former being derived from Greek, and the latter from Latin.

^ Arithmetic is usually divided into Abstract Arithmetic and Concrete Arithmetic, the former dealing with numbers and the latter with concrete objects.

.Complex numbers are conveniently treated in connexion not only with the theory of equations but also with analytical trigonometry, which suggests the graphic representation of a+b,l - by a line of length (a 2 +b 2)i drawn in a direction different from that of the line along which real numbers are represented.^ Equation Solver This is an equation solver for binomials and trinomail(yes, a quadratic is a trinomial) and also solves for complex solutions, (imaginary numbers) .
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

^ Several different representations of real numbers have been proposed, but by far the most widely used is the floating-point representation.
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^ Similarly, if the real number .0314159 is represented as 3.14 × 10 -2 , then it is in error by .159 units in the last place.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

References.-W. K. Clifford, The Common Sense of the Exact Sciences (1885), chapters i. and iii., forms a good introduction to algebra. .As to the teaching of algebra, see references under Arithmetic to works on the teaching of elementary mathematics.^ For an account of German methods, see W. King, Report on Teaching of Arithmetic and Mathematics in the Higher Schools of Germany (1903).

^ This work contains references to Grube's system, which has been much discussed in America: for a brief explanation, see L. Seeley, The Grube Method of Teaching Arithmetic (1890).

^ There is a certain difference between the use of words referring to equality and identity in > > < < arithmetic and in algebra respectively; what is an equality in the former becoming an identity in the latter.

Among school-books may be mentioned those of W. M. Baker and A. A. Bourne, W. G. Borchardt, W. D. Eggar, F. Gorse, H. S. Hall and S. R. Knight, A. E. F. Layng, R. B. Morgan. G. Chrystal, Introduction to Algebra (1898); H. B. Fine, A College Algebra (1905); C. Smith, A Treatise on Algebra (1st ed. 1888, 3rd ed. .1892), are more suitable for revision purposes; the second of these deals rather fully with irrational numbers.^ Rather than using all these digits, floating-point hardware normally operates on a fixed number of digits.
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^ If it has no more then 2 irrational or imaginary roots and at least one real root, my program will return all x intercepts, no matter the number of terms.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

.For the algebraic theory of number, and the convergence of sequences and of series, see T. J. I'A. Bromwich, Introduction to the Theory of Infinite Series (1908); H. S. Carslaw, Introduction to the Theory of Fourier's Series (1906); H. B. Fine, The Number-System of Algebra (1891); H. P. Manning, Irrational Numbers (1906); J. Pierpont, Lectures on the Theory of Functions of Real Variables (1905).^ These examples can be summarized by saying that optimizers should be extremely cautious when applying algebraic identities that hold for the mathematical real numbers to expressions involving floating-point variables.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ Again, In §§61 75 And 84 88 We Have Considered Various Kinds Of Numbers Other Than Those In The Original Number Series.

^ Fine point: Although the default in IEEE arithmetic is to round overflowed numbers to , it is possible to change the default (see Rounding Modes ) .
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

For general reference, G. Chrystal, TextBook of Algebra (pt. i. 5th ed. 1904, pt. ii. 2nd ed. .1900) is indispensable; unfortunately, like many of the works here mentioned, it lacks a proper index.^ Many programmers like to believe that they can understand the behavior of a program and prove that it will work correctly without reference to the compiler that compiles it or the computer that runs it.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ As the section Languages and Compilers mentions, many programming languages don't specify that each occurrence of an expression like 10.0*x in the same context should evaluate to the same value.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

.Reference may also be made to the special articles mentioned at the commencement of the present article, as well as to the articles on Differences, Calculus Of; Infinitesimal Calculus; Interpolation; Vector Analysis.^ For the latter, and for systems of notation, reference may also be made to Peacock's article " Arithmetic " in the Encyclopaedia Metropolitana, which contains a detailed account of the Greek system.

^ The symbols - and = mean respectively that the first quantity mentioned is to be reduced or divided by the second; but there is some vagueness about + and X. In the present article a+b will mean that a is taken first, and b added to it; but a X b will mean that b is taken first, and is then multiplied by a.

.The following may also be consulted: - E. Borel and J. Drach, Introduction a l'etude de la theorie des nombres et de l'algebre superieure (1895); C. de Comberousse, Cours de mathematiques, vols.^ Ce petit programme (envrion 400 octets) trace la compose des fonctions f et g ( f o g et g o f ) !
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

i. and iii. (1884-1887); H. Laurent, Traite d'analyse, vol. i. (1885); E. Netto, Vorlesungen fiber Algebra (vol. i. 1896, vol. ii. 1900); S. Pincherle, Algebra complementare (1893); G. Salmon, Lessons introductory to the Modern Higher Algebra (4th ed., 1885); J. A. Serret, Cours d'algebre superieure (4th ed., 2 vols., 1877); O. Stolz and J. A. Gmeiner, Theoretische Arithmetik (pt. i. 1900, pt. ii. 1902) and Einleitung in die Funktionen-theorie (pt. i. 1904, pt. ii. 1905) - these being developments from O. Stolz, Vorlesungen fiber allgemeine Arithmetik (pt. i. 1885, pt. ii. .1886); J. Tannery, Introduction a la theorie des fonctions d'une variable (1st ed.^ Ce petit programme (envrion 400 octets) trace la compose des fonctions f et g ( f o g et g o f ) !
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

1886, 2nd ed. 1904); H. Weber, Lehrbuch der Algebra, 2 vols. (1st ed. 1895-1896, 2nd ed. 1898-1899; vol. i. of 2nd ed. transl. by Griess as Traite d'algcbre superieure, 1898). For a fuller bibliography, see Encyclopadie der math. Wissenschaften (vol. i., 1898). A list of early works on algebra is given in Encyclopaedia Britannica, 9th ed., vol. i. p. 518.
.(W. F. SH.) B. Special Kinds Of Algebra I. A special algebra is one which differs from ordinary algebra in the laws of equivalence which its symbols obey.^ Special Thanks To: Edwin Howard For His Help*** This is the only algebra II program you'll ever need because it has everything all in one big package.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

^ AA - The Algebra II PRGM This program is actually 12 different programs all in one and at a reasonable size.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

^ An optimizer that believed floating-point arithmetic obeyed the laws of algebra would conclude that C = [ T - S ] - Y = [( S + Y )- S ] - Y = 0, rendering the algorithm completely useless.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

Theoretically, no limit can be assigned to the number of possible algebras; the varieties actually known use, for the most part, the same signs of operation, and differ among themselves principally by their rules of multiplication.
.2. Ordinary algebra developed very gradually as a kind of shorthand, devised to abbreviate the discussion of arithmetical problems and the statement of arithmetical facts.^ The arithmetical fact is that I i and 9 may be regrouped as 12 and 8, and the statement "IId.+9d.

^ In statements like Theorem 3 that discuss the relative error of an expression, it is understood that the expression is computed using floating-point arithmetic.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

.Although the distinction is one which cannot be ultimately maintained, it is convenient to classify the signs of algebra into symbols of quantity (usually figures or letters), symbols of operation, such as +, i, and symbols of distinction, such as brackets.^ A similar analysis of ( x x ) ( y y ) cannot result in a small value for the relative error, because when two nearby values of x and y are plugged into x 2  - y 2 , the relative error will usually be quite large.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ The operation is the converse or repetition; it is usually called partition , as representing division into a number of equal shares.

^ For multiplication by a proper fraction or a decimal, it is sometimes convenient, especially when we are dealing with mixed quantities, to convert the multiplier into the sum or difference of a number of fractions, each of which has i as its numerator.

.Even when the formal evolution of the science was fairly complete, it was taken for granted that its symbols of quantity invariably stood for numbers, and that its symbols of operation were restricted to their ordinary arithmetical meanings.^ One reason for completely specifying the results of arithmetic operations is to improve the portability of software.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ Let p be the floating-point precision, with the restriction that p is even when  >  2 , and assume that floating-point operations are exactly rounded.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

.It could not escape notice that one and the same symbol, such as -1 (a - b), or even (a - b), sometimes did and sometimes did not admit of arithmetical interpretation, according to the values attributed to the letters involved.^ To compute m x from mx involves rounding off the low order k digits (the ones marked with b ) so (32) m x = mx - x mod( k ) + r k The value of r is 1 if .bb...b is greater than and 0 otherwise.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ The implementation of library functions such as sin and cos is even more difficult, because the value of these transcendental functions aren't rational numbers.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

This led to a prolonged controversy on the nature of negative and imaginary quantities, which was ultimately settled in a very curious way. .The progress of analytical geometry led to a geometrical interpretation both of negative and also of imaginary quantities; and when a " meaning " or, more properly, an interpretation, had thus been found for the symbols in question, a reconsideration of the old algebraic problem became inevitable, and the true solution, now so obvious, was eventually obtained.^ When three or more numbers or quantities are added together, the result should always be checked by adding both upwards and downwards.

^ Now on the off chance that dosen't work, the equation will be solved for the imaginary solution!
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

^ In order to apply arithmetical processes to a quantity expressed in two or more denominations, we must first express it in terms of a single denomination by means of a varying scale of notation.

.It was at last realized that the laws of algebra do not depend for their validity upon any particular interpretation, whether arithmetical, geometrical or other; the only question is whether these laws do or do not involve any logical contradiction.^ In order to rely on examples such as these, a programmer must be able to predict how a program will be interpreted, and in particular, on an IEEE system, what the precision of the destination of each arithmetic operation may be.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ On the other hand, if the formula x y  =  e ylog x is used to define ** for real arguments, then depending on the log function, the result could be a NaN (using the natural definition of log( x ) = NaN when x < 0).
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ These examples can be summarized by saying that optimizers should be extremely cautious when applying algebraic identities that hold for the mathematical real numbers to expressions involving floating-point variables.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

.When this fundamental truth had been fully grasped, mathematicians began to inquire whether algebras might not be discovered which obeyed laws different from those obtained by the generalization of arithmetic.^ If both operands are NaNs, then the result will be one of those NaNs, but it might not be the NaN that was generated first.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ An optimizer that believed floating-point arithmetic obeyed the laws of algebra would conclude that C = [ T - S ] - Y = [( S + Y )- S ] - Y = 0, rendering the algorithm completely useless.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

The answer to this question has been so manifold as to be almost embarrassing. .All that can be done here is to give a sketch of the more important and independent special algebras at present known to exist.^ Special Thanks To: Edwin Howard For His Help*** This is the only algebra II program you'll ever need because it has everything all in one big package.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

^ Avoiding this kind of "optimization" is so important that it is worth presenting one more very useful algorithm that is totally ruined by it.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ Instead, these two programs have all the function that you need with no-nonsense interface that gives you all you need and nothing more.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

3. Although the results of ordinary algebra will be taken for granted, it is convenient to give the principal rules upon which it is based. .They are (a+b)-?-c=a+(b+c) (A) (aXb)Xc=aX(bXc) (A') a+b=b+a (c) aXb=bXa (c') a(b c) =ab-Fac (D) (a - b)+b=a (I) (a=b)Xb=a (I') These formulae express the associative and commutative laws of the operations + and X, the distributive law of X, and the definitions of the inverse symbols - and =, which are assumed to be unambiguous.^ We therefore only require a definite law for the formation of the successive names or symbols.

