# Encyclopedia

A number is a mathematical object used in counting and measuring. A notational symbol which represents a number is called a numeral, but in common usage the word number is used for both the abstract object and the symbol, as well as for the word for the number. In addition to their use in counting and measuring, numerals are often used for labels (telephone numbers), for ordering (serial numbers), and for codes (ISBNs). In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers.

Certain procedures which take one or more numbers as input and produce a number as output are called numerical operations. Unary operations take a single input number and produce a single output number. For example, the successor operation adds one to an integer, thus the successor of 4 is 5. More common are binary operations which take two input numbers and produce a single output number. Examples of binary operations include addition, subtraction, multiplication, division, and exponentiation. The study of numerical operations is called arithmetic.

The branch of mathematics that studies structure in number systems, by means of topics such as groups, rings and fields, is called abstract algebra.

## Types of numbers

Numbers can be classified into sets, called number systems. (For different methods of expressing numbers with symbols, such as the Roman numerals, see numeral systems.)

### Natural numbers

The most familiar numbers are the natural numbers or counting numbers: one, two, three, and so on.

In the base ten number system, in almost universal use today for arithmetic operations, the symbols for natural numbers are written using ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. In this base ten system, the rightmost digit of a natural number has a place value of one, and every other digit has a place value ten times that of the place value of the digit to its right. The symbol for the set of all natural numbers is N, also written $\mathbb\left\{N\right\}$.

In set theory, which is capable of acting as an axiomatic foundation for modern mathematics, natural numbers can be represented by classes of equivalent sets. For instance, the number 3 can be represented as the class of all sets that have exactly three elements. Alternatively, in Peano Arithmetic, the number 3 is represented as sss0, where s is the "successor" function. Many different representations are possible; all that is needed to formally represent 3 is to inscribe a certain symbol or pattern of symbols 3 times.

### Integers

Negative numbers are numbers that are less than zero. They are the opposite of positive numbers. For example, if a positive number indicates a bank deposit, then a negative number indicates a withdrawal of the same amount. Negative numbers are usually written by writing a negative sign (also called a minus sign) in front of the number they are the opposite of. Thus the opposite of 7 is written −7. When the set of negative numbers is combined with the natural numbers and zero, the result is the set of integer numbers, also called integers, Z (German Zahl, plural Zahlen), also written $\mathbb\left\{Z\right\}$.

### Rational numbers

A rational number is a number that can be expressed as a fraction with an integer numerator and a non-zero natural number denominator. The fraction m/n or

$m \over n \,$

represents m equal parts, where n equal parts of that size make up one whole. Two different fractions may correspond to the same rational number; for example 1/2 and 2/4 are equal, that is:

$\left\{1 \over 2\right\} = \left\{2 \over 4\right\}.\,$

If the absolute value of m is greater than n, then the absolute value of the fraction is greater than 1. Fractions can be greater than, less than, or equal to 1 and can also be positive, negative, or zero. The set of all rational numbers includes the integers, since every integer can be written as a fraction with denominator 1. For example −7 can be written −7/1. The symbol for the rational numbers is Q (for quotient), also written $\mathbb\left\{Q\right\}$.

### Real numbers

File:REAL
The real numbers contain the irrational, rational, integers and natural numbers, and transcendental numbers.

The real numbers include all of the measuring numbers. Real numbers are usually written using decimal numerals, in which a decimal point is placed to the right of the digit with place value one. Each digit to the right of the decimal point has a place value one-tenth of the place value of the digit to its left. Thus

$123.456\,$

represents 1 hundred, 2 tens, 3 ones, 4 tenths, 5 hundredths, and 6 thousandths. In saying the number, the decimal is read "point", thus: "one two three point four five six ". In the US and UK and a number of other countries, the decimal point is represented by a period, whereas in continental Europe and certain other countries the decimal point is represented by a comma. Zero is often written as 0.0 when necessary to indicate that it is to be treated as a real number rather than as an integer. Negative real numbers are written with a preceding minus sign:

$-123.456.\,$

Every rational number is also a real number. To write a fraction as a decimal, divide the numerator by the denominator. It is not the case, however, that every real number is rational. If a real number cannot be written as a fraction of two integers, it is called irrational. A decimal that can be written as a fraction either ends (terminates) or forever repeats, because it is the answer to a problem in division. Thus the real number 0.5 can be written as 1/2 and the real number 0.333... (forever repeating threes) can be written as 1/3. On the other hand, the real number π (pi), the ratio of the circumference of any circle to its diameter, is

$\pi = 3.14159265358979....\,$

Since the decimal neither ends nor forever repeats, it cannot be written as a fraction, and is an example of an irrational number. Other irrational numbers include

$\sqrt\left\{2\right\} = 1.41421356237 ...\,$

(the square root of 2, that is, the positive number whose square is 2).