^ These Are Examples Of The Associative Law For Multiplication (§ 58 (Iv)).

.The special symbols o and I are used to denote a - a and a=a. They behave exactly like the corresponding symbols in arithmetic; and it follows from this that whatever " meaning " is attached to the symbols of quantity, ordinary algebra includes arithmetic, or at least an image of it.^ X3 and 3X4 mean different things, but the operations which they denote produce the same result.

^ The ordinary notation of the Babylonians was denary, but they also used a sexagesimal scale, i.e.

^ In this section, we classify existing implementations of IEEE 754 arithmetic based on the precisions of the destination formats they normally use.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

.Every ordinary algebraic quantity may be regarded as of the form a+0 - I, where a, 6 are " real "; that is to say, every algebraic equivalence remains valid when its symbols of quantity are interpreted as complex numbers of the type a-1-(-V - I (cf.^ (The words in brackets are, inserted to avoid the difficulty, at this stage, of saying that every number is a factor of o, though it is of course true that o.

^ The pair of compartments on either side may, as here, contain numerical quantities, or may contain numbers.

^ If you get a non-real answer, you need to set your calculator to complex number mode.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

Number). .But the symbols of ordinary algebra do not necessarily denote numbers; they may, for instance, be interpreted as coplanar points or vectors.^ The symbols denoting a number are called its digits.

^ First of all, there are algebraic identities that are valid for floating-point numbers.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ Perhaps they have in mind that floating-point numbers model real numbers and should obey the same laws that real numbers do.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

.Evolution and involution are usually regarded as operations of ordinary algebra; this leads to a notation for powers and roots, and a theory of irrational algebraic quantities analogous to that of irrational numbers.^ It is desirable, wherever possible, to perform operations on numbers or numerical quantities from the left, rather than from the right.

^ In most cases the quantity in the second column may be regarded as increasing or decreasing continuously as the number in the first column increases, and it has intermediate values corresponding to intermediate (i.e.

^ The operation is the converse or repetition; it is usually called partition , as representing division into a number of equal shares.

4. The only known type of algebra which does not contain arithmetical elements is substantially due to George Boole. Although originally suggested by formal logic, it is most simply interpreted as an algebra of regions in space. .Let i denote a definite region of space; and let a, b, &c., stand for definite parts of i. Let a+b denote the region made up of a and b together (the common part, if any, being reckoned only once), and let a X b or ab mean the region common to a and b. Then a+a = as = a; hence numerical coefficients and indices are not required.^ Hence, so long as the denominator remains unaltered, we can deal with, exactly as if they were numbers, any operations being performed on the numerators.

^ Hence we must express I, which itself means $ times, as being 7 times something.

^ The Babylonians expressed numbers less than r by the numerator of a fraction with denominator 60; the numerator only being written.

The inverse symbols -, = are ambiguous, and in fact are rarely used. .Each symbol a is associated with its supplement a which satisfies the equivalences a+a = i, aa = o, the latter of which means that a and a have no region in common.^ But in the latter case it must always be understood that there is some unit concerned, and the results have no meaning until the unit is reintroduced.

.Finally, there is a law of absorption expressed by a+ab = a. From every proposition in this algebra a reciprocal one may be deduced by interchanging + and X, and also the symbols o and i. For instance, x+y = x+xy and xy = x(x+y) are reciprocal.^ On the grouping system we may in the first instance consider that we have separate symbols for numbers from " one " to " nine," but that when we reach ten objects we put them in a group and denote this group by the symbol used for " one," but printed in a different type or written of a different size or (in teaching) of a different colour.

^ This has one use that I can think of, Algebra 2 & factoring trinomials (what multiplies to AC and adds to B), but I'm sure there are others.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

^ But in algebra a X b = b X a is called an identity, in the sense that it is true whatever a and b may be; while n X X = A is called an equation , as being true, when n and A are given, for one value only of X. Similarly the numbers represented by and a are not identical, but are equal.

The operations + and X obey all the ordinary laws A, C, D (� 3) .
.5. A point A in space may be associated with a (real, positive, or negative) numerical quantity a, called its weight, and denoted by the symbol aA. The sum of two weighted points aA, (3B is, by definition, the point (a+13) G, where G divides AB so that AG: GB =0: a.^ The positive quantity or number obtained from a negative quantity or number by omitting the " - " is called its numerical value.

^ The symbols denoting a number are called its digits.

^ The pair of compartments on either side may, as here, contain numerical quantities, or may contain numbers.

It can be proved by geometry that (aA-H3B) +yC = aA+(aB+- y C) = (a + 1 3+ 7) P, where P is in fact the centroid of masses a, 13, y placed at A, B, C respectively. So, in general, if we put aA+ 1 3B+yC+...+AL = (a+13+y+...+X)X.
X is, in general, a determinate point, the barycentre of aA, 3B, &c. (or of A, B, &c. for the weights a, 0, &c.). If (a+(3+... +X) happens to be zero, X lies at infinity in a determinate direction; unless - aA is the barycentre of (3B, yC, ... XL, in which case aA+0B+ ... +XL vanishes identically, and X is indeterminate. .If ABCD is a tetrahedron of reference, any point P in space is determined by an equation of the form (a+13+ - y+5) P = aA+sB +yC +SD: a, a, y, b are, in fact, equivalent to a set of homogeneous coordinates of P. For constructions in a fixed plane three points of reference are sufficient.^ Write equation in point-slope form.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

^ This Is a simple program that uses Cramer's Rule To figure out the intersection point of two linear equations and provides you with the determinants and the intersecting point.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

^ It find the slope of two points on a 2D coordinate plane while only taking up 76 bytes.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

It is remarkable that Mobius employs the symbols AB, ABC, Abcd In Their Ordinary Geometrical Sense As Lengths, Areas And Volumes, Except That He Distinguishes Their Sign; Thus Ab = Ba, Abc= Acb, And So On. If He Had Happened To Think Of Them As " Products," He Might Have Anticipated Grassmann'S Discovery Of The Extensive Calculus. .From A Merely Formal Point Of View, We Have In The Barycentric Calculus A Set Of " Special Symbols Of Quantity " Or " Extraordinaries " A, B, C, &C., Which Combine With Each Other By Means Of Operations And Which Obey The Ordinary Rules, And With Ordinary Algebraic Quantities By Operations X And =, Also According To The Ordinary Rules, Except That Division By An Extraordinary Is Not Used.^ This, of course, is unintelligible on the grouping system of treating number; on the counting system it merely means that we count backwards from o, just as we might count inches backwards from a point marked o on a scale.

^ A set of written symbols is sometimes read in more than one way, while on the other hand two different sets of symbols (at any rate if denoting numerical quantities) may be read in the same way.

^ A B C of numerical quantities, merely correspond with each other, the correspondence being the result of some relation.

Eo
El
E2
E3
E L
Eo
E3
E2
E 2
E3
Eo
El
E 3
E2
E;
Eo
6. A quaternion is best defined as a symbol of the type q = Za s e s = aoeo + ales = ale, + a3e3, where eo, ... e 3 are independent extraordinaries and a o, ... a 3 ordinary algebraic quantities, which ma S' g q ? may be called the co-ordinates of q. The sum and product of two quaternions are defined by the formulae mi ase + F+lases = (a s + 133) es 2arer X ZO,es = Fiarfseres, where the products e,e, are further reduced according to the following multiplication table, in which, for example, the eo e1 e2 e3 second line is to be read eieo = e1, e 1 2 = - eo, e i e 2 = es, eie3 = - e2.
.The effect of these definitions is that the sum and the product of two quaternions are also quaternions; that addition is associative and commutative; and that multiplication is associative and distributive, but not commutative.^ Or we might say that, since multiplication is a form of addition, and division a form of subtraction, there are really only two fundamental processes, viz.

^ To multiply two decimals exactly, we multiply them as if the point were absent, and then insert it so that the number of figures after the point in the product shall be equal to the sum of the numbers of figures after the points in the original decimals.

^ Distributive Law, that multiplications and divisions may be distributed over additions and subtractions, e.g.

Thus e 1 e 2 = - e2ei, and if q, q are any two quaternions, qq is generally different from q'q. The symbol e 0 behaves exactly like i in ordinary algebra; Hamilton writes I, i, j, k instead of eo, el, e2, es, and in this notation all the special rules of operation may he summed up by the equalities = - I. Putting q=a+,61+yj+bk, Hamilton calls a the scalar part of q, and denotes it by Sq; he also writes Vq for 01+yj+b � , which is called the vector part of q. Thus every quaternion may be written in the form q = Sq+Vq, where either Sq or Vq may separately vanish; so that ordinary algebraic quantities (or scalars, as we shall call them) and pure vectors may each be regarded as special cases of quaternions.
.The equations q'+x = q and y+q' = q are satisfied by the same quaternion, which is denoted by q - q'. On the other hand, the equations q'x = q and yq' = q have, in general, different solutions.^ X3 and 3X4 mean different things, but the operations which they denote produce the same result.

^ A set of written symbols is sometimes read in more than one way, while on the other hand two different sets of symbols (at any rate if denoting numerical quantities) may be read in the same way.

^ Generally, to find the sum or difference of two or more fractional numbers, we must replace them by other fractional numbers having the same denominator; it is usually most convenient to take as this denominator the L.C.M. of the original fractional numbers (cf.

It is the value of y which is generally denoted by q= q'; a special symbol for x is desirable, but has not been established. .If we put qo= Sq' - Vq', then qo is called the conjugate of q', and the scalar q'qo = qoq' is called the norm of q' and written Nq'. With this notation the values of x and y may be expressed in the forms x q q /N q ', gg /Nq', which are free from ambiguity, since scalars are commutative with quaternions.^ The representation of numbers by spoken sounds is called numeration; their representation by written signs is called notation.

^ A fraction written in this way is called a decimal fraction; or we might define a decimal fraction as a fraction having a power of To for its denominator, there being a special notation for writing such fractions.

^ If we write 74 in the form 47 we may say that the value of a fraction is not altered by multiplying or dividing the numerator and denominator by any number.

The values of x and y are different, unless V (qq o) = o.
.In the applications of the calculus the co-ordinates of a quaternion are usually assumed to be numerical; when they are complex, the quaternion is further distinguished by Hamilton as a biquaternion. Clifford's biquaternions are quantities Eq+nr, where q, r are quaternions, and E, n are symbols (commutative with quaternions) obeying the laws E 2 = E, n 2 =,g, = 1 j E=0 (cf.^ Perhaps they have in mind that floating-point numbers model real numbers and should obey the same laws that real numbers do.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ Applications to Physics are numerous, but are usually only of special interest.

^ Simple " practice involves an application of the commutative law.

Quaternions).
7. In the extensive calculus of the nth category, we have, first of all, n independent " units," el, e2, ... e,,. From these are derived symbols of the type A i = ales+a2e2+... +a,,en =t ae, which we shall call extensiveuantities of the first species q f ? (and, when necessary, of the nth category). The co- a l, ... a n are scalars, and in particular applications may be restricted to real or complex numerical values. If B 1 = E(3e, there is a law of addition expressed by A1+B1 = (a, +�ei =B1+Ai; this law of addition is associative as well as commutative. The inverse operation is free from ambiguity, and, in fact, A 1 - B 1 = E (a i - 130ei.
To multiply A 1 by a scalar, we apply the rule A = A1E = E (Eat) ea, and similarly for division by a scalar.
.All this is analogous to the corresponding formulae in the barycentric calculus and in quaternions; it remains to consider the multiplication of two or more extensive quantities The binary products of the units i are taken to satisfy the equalities e, 2 =o, i ej = - eeei; this reduces them to.^ The result of the multiplication is called the product of the unit by the number of times it is taken.

^ Correspondence of Numerical Quantities.-Again, in § 89, we have considered cases of multiple-tables of numerical quantities, where each quantity in one series is equivalent to the corresponding quantity in the other series.

^ If as our unit we take i of A = 1% of A, the above quantity might equally be written 2376 X = 21:367-0 s_.

1n(n - I) distinct values, exclusive of zero. .These values are assumed to be independent, so we have 2n(n - I) derived units of the second species or order. Associated with these new units there is a system of extensive quantities of the second species, represented by symbols of the type = Ei 121 [i=I, 2,...zn(n - I)]. where E 1 (2) ,E2 2), &c., are the derived units of the second species.^ These special values are all encoded with exponents of either e max  + 1 or e min - 1 (it was already pointed out that 0 has an exponent of e min - 1).
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ On single/double systems, four of the five options listed above coincide, and there is no need to differentiate fast and exact width types.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ You don't have to enter the terms in any special order, and there can be several terms of the same degree.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

If A 1 = X a i e i, B i = /if i e i, the distributive law of multiplication is preserved by assuming A1B1=E(a0 i 3)eiej; it follows that A 1 B 1 = - B 1 A 1, and that A l 2 = o.
.By assuming the truth of the associative law of multiplication, and taking account of the reducing formulae for binary products, - 'el ' 'e2 ' 'e3 we may construct derived units of the third, fourth ...^ Similarly we may take the farthing as a unit, and invent smaller units, represented either by tokens or by no material objects at all.

^ Hence we may insert in the power-series numbers with fractional indices, provided that the multiplication of these numbers follows the same law.

^ If we take several different units, and write down their successive multiples in parallel columns, preceded by the numberseries, we obtain a multiple-table such as the following: It is to be considered that each column may extend downwards indefinitely.

nth species. .Every unit of the rth species which does not vanish is the product of
r different units of the first species; two such units are independent unless they are permutations of the same set of primary units e i , in which case they are equal or opposite according to the usual rule employed in determinants.^ For example sums are a special case of inner products, and the sum ((2 × 10 -30 + 10 30 ) - 10 30 ) - 10 -30 is exactly equal to 10 -30 , but on a machine with IEEE arithmetic the computed result will be -10 -30 .
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ This Is a simple program that uses Cramer's Rule To figure out the intersection point of two linear equations and provides you with the determinants and the intersecting point.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

^ One can prove that the sum, difference, product, or quotient of two p -bit numbers, or the square root of a p -bit number, rounded first to q bits and then to p bits gives the same value as if the result were rounded just once to p bits provided q 2 p + 2.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

Thus, for instance
ei.e2e3 = l e 2 .e 3 = ele2e3 = - e2eie3 = e2e3e1; and, in general, the number of distinct units of the rth species in the nth category (r<n) is C n, r. Finally, it is assumed that (in the nth category)
e' 1 e 2 e 3 ... e n = I, the suffixes being in their natural order.
.Let A n =ZaE (r) and B 8 =ZOE' 8) be two extensive quantities of species r and s; then if +s<n, they may be multiplied by the rule ArB, =E(0)E(r)E(8) where the products E (r) E) may be expressed as derived units of species (r+s).^ Comparison, Addition and Subtraction of Fractions.-The quantities 4 of A and 7 of A are expressed in terms of different units.

^ S. The simplest case, in which the quantity can be expressed as an integral number of the largest units B involved, has already been considered (§§ 37, 42).

^ In order to apply arithmetical processes to a quantity expressed in two or more denominations, we must first express it in terms of a single denomination by means of a varying scale of notation.

The product B 8 A r is equal or opposite to ArB8, according as rs is even or odd. .This process may be extended to the product of three or more factors such as A r B B C t ..^ There is a more interesting use for trap handlers that comes up when computing products such as that could potentially overflow.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ The process depends on (ii) of § 45, in the extended form that, if x is a factor of a and b, it is a factor of pa-qb, where p and q are any integers.

^ But we cannot apply it to finding the L.C.M. of three or more numbers; if we cannot resolve the numbers into their prime factors, we must find the L.C.M. of the first two, then the L.C.M. of this and the next number, and so on.

� provided that
r+s+t+ ... does not exceed n. The law is associative; thus, for instance, (AB)C=A(BC). But the commutative law does not always hold; thus, indicating species, as before, by suffixes, A r B B C t =(-1) ra+8t+tr C t B B A r, with analo gous rules for other cases.
.If r+s>n, a product such as E r E 3, worked out by the previous rules, comes out to be zero.^ There is a more interesting use for trap handlers that comes up when computing products such as that could potentially overflow.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ Linear Works out Linear equations just about any way they come.with one exception....
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

^ So all you have to do is enter the the numbers for value A, B, and C. Then, it will compute the answers for you following the rules, such as dividing out all common factors.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

.A characteristic feature of the calculus is that a meaning can be attached to a symbol of this kind by adopting a new rule, called that of regressive multiplication, as distinguished from the foregoing, which is progressive. The new rule requires some preliminary explanation.^ The combination of features required or recommended by the C99 standard supports some of the five options listed above but not all.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ The C99 standard improves predictability to some degree at the expense of requiring programmers to write multiple versions of their programs, one for each FLT_EVAL_METHOD .
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

.If E is any extensive unit, there is one other unit E', and only one, such that the (progressive) product EE' =- r.^ There is a more interesting use for trap handlers that comes up when computing products such as that could potentially overflow.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ This program also features other such equations where theres are unkowns on both sides of the equal sign.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

^ This has one use that I can think of, Algebra 2 & factoring trinomials (what multiplies to AC and adds to B), but I'm sure there are others.
  • TI-83/84 Plus BASIC Math Programs (Algebra) - ticalc.org 10 February 2010 11:011 UTC www.ticalc.org [Source type: FILTERED WITH BAYES]

This unit is called the supplement of E, and denoted by IE. For example, when n= 4, I e1 = e2e3e4, Ieie2 = e3e4, 1e2e3e4 = - ei, and so on. Now when r+s>n, the product E r E B is defined to be that unit of which the supplement is the progressive product IE,4E 8. For instance, if n= 4, E r = e l e 3, Es= e 2 e 3 e 4 , we have IErIE8 = (-e2e4) (- e 1) = ele2e4 = l e3, consequently, by the rule of regressive multiplication, eie3�e2e3e4 = e3.
Applying the distributive law, we obtain, when r+s>n, ArB 8 = EaErE i 3E 8 = E(a(3)ErEs, where the regressive products E r E B are to be reduced to units of species (r+s-n) by the foregoing rule.
If A=ZaE, then, by definition, IA=EajE, and hence AI(B+C) =AIB+AIC.
.Now this is formally analogous to the distributive law of multiplication; and in fact we may look upon AFB as a particular way of multiplying A and B (not A and B).^ (A fused multiply-add can also foil the splitting process of Theorem 6, although it can be used in a non-portable way to perform multiple precision multiplication without the need for splitting.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

The symbol AB, from this point of view, is called the inner product of A and B, as distinguished from the outer product IAB. An inner product may be either progressive or regressive. In the course of reducing such expressions as (AB)C, (AB){C(DE)} and the like, where a chain of multiplications has to be performed in a certain order, the multiplications may be all progressive, or all regressive, or partly, one, partly the other. .In the first two cases the product is said to be pure, in the third case mixed. A pure product is associative; a mixed product, speaking generally, is not.^ In either case, the first rounding of the product will deliver a result that differs from m by at most 1/2, and again by previous arguments, the second rounding will round to m .
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ In either case, the first rounding of the product will deliver a result that differs from m by at most 1/4, and by previous arguments, the second rounding will round to m .
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ In order to produce the exactly rounded product of two p -digit numbers, a multiplier needs to generate the entire 2 p bits of product, although it may throw bits away as it proceeds.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