Thus 1.0 and 0.999... are two different decimal numerals representing the natural number 1. There are infinitely many other ways of representing the number 1, for example 2/2, 3/3, 1.00, 1.000, and so on.

Every real number is either rational or irrational. Every real number corresponds to a point on the number line. The real numbers also have an important but highly technical property called the least upper bound property. The symbol for the real numbers is R or $\mathbb\left\{R\right\}$.

When a real number represents a measurement, there is always a margin of error. This is often indicated by rounding or truncating a decimal, so that digits that suggest a greater accuracy than the measurement itself are removed. The remaining digits are called significant digits. For example, measurements with a ruler can seldom be made without a margin of error of at least 0.01 meters. If the sides of a rectangle are measured as 1.23 meters and 4.56 meters, then multiplication gives an area for the rectangle of 5.6088 square meters. Since only the first two digits after the decimal place are significant, this is usually rounded to 5.61.

In abstract algebra, the real numbers are up to isomorphism uniquely characterized by being the only complete ordered field. They are not, however, an algebraically closed field.

### Complex numbers

Moving to a greater level of abstraction, the real numbers can be extended to the complex numbers. This set of numbers arose, historically, from the question of whether a negative number can have a square root. This led to the invention of a new number: the square root of negative one, denoted by i, a symbol assigned by Leonhard Euler, and called the imaginary unit. The complex numbers consist of all numbers of the form

$\,a + b i$

where a and b are real numbers. In the expression a + bi, the real number a is called the real part and bi is called the imaginary part. If the real part of a complex number is zero, then the number is called an imaginary number or is referred to as purely imaginary; if the imaginary part is zero, then the number is a real number. Thus the real numbers are a subset of the complex numbers. If the real and imaginary parts of a complex number are both integers, then the number is called a Gaussian integer. The symbol for the complex numbers is C or $\mathbb\left\{C\right\}$.

In abstract algebra, the complex numbers are an example of an algebraically closed field, meaning that every polynomial with complex coefficients can be factored into linear factors. Like the real number system, the complex number system is a field and is complete, but unlike the real numbers it is not ordered. That is, there is no meaning in saying that i is greater than 1, nor is there any meaning in saying that that i is less than 1. In technical terms, the complex numbers lack the trichotomy property.

Complex numbers correspond to points on the complex plane, sometimes called the Argand plane.

Each of the number systems mentioned above is a proper subset of the next number system. Symbolically, NZQRC.

### Computable numbers

Moving to problems of computation, the computable numbers are determined in the set of the real numbers. The computable numbers, also known as the recursive numbers or the computable reals, are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm. Equivalent definitions can be given using μ-recursive functions, Turing machines or λ-calculus as the formal representation of algorithms. The computable numbers form a real closed field and can be used in the place of real numbers for many, but not all, mathematical purposes.

### Other types

Hyperreal and hypercomplex numbers are used in non-standard analysis. The hyperreals, or nonstandard reals (usually denoted as *R), denote an ordered field which is a proper extension of the ordered field of real numbers R and which satisfies the transfer principle. This principle allows true first order statements about R to be reinterpreted as true first order statements about *R.

Superreal and surreal numbers extend the real numbers by adding infinitesimally small numbers and infinitely large numbers, but still form fields.

The idea behind p-adic numbers is this: While real numbers may have infinitely long expansions to the right of the decimal point, these numbers allow for infinitely long expansions to the left. The number system which results depends on what base is used for the digits: any base is possible, but a system with the best mathematical properties is obtained when the base is a prime number.

For dealing with infinite collections, the natural numbers have been generalized to the ordinal numbers and to the cardinal numbers. The former gives the ordering of the collection, while the latter gives its size. For the finite set, the ordinal and cardinal numbers are equivalent, but they differ in the infinite case.

There are also other sets of numbers with specialized uses. Some are subsets of the complex numbers. For example, algebraic numbers are the roots of polynomials with rational coefficients. Complex numbers that are not algebraic are called transcendental numbers.

Sets of numbers that are not subsets of the complex numbers are sometimes called hypercomplex numbers. They include the quaternions H, invented by Sir William Rowan Hamilton, in which multiplication is not commutative, and the octonions, in which multiplication is not associative. Elements of function fields of non-zero characteristic behave in some ways like numbers and are often regarded as numbers by number theorists.

In addition, various specific kinds of numbers are studied in sets of natural and integer numbers.

An even number is an integer that is "evenly divisible" by 2, i.e., divisible by 2 without remainder; an odd number is an integer that is not evenly divisible by 2. (The old-fashioned term "evenly divisible" is now almost always shortened to "divisible".) A formal definition of an odd number is that it is an integer of the form n = 2k + 1, where k is an integer. An even number has the form n = 2k where k is an integer.

A perfect number is defined as a positive integer which is the sum of its proper positive divisors, that is, the sum of the positive divisors not including the number itself. Equivalently, a perfect number is a number that is half the sum of all of its positive divisors, or σ(n) = 2 n. The first perfect number is 6, because 1, 2, and 3 are its proper positive divisors and 1 + 2 + 3 = 6. The next perfect number is 28 = 1 + 2 + 4 + 7 + 14. The next perfect numbers are 496 and 8128 (sequence A000396 in OEIS). These first four perfect numbers were the only ones known to early Greek mathematics.

A figurate number is a number that can be represented as a regular and discrete geometric pattern (e.g. dots). If the pattern is polytopic, the figurate is labeled a polytopic number, and may be a polygonal number or a polyhedral number. Polytopic numbers for r = 2, 3, and 4 are:

A relation number is defined as the class of relations consisting of all those relations that are similar to one member of the class. [1]

## Numerals

Numbers should be distinguished from numerals, the symbols used to represent numbers. Boyer showed that Egyptians created the first ciphered numeral system. Greeks followed by mapping their counting numbers onto Ionian and Doric alpabets. The number five can be represented by both the base ten numeral '5', by the Roman numeral 'V' and ciphered letters. Notations used to represent numbers are discussed in the article numeral systems. An important development in the history of numerals was the development of a positional system, like modern decimals, which can represent very large numbers. The Roman numerals require extra symbols for larger numbers.

## History

### History of integers

#### The first use of numbers

It is speculated that the first known use of numbers dates back to around 30,000 BC. Bones and other artifacts have been discovered with marks cut into them which many consider to be tally marks. The uses of these tally marks may have been for counting elapsed time, such as numbers of days, or keeping records of quantities, such as of animals.

Tallying systems have no concept of place-value (such as in the currently used decimal notation), which limit its representation of large numbers and as such is often considered that this is the first kind of abstract system that would be used, and could be considered a Numeral System.

The first known system with place-value was the Mesopotamian base 60 system (ca. 3400 BC) and the earliest known base 10 system dates to 3100 BC in Egypt. [1]

#### History of zero

The use of zero as a number should be distinguished from its use as a placeholder numeral in place-value systems. Many ancient texts used zero. Babylonians and Egyptian texts used it. Egyptians used the word nfr to denote zero balance in double entry accounting entries. Indian texts used a Sanskrit word Shunya to refer to the concept of void; in mathematics texts this word would often be used to refer to the number zero. [2]. In a similar vein, Pāṇini (5th century BC) used the null (zero) operator (ie a lambda production) in the Ashtadhyayi, his algebraic grammar for the Sanskrit language. (also see Pingala)

Records show that the Ancient Greeks seemed unsure about the status of zero as a number: they asked themselves "how can 'nothing' be something?" leading to interesting philosophical and, by the Medieval period, religious arguments about the nature and existence of zero and the vacuum. The paradoxes of Zeno of Elea depend in large part on the uncertain interpretation of zero. (The ancient Greeks even questioned if 1 was a number.)

The late Olmec people of south-central Mexico began to use a true zero (a shell glyph) in the New World possibly by the 4th century BC but certainly by 40 BC, which became an integral part of Maya numerals and the Maya calendar. Mayan arithmetic used base 4 and base 5 written as base 20. Sanchez in 1961 reported a base 4, base 5 'finger' abacus.