.The outer and inner products of two extensive quantities A, B, are in many ways analogous to the quaternion symbols Vab and Sab respectively.^ Theorem 6 gives a way to express the product of two working precision numbers exactly as a sum.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ Multiplying two quantities with a small relative error results in a product with a small relative error (see the section Rounding Error ).
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

.As in quaternions, so in the extensive calculus, there are numerous formulae of transformation which enable us to deal with extensive quantities without expressing them in terms of the primary units.^ The answer is that it does matter, because accurate basic operations enable us to prove that formulas are "correct" in the sense they have a small relative error.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ Unlike the quadratic formula, this improved form still has a subtraction, but it is a benign cancellation of quantities without rounding error, not a catastrophic one.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ There is a companion formula for expressing a sum exactly.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

Only a few illustrations can be given here, Let a, b, c, d, e, be quantities of the first species in the fourth category; A, B, C ... quantities of the third species in the same category. Then (de)(abc) = (abde)c+(cade)b+(bcde)a = (abce)d - (abcd)e, (ab) (AB) = (aA) (bB) - (aB) (bA) abic = (alc) b - (bjc)a, (ablcd) = (ajc) (bjd) - (af d) (bIc). These may be compared and contrasted with such quaternion formulae as S(VabVcd) =SadSbc-SacSbd dSabc = aSbcd - bScda+cSadb where a, b, c, d denote arbitrary vectors.
.8. An n-tuple linear algebra (also called a complex number system) deals with quantities of the type A=/aiei derived from n special units e l, e 2 ...^ An algorithm that involves thousands of operations (such as solving a linear system) will soon be operating on numbers with many significant bits, and be hopelessly slow.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ On the other hand, the VAX TM reserves some bit patterns to represent special numbers called reserved operands .
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ Computer Solution of Linear Algebraic Systems , Prentice-Hall, Englewood Cliffs, NJ. .
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

en.
The sum and product of two quantities are defined in the first instance by the formulae zae -IE(3e = E (a +0) e, Za,ei X E ai e j = (a iai) eie9, so that the laws A, C, D of � 3 are satisfied. The binary products e i e j , however, are expressible as linear functions of the units e i by means of a " multiplication table " which defines the special characteristics of the algebra in question. Multiplication may or may not be commutative, and in the same way it may or may not be associative. The types of linear associative algebras, not assumed to be commutative, have been enumerated (with some omissions) up to sextuple algebras inclusive by B. Pierce. Quaternions afford an example of a quadruple algebra of this kind; ordinary algebra is a special case of a duplex linear algebra. If, in the extensive calculus of the nth category, all the units (including i and the derived units E) are taken to be homologous instead of being distributed into species, we may regard it as a (2'-I)-tuple linear algebra, which, however, is not wholly associative. It should be observed that while the use of special units, or extraordinaries, in a linear algebra is convenient, especially in applications, it is not indispensable. Any linear quantity may be denoted by a symbol (a 1, a2, ... a n) in which only its scalar coefficients occur; in fact, the special units only serve, in the algebra proper, as umbrae or regulators of certain operations on scalars (see Number). .This idea finds fuller expression in the algebra of matrices, as to which it must suffice to say that a matrix is a symbol consisting of a rectangular array of scalars, and that matrices may be combined by a rule of addition which obeys the usual laws, and a rule of multiplication which is distributive and associative, but not, in general, commutative.^ When a multiplication or division involves a signed zero, the usual sign rules apply in computing the sign of the answer.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ Of course, to find this solution, the programmer must know that double expressions may be evaluated in extended precision, that the ensuing double-rounding problem can cause the algorithm to malfunction, and that extended precision may be used instead according to Theorem 14.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ For a more complicated program, it may be impossible to systematically account for the effects of double-rounding, not to mention more general combinations of double and extended double precision computations.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

Various special algebras (for example, quaternions) may be expressed in the notation of the algebra of matrices.
9. In ordinary algebra we have the disjunctive law that if ab = o, then either a = o or b= o. This applies also to quaternions, but not to extensive quantities, nor is it true for linear algebras in general. .One of the most important questions in investigating a linear algebra is to decide the necessary relations between a and b in order that this product may be zero.^ In order to produce the exactly rounded product of two p -digit numbers, a multiplier needs to generate the entire 2 p bits of product, although it may throw bits away as it proceeds.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ When only the order of magnitude of rounding error is of interest, ulps and may be used interchangeably, since they differ by at most a factor of .
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

Io. .The algebras discussed up to this point may be considered as independent in the sense that each of them deals with a class of symbols of quantity more or less homogeneous, and a set of operations applying to them all.^ First of all, there are algebraic identities that are valid for floating-point numbers.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ This is probably because designers like "orthogonal" instruction sets, where the precisions of a floating-point instruction are independent of the actual operation.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ Rather than using all these digits, floating-point hardware normally operates on a fixed number of digits.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

.But when an algebra is used with a particular interpretation, or even in the course of its formal development, it frequently happens that new symbols of operation are, so to speak, superposed upon the algebra, and are found to obey certain formal laws of combination of their own.^ But accurate operations are useful even in the face of inexact data, because they enable us to establish exact relationships like those discussed in Theorems 6 and 7.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ This example suggests that when using the round up rule, computations can gradually drift upward, whereas when using round to even the theorem says this cannot happen.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ Many programmers may not realize that even a program that uses only the numeric formats and operations prescribed by the IEEE standard can compute different results on different systems.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

.For instance, there are the symbols A, D, E used in the calculus of finite differences; Aronhold's symbolical method in the calculus of invariants; and the like.^ The fact is that there are useful algorithms (like the Kahan summation formula) that exploit the fact that ( x + y ) + z x + ( y + z ), and work whenever the bound .
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

.In most cases these subsidiary algebras, as they may be called, are inseparable from the applications in which they are used; but in any attempt at a natural classification of algebra (at present a hopeless task), they would have to be taken into account.^ The most useful of these are the portable algorithms for performing simulated multiple precision arithmetic mentioned in the section Exactly Rounded Operations .
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ Those explanations that are not central to the main argument have been grouped into a section called "The Details," so that they can be skipped if desired.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ When only the order of magnitude of rounding error is of interest, ulps and may be used interchangeably, since they differ by at most a factor of .
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

Even in ordinary algebra the notation for powers and roots disturbs the symmetry of the rational theory; and when a schoolboy illegitimately extends the distributive law by writing -V (a+b)a+J b, he is unconsciously emphasizing this want of complete harmony.

Authorities

- A. de Morgan, " On the Foundation of Algebra," Trans. Camb. P.S. (vii., viii., 1839-1844); G. Peacock, Symbolical Algebra (Cambridge, 1845); G. Boole, Laws of Thought (London, 1854); E. Schroder, Lehrbuch der Arithmetik u. Algebra (Leipzig, 1873), Vorlesungen fiber die Algebra der Logik (ibid., 1890-1895); A. F. Mobius, Der barycentrische Calcul (Leipzig, 1827) (reprinted in his collected works, vol. i., Leipzig, 1885); W. R. Hamilton, Lectures on Quaternions (Dublin, 1853), Elements of Quaternions (ibid., 1866); H. Grassmann, Die lineale Ausdehnungslehre (Leipzig, 1844), Die Ausdehnungslehre (Berlin, 1862) (these are reprinted with valuable emendations and notes in his Gesammelte math. u. phys. Werke, vol. i., Leipzig (2 parts), 1894, 1896), and papers in Grunert's Arch. vi., Crelle, xlix. lxxxiv., Math. Ann. vii. xii.; B. and C. S. Peirce, " Linear Associative Algebra," Amer. Journ. Math. iv. (privately circulated, 1871); A. Cayley, on Matrices, Phil. Trans. cxlviii., on Multiple Algebra, Quart. M. Journ. xxii.; J. J. Sylvester, on Universal Algebra (i.e. Matrices), Amer. Journ. Math. vi.; H. J. S. Smith, on Linear Indeterminate Equations, Phil. Trans. cli.; R. S. Ball, Theory of Screws (Dublin, 1876); and papers in Phil. Trans. clxiv., and Trans. R. Ir. Ac. xxv.; W. K. Clifford, on Biquaternions, Proc. L. M. S. iv.; A. Buchheim, on Extensive Calculus and its Applications, Proc. L. M. S. xv.-xvii.; H. Taber, on Matrices, Amer. J. M. xii.; K. Weierstrass, " Zur Theorie der aus n Haupteinheiten gebildeten complexen Grossen," Gotting. Nachr. (1884); G. Frobenius, on Bilinear Forms, Crelle, lxxxiv., and Berl. Ber. (1896); L. Kronecker, on Complex Numbers and Modular Systems, Berl. Ber. (1888); G. Scheffers, "Complexe Zahlensysteme," Math. Ann. xxxix. (this contains a (biblio raphy up to 1890); S. Lie, Vorlesungen fiber continuirliche Gruppen (Leipzig, 1893), ch. xxi.; A. M'Aulay, " Algebra after Hamilton, or Multenions," Proc. R. S. E., 1908, 28, p. 503. For a more complete account see H. Hankel Theorie der complexen Zahlensysteme (Leipzig, 1867); O. Stolz, Vorlesungen fiber allgemeine Arithmetik (ibid., 1883); A. N. Whitehead, A Treatise on Universal Algebra, with Applications (vol. i., Cambridge, 1898) (a very comprehensive work, to which the writer of this article is in many ways indebted); and the Encyclopadie d. math. Wissenschaften (vol. i., Leipzig, 1898), &c., �� A I (H. Schubert), A 4 (E. Study), and B I c (G. Landsberg). For the history of the development of ordinary algebra M. Cantor's Vorlesungen fiber Geschichte der Mathematik is the standard authority. (G. B. M.) C. History Various derivations of the word " algebra," which is of Arabian origin, have been given by different writers. The first mention of the word is to be found in the title Ety= of a work by Mahommed ben Musa al-Khwarizmi (Hovarezmi), who flourished about the beginning of the 9th century. The full title is ilm al jebr wa'l-mugabala, which contains the ideas of restitution and comparison, or opposition and comparison, or resolution and equation, jebr being derived from the verb jabara, to reunite, and mugabala, from gabala, to make equal. (The root jabara is also met with in the word algebrista, which means a " bone-setter," and is still in common use in Spain.) The same derivation is given by Lucas Paciolus (Luca Pacioli), who reproduces the phrase in the transliterated form alghebra e almucabala, and ascribes the invention of the art to the Arabians.
Other writers have derived the word from the Arabic particle al (the definite article), and geber, meaning " man." Since, however, Geber happened to be the name of a celebrated Moorish philosopher who flourished in about the iith or 12th century, it has been supposed that he was the founder of algebra, which has since perpetuated his name. The evidence of Peter Ramus (1515-1572) on this point is interesting, but he gives no authority for his singular statements. In the preface to his Arithmeticae libri duo et totidem Algebrae (1560) he says: " The name Algebra is Syriac, signifying the art or doctrine of an excellent man. For Geber, in Syriac, is a name applied to men, and is sometimes a term of honour, as master or doctor among us. There was a certain learned mathematician who sent his algebra, written in the Syriac language, to Alexander the Great, and he named it almucabala, that is, the book of dark or mysterious things, which others would rather call the doctrine of algebra. To this day the same book is in great estimation among the learned in the oriental nations, and by the Indians, who cultivate this art, it is called aljabra and alboret; though the name of the author himself is not known." .The uncertain authority of these statements, and the plausibility of the preceding explanation, have caused philologists to accept the derivation from al and jabara. Robert Recorde in his Whetstone of Witte (1557) uses the variant algeber, while John Dee (1527-1608) affirms that algiebar, and not algebra, is the correct form, and appeals to the authority of the Arabian Avicenna.^ These are useful even if every floating-point variable is only an approximation to some actual value.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ A simple way to correct for this is to store the partial summand in a double precision variable and to perform each addition using double precision.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

.Although the term " algebra " is now in universal use, various other appellations were used by the Italian mathematicians during the Renaissance.^ Depending on the programming language being used, the trap handler might be able to access other variables in the program as well.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

.Thus we find Paciolus calling it l'Arte Magiore; ditta dal vulgo la Regula de la Cosa over Alghebra e Almucabala. The name l'arte magiore, the greater art, is designed to distinguish it from l'arte minore, the lesser art, a term which he applied to the modern arithmetic.^ Thus, on these systems, we can't predict the behavior of the program simply by reading its source code and applying a basic understanding of IEEE 754 arithmetic.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

His second variant, la regula de la cosa, the rule of the thing or unknown quantity, appears to have been in common use in Italy, and the word cosa was preserved for several centuries in the forms toss or algebra, cossic or algebraic, cossist or algebraist, &c. Other Italian writers termed it the Regula rei et census, the rule of the thing and the product, or the root and the square. .The principle underlying this expression is probably to be found in the fact that it measured the limits of their attainments in algebra, for they were unable to solve equations of a higher degree than the quadratic or square.^ In fact, the ANSI C standard explicitly allows a compiler to evaluate a floating-point expression to a precision wider than that normally associated with its type.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

.Franciscus Vieta (Francois Viete) named it Specious Arithmetic, on account of the species of the quantities involved, which he represented symbolically by the various letters of the alphabet.^ The section Base explained that e min - 1 is used for representing 0, and Special Quantities will introduce a use for e max + 1.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

.Sir Isaac Newton introduced the term Universal Arithmetic, since it is concerned with the doctrine of operations, not affected on numbers, but on general symbols.^ Since the result of an operation in interval arithmetic is an interval, in general the input to an operation will also be an interval.
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^ This term was introduced by Forsythe and Moler [1967], and has generally replaced the older term mantissa .
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ A number of claims have been made in this paper concerning properties of floating-point arithmetic.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

Notwithstanding these and other idiosyncratic appellations, European mathematicians have adhered to the older name, by which the subject is now universally known.
It is difficult to assign the invention of any art or science definitely to any particular age or race. The few fragmentary records, which have come down to us from past civilizations, must not be regarded as representing the totality of their knowledge, and the omission of a science or art does not necessarily imply that the science or art was unknown. .It was formerly the custom to assign the invention of algebra to the Greeks, but since the decipherment of the Rhind papyrus by Eisenlohr this view has changed, for in this work there are distinct signs of an algebraic analysis.^ Since the sign bit can take on two different values, there are two zeros, +0 and -0.
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.The particular problem - a heap (hau) and its seventh makes 19 - is solved as we should now solve a simple equation; but Ahmes varies his methods in other similar problems.^ Consider the problem of solving a system of linear equations, .
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

This discovery carries the invention of algebra back to about 1700 B.C., if not earlier.
It is probable that the algebra of the Egyptians was of a most rudimentary nature, for otherwise we should expect to find traces of it in the works of the Greek geometers, of whom Thales of Miletus (640-546 B.C.) was the first.
.Notwithstanding the prolixity of writers and the number of the writings, all attempts at extracting an algebraic analysis from their geometrical theorems and problems have been fruitless, and it is generally conceded that their analysis was geometrical and had little or no affinity to algebra.^ First of all, there are algebraic identities that are valid for floating-point numbers.
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^ We will verify the theorem when no guard digits are used; the general case is similar.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ In the = 16, p = 1 system, all the numbers between 1 and 15 have the same exponent, and so no shifting is required when adding any of the ( ) = 105 possible pairs of distinct numbers from this set.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

The first extant work which approaches to a treatise on algebra is by Diophantus, an Alexandrian mathematician, who flourished about A.D. 350. The original, which consisted of a preface and thirteen books, is not lost, but we have a Latin translation of the first six books and a fragment of another on polygonal numbers by Xylander of Augsburg (1575), and Latin and Greek translations by Gaspar Bachet de Merizac (1621-1670). Other editions have been published, of which we may mention Pierre Fermat's (1670), T. L. Heath's (1885) and P. Tannery's (1893-1895). In the preface to this work, which is dedicated to one Dionysius, Diophantus explains his notation, naming the square, cube and fourth powers, dynamis, cubus, dynamodinimus, and so on, according to the sum in the indices. .The unknown he terms arithmos, the number, and in solutions he marks it by the final s; he explains the generation of powers, the rules for multiplication and division of simple quantities, but he does not treat of the addition, subtraction, multiplication and division of compound quantities.^ The IEEE standard requires that the result of addition, subtraction, multiplication and division be exactly rounded.
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^ When a multiplication or division involves a signed zero, the usual sign rules apply in computing the sign of the answer.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ In the second expression these are exact (i.e., x ), and the final division commits just one additional rounding error.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

.He then proceeds to discuss various artifices for the simplification of equations, giving methods which are still in common use.^ The term IEEE Standard will be used when discussing properties common to both standards.
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.In the body of the work he displays considerable ingenuity in reducing his problems to simple equations, which admit either of direct solution, or fall into the class known as indeterminate equations.^ Many problems, such as numerical integration and the numerical solution of differential equations involve computing sums with many terms.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

.This latter class he discussed so assiduously that they are often known as Diophantine problems, and the methods of resolving them as the Diophantine analysis (see Equation, Indeterminate). It is difficult to believe that this work of Diophantus arose spontaneously in a period of general stagnation.^ Many programmers like to believe that they can understand the behavior of a program and prove that it will work correctly without reference to the compiler that compiles it or the computer that runs it.
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^ Another way to see this is to try and duplicate the analysis that worked on ( x y ) ( x y ), yielding .
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

.It is more than likely that he was indebted to earlier writers, whom he omits to mention, and whose works are now lost; nevertheless, but for this work, we should be led to assume that algebra was almost, if not entirely, unknown to the Greeks.^ We are now in a position to answer the question, Does it matter if the basic arithmetic operations introduce a little more rounding error than necessary?
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.The Romans, who succeeded the Greeks as the chief civilized power in Europe, failed to set store on their literary and scientific treasures; mathematics was all but neglected; and beyond a few improvements in arithmetical computations, there are no material advances to be recorded.^ Although most modern computers have a guard digit, there are a few (such as Cray systems) that do not.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ Although there are infinitely many integers, in most programs the result of integer computations can be stored in 32 bits.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ However, there is a small snag, because the computation of (1 - x )/(1 + x ) will cause the divide by zero exception flag to be set, even though arccos(-1) is not exceptional.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

In the chronological development of our subject we have now to turn to the Orient. Investigation of the writings of Indian mathematicians has exhibited a fundamental distinction between the Greek and Indian mind, the former being pre-eminently geometrical and speculative, the latter arithmetical and mainly practical. We find that geometry was neglected except in so far as it was of service to astronomy; trigonometry was advanced, and algebra improved far beyond the attainments of Diophantus.
The earliest Indian mathematician of whom we have certain knowledge is Aryabhatta, who flourished about the beginning of the 6th century of our era. The fame of this astronomer and mathematician rests on his work, the Aryabhattiyam, the third chapter of which is devoted to mathematics. Ganessa, an eminent astronomer, mathematician and scholiast of Bhaskara, quotes this work and makes separate mention of the cuttaca (" pulveriser "), a device for effecting the solution of indeterminate equations. .Henry Thomas Colebrooke, one of the earliest modern investigators of Hindu science, presumes that the treatise of Aryabhatta extended to determinate quadratic equations, indeterminate equations of the first degree, and probably of the second.^ On hardware that can do an add and multiply in parallel, an optimizer would probably move the addition operation ahead of the second multiply, so that the add can proceed in parallel with the first multiply.
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^ This can happen when the result as rounded to extended double precision is a "halfway case", i.e., it lies exactly halfway between two double precision numbers, so the second rounding is determined by the round-ties-to-even rule.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

An astronomical work, called the Surya-siddhanta (" knowledge of the Sun "), of uncertain authorship and probably belonging to the 4th or 5th century, was considered of great merit by the Hindus, who ranked it only second to the work of Brahmagupta, who flourished about a century later. It is of great interest to the historical student, for it exhibits the influence of Greek science upon Indian mathematics at a period prior to Aryabhatta. After an interval of about a century, during which mathematics attained its highest level, there flourished Brahmagupta (b. A.D. 598), whose work entitled Brahma-sphuta-siddhanta (" The revised system of Brahma ") contains several chapters devoted to mathematics. Of other Indian writers mention may be made of Cridhara, the author of a Ganita-sara (" Quintessence of Calculation "), and Padmanabha, the author of an algebra.
A period of mathematical stagnation then appears to have possessed the Indian mind for an interval of several centuries, for the works of the next author of any moment stand but little in advance of Brahmagupta. We refer to Bhaskara Acarya, whose work the Siddhanta-ciromani (" Diadem of an Astronomical System "), written in 1150, contains two important chapters, the Lilavati (" the beautiful [science or art] ") and Viga-ganita (" root-extraction "), which are given up to arithmetic and algebra.
English translations of the mathematical chapters of the Brahma-siddhanta and Siddhanta-ciromani by H. T. Colebrooke (1817), and of the Surya-siddhanta by E. Burgess, with annotations by W. D. Whitney (1860), may be consulted for details.
The question as to whether the Greeks borrowed their algebra from the Hindus or vice versa has been the subject of much discussion. .There is no doubt that there was a constant traffic between Greece and India, and it is more than probable that an exchange of produce would be accompanied by a transference of ideas.^ Although formula (7) is much more accurate than (6) for this example, it would be nice to know how well (7) performs in general.
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^ There is more than one way to split a number.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ Even if there are over/underflows, the calculation is more accurate than if it had been computed with logarithms, because each p k was computed from p k - 1 using a full precision multiply.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

.Moritz Cantor suspects the influence of Diophantine methods, more particularly in the Hindu solutions of indeterminate equations, where certain technical terms are, in all probability, of Greek origin.^ Many problems, such as numerical integration and the numerical solution of differential equations involve computing sums with many terms.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

However this may be, it is certain that the Hindu algebraists were far in advance of Diophantus. .The deficiencies of the Greek symbolism were partially remedied; subtraction was denoted by placing a dot over the subtrahend; multiplication, by placing bha (an abbreviation of bhavita, the product ") after the factors; division, by placing the divisor under the dividend; and square root, by inserting ka (an abbreviation of karana, irrational) before the quantity.^ Under IBM System/370 FORTRAN, the default action in response to computing the square root of a negative number like -4 results in the printing of an error message.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ It gives an algorithm for addition, subtraction, multiplication, division and square root, and requires that implementations produce the same result as that algorithm.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

The unknown was called yavattavat, and if there were several, the first took this appellation, and the others were designated by the names of colours; for instance, x was denoted by y�nd y by lea (from kalaka, black).
.A notable improvement on the ideas of Diophantus is to be found in the fact that the Hindus recognized the existence of two roots of a quadratic equation, but the negative roots were considered to be inadequate, since no interpretation could be found for them.^ Thanks to signed zero, x will be negative, so log can return a NaN. However, if there were no signed zero, the log function could not distinguish an underflowed negative number from 0, and would therefore have to return - .
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

It is also supposed that they anticipated discoveries of the solutions of higher equations. Great advances were made in the study of indeterminate equations, a branch of analysis in which Diophantus excelled. But whereas Diophantus aimed at obtaining a single solution, the Hindus strove for a general method by which any indeterminate problem could be resolved. .In this they were completely successful, for they obtained general solutions for the equations ax by = c, xy = ax+by+c (since rediscovered by Leonhard Euler) and cy 2 = ax e + b. A particular case of the last equation, namely, y 2 = ax e + 1, sorely taxed the resources of modern algebraists.^ This example illustrates a general fact, namely that infinity arithmetic often avoids the need for special case checking; however, formulas need to be carefully inspected to make sure they do not have spurious behavior at infinity (as x /( x 2  + 1) did).
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^ In general, the computation of and y will incur rounding error, so Ay         Ax (1)  -  b  =  A ( x (1) - x ), where x is the (unknown) true solution.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

It was proposed by Pierre de Fermat to Bernhard Frenicle de Bessy, and in 1657 to all mathematicians. .John Wallis and Lord Brounker jointly obtained a tedious solution which was published in 1658, and afterwards in 1668 by John Pell in his Algebra. A solution was also given by Fermat in his Relation. Although Pell had nothing to do with the solution, posterity has termed the equation Pell's Equation, or Problem, when more rightly it should be the Hindu Problem, in recognition of the mathematical attainments of the Brahmans.^ Many problems, such as numerical integration and the numerical solution of differential equations involve computing sums with many terms.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

.Hermann Hankel has pointed out the readiness with which the Hindus passed from number to magnitude and vice versa. Although this transition from the discontinuous to continuous is not truly scientific, yet it materially augmented the development of algebra, and Hankel affirms that if we define algebra as the application of arithmetical operations to both rational and irrational numbers or magnitudes, then the Brahmans are the real inventors of algebra.^ Several different representations of real numbers have been proposed, but by far the most widely used is the floating-point representation.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ Thus computing mx - ( mx - x ) in floating-point arithmetic precision is exactly equal to rounding x to p - k places, in the case when x + k x does not carry out.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ Perhaps they have in mind that floating-point numbers model real numbers and should obey the same laws that real numbers do.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

The integration of the scattered tribes of Arabia in the 7th century by the stirring religious propaganda of Mahomet was accompanied by a meteoric rise in the intellectual powers of a hitherto obscure race. The Arabs became the custodians of Indian and Greek science, whilst Europe was rent by internal dissensions. Under the rule of the Abbasids, Bagdad became the centre of scientific thought; physicians and astronomers from India and Syria flocked to their court; Greek and Indian manuscripts were translated (a work commenced by the Caliph Mamun (813-833) and ably continued by his successors); and in about a century the Arabs were placed in possession of the vast stores of Greek and Indian learning. Euclid's Elements were first translated in the reign of Harun-al-Rashid (786-809), and revised by the order of Mamun. But these translations were regarded as imperfect, and it remained for Tobit ben Korra (836-901) to produce a satisfactory edition. Ptolemy's Almagest, the works of Apollonius, Archimedes, Diophantus and portions of the Brahmasiddhanta, were also translated. The first notable Arabian mathematician was Mahommed ben Musa al-Khwarizmi, who flourished in the reign of Mamun. His treatise on algebra and arithmetic (the latter part of which is only extant in the form of a Latin translation, discovered in 1857) contains nothing that was unknown to the Greeks and Hindus; it exhibits methods allied to those of both races, with the Greek element predominating. .The part devoted to algebra has the title al-jebr wa'lmugabala, and the arithmetic begins with " Spoken has Algoritmi," the name Khwarizmi or Hovarezmi having passed into the word Algoritmi, which has been further transformed into the more modern words algorism and algorithm, signifying a method of computing.^ A splitting method that is easy to compute is due to Dekker [1971], but it requires more than a single guard digit.
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Tobit ben Korra (836-901), born at Harran in Mesopotamia, an accomplished linguist, mathematician and astronomer, rendered conspicuous service by his translations of various Greek authors. His investigation of the properties of amicable numbers and of the problem of trisecting an angle, are of importance. .The Arabians more closely resembled the Hindus than the Greeks in the choice of studies; their philosophers blended speculative dissertations with the more progressive study of medicine; their mathematicians neglected the subtleties of the conic sections and Diophantine analysis, and applied themselves more particularly to perfect the system of numerals (see Numeral), arithmetic and astronomy.^ Clearly most numerical software does not require more of the arithmetic than that the relative error in each operation is bounded by the "machine epsilon".
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^ Computer system designers rarely get guidance from numerical analysis texts, which are typically aimed at users and writers of software, not at computer designers.
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^ Categories and Subject Descriptors: (Primary) C.0 [Computer Systems Organization]: General -- instruction set design ; D.3.4 [Programming Languages]: Processors -- compilers, optimization ; G.1.0 [Numerical Analysis]: General -- computer arithmetic, error analysis, numerical algorithms (Secondary) .
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It thus came about that while some progress was made in algebra, the talents of the race were bestowed on astronomy and trigonometry. Fahri des al Karhi, who flourished about the beginning of the i 1 th century, is the author of the most important Arabian work on algebra. He follows the methods of Diophantus; his work on indeterminate equations has no resemblance to the Indian methods, and contains nothing that cannot be gathered from Diophantus. .He solved quadratic equations both geometrically and algebraically, and also equations of the form x 2 "+ax n +b=o; he also proved certain relations between the sum of the first n natural numbers, and the sums of their squares and cubes.^ First of all, there are algebraic identities that are valid for floating-point numbers.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ One can prove that the sum, difference, product, or quotient of two p -bit numbers, or the square root of a p -bit number, rounded first to q bits and then to p bits gives the same value as if the result were rounded just once to p bits provided q 2 p + 2.
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^ Every time two n bit numbers with widely spaced exponents are added, the number of bits in the sum is n + the space between the exponents.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

Cubic equations were solved geometrically by determining the intersections of conic sections. .Archimedes' problem of dividing a sphere by a plane into two segments having a prescribed ratio,was first expressed as a cubic equation by Al Mahani, and the first solution was given by Abu Gafar al Hazin.^ Current implementations of IEEE 754 arithmetic can be divided into two groups distinguished by the degree to which they support different floating-point formats in hardware.
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The determination of the side of a regular heptagon which can be inscribed or circumscribed to a given circle was reduced to a more complicated equation which was first successfully resolved by Abul Gud. The method of solving equations geometrically was considerably developed by Omar Khayyam of Khorassan, who flourished in the 1 r th century. This author questioned the possibility of solving cubics by pure algebra, and biquadratics by geometry. .His first contention was not disproved until the 15th century, but his second was disposed of by Abul Wefa (940-998), who succeeded in solving the forms x 4 =a and x4-%ax3=b. Although the foundations of the geometrical resolution of cubic equations are to be ascribed to the Greeks (for Eutocius assigns to Menaechmus two methods of solving the equation x 3 = a and x 3 = 2a 3), yet the subsequent development by the Arabs must be regarded as one of their most important achievements.^ Extended-based systems run most efficiently when expressions are evaluated in extended precision registers whenever possible, yet values that must be stored are stored in the narrowest precision required.
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^ One method of computing the difference between two floating-point numbers is to compute the difference exactly and then round it to the nearest floating-point number.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ Also, because the scaled values of m and q satisfy m /2 < q < 2 m , the corresponding value of n must have one of two forms depending on which of m or q is larger: if q   <  m , then evidently 1 < n < 2, and since n is a sum of two powers of two, n = 1 + 2 - k for some k ; similarly, if q > m , then 1/2 < n < 1, so n = 1/2 + 2 -( k + 1) .
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.The Greeks had succeeded in solving an isolated example; the Arabs accomplished the general solution of numerical equations.^ Many problems, such as numerical integration and the numerical solution of differential equations involve computing sums with many terms.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

Considerable attention has been directed to the different styles in which the Arabian authors have treated their subject. Moritz Cantor has suggested that at one time there existed two schools, one in sympathy with the Greeks, the other with the Hindus; and that, although the writings of the latter were first studied, they were rapidly discarded for the more perspicuous Grecian methods, so that, among the later Arabian writers, the Indian methods were practically forgotten and their mathematics became essentially Greek in character.
Turning to the Arabs in the West we find the same enlightened spirit; Cordova, the capital of the Moorish empire in Spain, was as much a centre of learning as Bagdad. The earliest known Spanish mathematician is Al Madshritti (d. 1007), whose fame rests on a dissertation on amicable numbers, and on the schools which were founded by his pupils at Cordova, Dania and Granada. Gabir ben Aflah of Sevilla, commonly called Geber, was a celebrated astronomer and apparently skilled in algebra, for it has been supposed that the word " algebra " is compounded from his name.
When the Moorish empire began to wane the brilliant intellectual gifts which they had so abundantly nourished during three or four centuries became enfeebled, and after that period they failed to produce an author comparable with those of the 7th to the 11th centuries.
In Europe the decline of Rome was succeeded by a period, lasting several centuries, during which the sciences and arts were all but neglected. Political and ecclesiastical dissensions occupied the greatest intellects, and the only progress to be recorded is in the art of computing or arithmetic, and the trans pons asinorum of the earlier mathematicians. The first step in this direction was made by Scipio q Ferro (d. 1526), who solved the equation x 3 +ax=b. Of his discovery we know nothing except that he declared it to his pupil Antonio Marie Floridas. An imperfect solution of the equation x 3 +-- px 2 was discovered by Nicholas Tartalea (Tartaglia) in 1530, and his pride in this achievement led him into conflict with Floridas, who proclaimed his own knowledge of the form resolved by Ferro. Mutual recriminations led to a public discussion in 1535, when Tartalea completely vindicated the general applicability of his methods and exhibited the inefficiencies of that of Floridas. This contest over, Tartalea redoubled his attempts to generalize his methods, and by 1541 he possessed the means for solving any form of cubic equation. His discoveries had made him famous all over Italy, and he was earnestly solicited to publish his methods; but he abstained from doing so, saying that he intended to embody them in a treatise on algebra which he was preparing. At last he succumbed to the repeated requests of Girolamo or Geronimo Cardano, who swore that he would regard them as an inviolable secret. Cardan or Cardano, who was at that time writing his great work, the Ars Magna, could not restrain the temptation of crowning his treatise with such important discoveries, and in 1 545 he broke his oath and gave to the world Tartalea's rules for solving cubic equations. Tartalea, thus robbed of his most cherished possession, was in despair. Recriminations ensued until his death in 1557, and although he sustained his claim for priority, posterity has not conceded to him the honour of his discovery, for his solution is now known as Cardan's Rule. Cubic equations having been solved, biquadratics soon followed suit. As early as 1539 Cardan had solved certain particular cases, but it remained for his pupil, Lewis (Ludovici) Ferrari, to devise a general method. .His solution, which is sometimes erroneously ascribed to Rafael Bombelli, was published in the Ars Magna. In this work, which is one of the most valuable contributions to the literature of algebra, Cardan shows that he was familiar with both real positive and negative roots of equations whether rational or irrational, but of imaginary roots he was quite ignorant, and he admits his inability to resolve the so-called lation of Arabic manuscripts.^ However, square root is continuous if a branch cut consisting of all negative real numbers is excluded from consideration.
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^ In that case, if x is small but not quite small enough that 1.0   +   x rounds to 1.0 in single precision, then the value returned by log1p(x) can exceed the correct value by nearly as much as x , and again the relative error can approach one.
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^ Similarly, if p k underflows, the counter would be decremented, and negative exponent would get wrapped around into a positive one.
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The first successful attempt to revive the study of algebra in Christendom was due to Leonardo of Pisa, an Italian merchant trading in the Mediterranean. .His travels and mercantile experience had led E t u eopre him to conclude that the Hindu methods of computing were in advance of those then in general use, and in 1202 he published his Liber Abaci, which treats of both algebra and arithmetic.^ A better method of computing the quotients is to use Smith's formula: .
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^ When the preceding code is compiled by the Sun WorkShop Compilers 4.2.1 Fortran 77 compiler for x86 systems using the -O optimization flag, the generated code computes 1.0   +   x exactly as described.
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^ If both operands are NaNs, then the result will be one of those NaNs, but it might not be the NaN that was generated first.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

In this work, which is of great historical interest, since it was published about two centuries before the art of printing was discovered, he adopts the Arabic notation for numbers, and solves many problems, both arithmetical and algebraical. But it contains little that is original, and although the work created a great sensation when it was first published, the effect soon passed away, and the book was practically forgotten. Mathematics was more or less ousted from the academic curricula by the philosophical inquiries of the schoolmen, and it was only after an interval of nearly three centuries that a worthy successor to Leonardo appeared. .This was Lucas Paciolus (Lucas de Burgo), a Minorite friar, who, having previously written works on algebra, arithmetic and geometry, published, in 1494, his principal work, entitled Summa de Arithmetica, Geometria, Proportioni et Proportionalita. In it he mentions many earlier writers from whom he had learnt the science, and although it contains very little that cannot be found in Leonardo's work, yet it is especially noteworthy for the systematic employment of symbols, and the manner in which it reflects the state of mathematics in Europe during this period.^ In the case of 0 0 , plausibility arguments can be made, but the convincing argument is found in "Concrete Mathematics" by Graham, Knuth and Patashnik, and argues that 0 0 = 1 for the binomial theorem to work.
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^ This is probably due to the fact that floating-point is given very little (if any) attention in the computer science curriculum.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

These works are the earliest printed books on mathematics. The renaissance of mathematics was thus effected in Italy, and it is to that country that the leading developments of the following century were due. The first difficulty to be overcome was the algebraical solution of cubic equations, the " irreducible case" (see Equation). .Fundamental theorems in the theory of equations are to be found in the same work.^ In the case of 0 0 , plausibility arguments can be made, but the convincing argument is found in "Concrete Mathematics" by Graham, Knuth and Patashnik, and argues that 0 0 = 1 for the binomial theorem to work.
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Clearer ideas of imaginary quantities and the " irreducible case " were subsequently published by Bombelli, in a work of which the dedication is dated 1572, though the book was not published until 1579.
Contemporaneously with the remarkable discoveries of the Italian mathematicians, algebra was increasing in popularity in Germany, France and England. Michael Stifel and Johann Scheubelius (Scheybl) (1494-1570) flourished in Germany, and although unacquainted with the work of Cardan and Tartalea, their writings are noteworthy for their perspicuity and the introduction of a more complete symbolism for quantities and operations. .Stifel introduced the sign (+) for addition or a positive quantity, which was previously denoted by plus, pia, or the letter p. Subtraction, previously written as minus, mene or the letter m, was symbolized by the sign (-) which is still in use.^ In other words, the evaluation of any expression containing a subtraction (or an addition of quantities with opposite signs) could result in a relative error so large that all the digits are meaningless (Theorem 1).
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The square root he denoted by (j), whereas Paciolus, Cardan and others used the letter R.
The first treatise on algebra written in English was by Robert Recorde, who published his arithmetic in 1552, and his algebra entitled The Whetstone of Witte, which is the second part of Arithmetik, in 1557. This work, which is written in the form of a dialogue, closely resembles the works of Stifel and Scheubelius, the latter of whom he often quotes. .It includes the properties of numbers; extraction of roots of arithmetical and algebraical quantities, solutions of simple and quadratic equations, and a fairly complete account of surds.^ A number of claims have been made in this paper concerning properties of floating-point arithmetic.
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^ An optimizer that believed floating-point arithmetic obeyed the laws of algebra would conclude that C = [ T - S ] - Y = [( S + Y )- S ] - Y = 0, rendering the algorithm completely useless.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

.He introduced the sign (=) for equality, and the terms binomial and residual. Of other writers who published works about the end of the 16th century, we may mention Jacques Peletier, or Jacobus Peletarius (De occulta parte Numerorum, quam Algebram vocant, 1558); Petrus Ramus (Arithmeticae Libri duo et totidem Algebrae, 1560), and Christoph Clavius, who wrote on algebra in 1580, though it was not published until 1608. At this time also flourished Simon Stevinus (Stevin) of Bruges, who published an arithmetic in 1585 and an algebra shortly afterwards.^ If x < 0 and y >  0 are within E , should they really be considered to be equal, even though they have different signs?
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These works possess considerable originality, and contain many new improvements in algebraic notation; the unknown (res) is denoted by a small circle, in which he places an integer corresponding to the power. He introduced the terms multinomial, trinomial, quadrinomial, &c., and considerably simplified the notation for decimals.
About the beginning of the 17th century various mathematical works by Franciscus Vieta were published, which were afterwards collected by Franz van Schooten and republished in 1646 at Leiden. These works exhibit great originality and mark an important epoch in the history of algebra. Vieta, who does not avail himself of the discoveries of his predecessors - the negative roots of Cardan, the revised notation of Stifel and Stevin, &c. - introduced or popularized many new terms and symbols, some of which are still in use. He denotes quantities by the letters of the alphabet, retaining the vowels for the unknown and the consonants for the knowns; he introduced the vinculum and among others the terms coefficient, affirmative, negative, pure and adfected equations. He improved the methods for solving equations, and devised geometrical constructions with the aid of the conic sections. His method for determining approximate values of the roots of equations is far in advance of the Hindu method as applied by Cardan, and is identical in principle with the methods of Sir Isaac Newton and W. G. Homer.
We have next to consider the works of Albert Girard, a Flemish mathematician. This writer, after having published an edition of Stevin's works in 1625, published in 1629 at Amsterdam a small tract on algebra which shows a considerable advance on the work of Vieta. Girard is inconsistent in his notation, sometimes following Vieta, sometimes Stevin; he introduced the new symbols ff for greater than and � for less than; he follows Vieta in using the plus (+) for addition, he denotes subtraction by Recorde's symbol for equality (=), and he had no sign for equality but wrote the word out. .He possessed clear ideas of indices and the generation of powers, of the negative roots of equations and their geometrical interpretation, and was the first to use the term imaginary roots. He also discovered how to sum the powers of the roots of an equation.^ Many problems, such as numerical integration and the numerical solution of differential equations involve computing sums with many terms.
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Passing over the invention of logarithms by John Napier, and their development by Henry Briggs and others, the next author of moment was an Englishman, Thomas Harriot, whose algebra (Artis analyticae praxis) was published posthumously by Walter Warner in 1631. Its great merit consists in the complete notation and symbolism, which avoided the cumbersome expressions of the earlier algebraists, and reduced the art to a form closely resembling that of to-day. .He follows Vieta in assigning the vowels to the unknown quantities and the consonants to the knowns, but instead of using capitals, as with Vieta, he employed the small letters; equality he denoted by Recorde's symbol, and he introduced the signs > and < for greater than and less than. His principal discovery is concerned with equations, which he showed to be derived from the continued multiplication of as many simple factors as the highest power of the unknown, and he was thus enabled to deduce relations between the coefficients and various functions of the roots.^ There is a small snag when = 2 and a hidden bit is being used, since a number with an exponent of e min will always have a significand greater than or equal to 1.0 because of the implicit leading bit.
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^ Thus the 2 error bound for the Kahan summation formula (Theorem 8) is not as good as using double precision, even though it is much better than single precision.
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^ The problem with this approach is that it is less accurate, and that it costs more than the simple expression , even if there is no overflow.
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.Mention may also be made of his chapter on inequalities, in which he proves that the arithmetic mean is always greater than the geometric mean.^ With a single guard digit, the relative error of the result may be greater than , as in 110 - 8.59.
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^ To show that the spacing between binary numbers is always greater than the spacing between decimal numbers, consider an interval [10 n , 10 n + 1 ].
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

^ To show that 9 digits are sufficient, it is enough to show that the spacing between binary numbers is always greater than the spacing between decimal numbers.
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William Oughtred, a contemporary of Harriot, published an algebra, Clavis mathematicae, simultaneously with Harriot's treatise. His notation is based on that of Vieta, but he introduced the sign X for multiplication, - for continued proportion, :: for proportion, ' and denoted ratio by one dot. This last character has since been entirely restricted to multiplication, and ratio is now denoted by two dots (:). His symbols for greater than and less than ("3 and .) have been completely superseded by Harriot's signs.
So far the development of algebra and geometry had been mutually independent, except for a few isolated applications of geometrical constructions to the solution of algebraical problems. Certain minds had long suspected the advantages which would accrue from the unrestricted application of algebra to geometry, but it was not until the advent of the philosopher Rene Descartes that the co-ordination was effected. In his famous Geometria (1637), which is really a treatise on the algebraic representation of geometric theorems, he founded the modern theory of analytical geometry (see Geometry), and at the same time he rendered signal service to algebra, more especially in the theory of equations. His notation is based primarily on that of Harriot; but he differs from that writer in retaining the first letters of the alphabet for the known quantities and the final letters for the unknowns.
The 17th century is a famous epoch in the progress of science, and the mathematics in no way lagged behind. The discoveries of Johann Kepler and Bonaventura Cavalieri were the foundation upon which Sir Isaac Newton and Gottfried Wilhelm Leibnitz erected that wonderful edifice, the Infinitesimal Calculus. Many new fields were opened up, but there was still continual progress in pure algebra. .Continued fractions, one of the earliest examples of which is Lord Brouncker's expression for the ratio of the circumference to the diameter of a circle (see Circle), were elaborately discussed by John Wallis and Leonhard Euler; the convergency of series treated by Newton, Euler and the Bernoullis; the binomial theorem, due originally to Newton and subsequently expanded by Euler and others, was used by Joseph Louis Lagrange as the basis of his Calcul des Fonctions. Diophantine problems were revived by Gaspar Bachet, Pierre Fermat and Euler; the modern theory of numbers was founded by Fermat and developed by Euler, Lagrange and others; and the theory of probability was attacked by Blaise Pascal and Fermat, their work being subsequently expanded by James Bernoulli, Abraham de Moivre, Pierre Simon Laplace and others.^ The IEEE standard uses denormalized 18 numbers, which guarantee (10) , as well as other useful relations.
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^ In the case of 0 0 , plausibility arguments can be made, but the convincing argument is found in "Concrete Mathematics" by Graham, Knuth and Patashnik, and argues that 0 0 = 1 for the binomial theorem to work.
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^ For example, some single/double systems provide a single instruction to multiply two numbers and add a third with just one final rounding.
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The germs of the theory of determinants are to be found in the works of Leibnitz; Etienne Bezout utilized them in 1764 for expressing the result obtained by the process of elimination known by his name, and since restated by Arthur Cayley.
.In recent times many mathematicians have formulated other kinds of algebras, in which the operators do not obey the laws of ordinary algebra.^ An optimizer that believed floating-point arithmetic obeyed the laws of algebra would conclude that C = [ T - S ] - Y = [( S + Y )- S ] - Y = 0, rendering the algorithm completely useless.
  • What Every Computer Scientist Should Know About Floating-Point Arithmetic 3 February 2010 14:24 UTC docs.sun.com [Source type: Academic]

This study was inaugurated by George Peacock, who was one of the earliest mathematicians to recognize the symbolic character of the fundamental principles of algebra. About the same time, D. F. Gregory, published a paper " on the real nature of symbolical algebra." In Germany the work of Martin Ohm (System der Mathematik, 1822) marks a step forward. Notable service was also rendered by Augustus de Morgan, who applied logical analysis to the laws of mathematics.
The geometrical interpretation of imaginary quantities had a far-reaching influence on the development of symbolic algebras. The attempts to elucidate this question by H. Kuhn (1750-1751) and Jean Robert Argand (1806) were completed by Karl Friedrich Gauss, and the formulation of various systems of vector analysis by Sir William Rowan Hamilton, Hermann Grassmann and others, followed. These algebras were essentially geometrical, and it remained, more or less, for the American mathematician Benjamin Peirce to devise systems of pure symbolic algebras; in this work he was ably seconded by his son Charles S. Peirce. In England, multiple algebra was developed by j ames Joseph Sylvester, who, in company with Arthur Cayley, expanded the theory of matrices, the germs of which are to be found in the writings of Hamilton (see above, under (B); and Quaternions).
The preceding summary shows the specialized 'nature which algebra has assumed since the 17th century. To attempt a history of the development of the various topics in this article is inappropriate, and we refer the reader to the separate articles.
REFERENCES. - The history of algebra is treated in all historical works on mathematics in general (see MATHEMATICS: References). Greek algebra can be specially studied in T. L. Heath's Diophantus. See also John Wallis, Opera Mathematica (1693-1699), and Charles Hutton, Mathematical and Philosophical Dictionary (1815), article " Algebra." (C. E.*)


Wiktionary

Up to date as of January 15, 2010

Definition from Wiktionary, a free dictionary

See also algebra, álgebra, and àlgebra

German

Noun

Algebra f. (genitive Algebra, no plural)
  1. algebra

Wikibooks

Up to date as of January 23, 2010

From Wikibooks, the open-content textbooks collection

This book is part of a series on Algebra:
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Welcome to the Wikibooks Algebra textbook! Take a moment to skim over the chapters and sections to get a better idea if this is the right text for you. Also, please familiarize yourself with the GNU Free Documentation License, as this book falls under its rules. This textbook is currently undergoing an expansion and reorganization, so keep checking back for new content. .Eventually it will become a comprehensive, well-organized textbook capable of being used for personal or academic use.^ Depending on the programming language being used, the trap handler might be able to access other variables in the program as well.
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  • Algebra is the study of symbolic mathematics and mathematical structures.
  • It is a major branch of mathematics.
  • You can perform a Google Search in this book.
  • Interested in a simple-English version of this book? Check out Algebra I in Simple English.

Contents

Chapter 1: Introduction to Mathematics

  1. Algebra will begin with an introduction into the world of Mathematics, starting with an explanation of what this book is about, whom it is for, and what you can expect to find within it. There will also be a variety of information and resources regarding math.
  2. About This Book Development stage: 25% (as of {{{2}}})
    1. What is math, exactly? - And why should I care?
    2. Who should read this book - And what to expect

Chapter 2: Numbers, Variables and Equations

Algebra As a Problem Solving Technique
.
  • You've been asked to buy peanuts at a baseball game; you have $12.50, and you want to know how many bags you can buy.^ For example, suppose you are debugging a program and want to know the value of a subexpression.
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    One bag is $2.75. This is an algebra problem! In this chapter you will begin learning how to use numbers and variables to find out unknown information. .This starts off with a closer look at numbers and operations, then covers variables and equations.
  1. NumbersDevelopment stage: 25% (as of {{{2}}})
    1. The Real Number System
    2. Sets and the Number Line
    3. Arithmetic
    4. Order of Operations
    5. Exponents
    6. Ways to Organize Numbers
  2. Variables Development stage: 25% (as of {{{2}}})
    1. Variables
  3. Equations Development stage: 25% (as of {{{2}}})
    1. Equations
    2. Solving Equations
    3. Word Problems
    4. Systems of Equations
  • Math is a method of solving problems.^ Consider the problem of solving a system of linear equations, .
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    ^ In order to rely on examples such as these, a programmer must be able to predict how a program will be interpreted, and in particular, on an IEEE system, what the precision of the destination of each arithmetic operation may be.
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    ^ The problem it solves is that when x is small, LN (1 x ) is not close to ln(1 + x ) because 1 x has lost the information in the low order bits of x .
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    You take information you know, and by manipulating it using mathematical principles, you can find information you don't know. Functions are the mathematical framework for solving problems. They have parameters, rules, and ways of being solved. This section will introduce you to common functions and how to use them.
  1. Introduction to Functions Development stage: 25% (as of {{{2}}})
    1. Functions
    2. Toolkit Functions
  2. Graphing Functions Development stage: 25% (as of {{{2}}})
    1. Graphing Functions
  1. To be merged/unsorted
    1. Binomial Theorem
    2. Factoring Polynomials
    3. Roots and Radicals
    4. The Pythagorean Theorem
    5. Completing the Square
    6. Quadratic Equation
    7. Complex Numbers
    8. Conic Sections
      1. Circle
      2. Ellipse
      3. Hyperbola
      4. Parabola
    9. Probability
    10. Statistics
    11. Theory
    12. Iteration

Simple English