By 130, Ptolemy, influenced by Hipparchus and the Babylonians, was using a symbol for zero (a small circle with a long overbar) within a sexagesimal numeral system otherwise using alphabetic Greek numerals. Because it was used alone, not as just a placeholder, this Hellenistic zero was the first documented use of a true zero in the Old World. In later Byzantine manuscripts of his Syntaxis Mathematica (Almagest), the Hellenistic zero had morphed into the Greek letter omicron (otherwise meaning 70).

Another true zero was used in tables alongside Roman numerals by 525 (first known use by Dionysius Exiguus), but as a word, nulla meaning nothing, not as a symbol. When division produced zero as a remainder, nihil, also meaning nothing, was used. These medieval zeros were used by all future medieval computists (calculators of Easter). An isolated use of their initial, N, was used in a table of Roman numerals by Bede or a colleague about 725, a true zero symbol.

An early documented use of the zero by Brahmagupta (in the Brahmasphutasiddhanta) dates to 628. He treated zero as a number and discussed operations involving it, including division. By this time (7th century) the concept had clearly reached Cambodia, and documentation shows the idea later spreading to China and the Islamic world.

#### History of negative numbers

The abstract concept of negative numbers was recognised as early as 100 BC - 50 BC. The Chinese Nine Chapters on the Mathematical Art (Jiu-zhang Suanshu) contains methods for finding the areas of figures; red rods were used to denote positive coefficients, black for negative. This is the earliest known mention of negative numbers in the East; the first reference in a western work was in the 3rd century in Greece. Diophantus referred to the equation equivalent to $4x + 20 = 0$ (the solution would be negative) in Arithmetica, saying that the equation gave an absurd result.

During the 600s, negative numbers were in use in India to represent debts. Diophantus’ previous reference was discussed more explicitly by Indian mathematician Brahmagupta, in Brahma-Sphuta-Siddhanta 628, who used negative numbers to produce the general form quadratic formula that remains in use today. However, in the 12th century in India, Bhaskara gives negative roots for quadratic equations but says the negative value "is in this case not to be taken, for it is inadequate; people do not approve of negative roots."

European mathematicians, for the most part, resisted the concept of negative numbers until the 17th century, although Fibonacci allowed negative solutions in financial problems where they could be interpreted as debts (chapter 13 of Liber Abaci, 1202) and later as losses (in Flos). At the same time, the Chinese were indicating negative numbers by drawing a diagonal stroke through the right-most nonzero digit of the corresponding positive number's numeral[citation needed]. The first use of negative numbers in a European work was by Chuquet during the 15th century. He used them as exponents, but referred to them as “absurd numbers”.

As recently as the 18th century, the Swiss mathematician Leonhard Euler believed that negative numbers were greater than infinity[citation needed], and it was common practice to ignore any negative results returned by equations on the assumption that they were meaningless, just as René Descartes did with negative solutions in a cartesian coordinate system.

### History of rational, irrational, and real numbers

#### History of rational numbers

It is likely that the concept of fractional numbers dates to prehistoric times. Even the Ancient Egyptians wrote math texts describing how to convert general fractions into their special notation. The RMP 2/n table and the Kahun Papyrus wrote out unit fraction series by using least common multiples. Classical Greek and Indian mathematicians made studies of the theory of rational numbers, as part of the general study of number theory. The best known of these is Euclid's Elements, dating to roughly 300 BC. Of the Indian texts, the most relevant is the Sthananga Sutra, which also covers number theory as part of a general study of mathematics.

The concept of decimal fractions is closely linked with decimal place value notation; the two seem to have developed in tandem. For example, it is common for the Jain math sutras to include calculations of decimal-fraction approximations to pi or the square root of two. Similarly, Babylonian math texts had always used sexagesimal fractions with great frequency.

#### History of irrational numbers

The earliest known use of irrational numbers was in the Indian Sulba Sutras composed between 800-500 BC.[citation needed] The first existence proofs of irrational numbers is usually attributed to Pythagoras, more specifically to the Pythagorean Hippasus of Metapontum, who produced a (most likely geometrical) proof of the irrationality of the square root of 2. The story goes that Hippasus discovered irrational numbers when trying to represent the square root of 2 as a fraction. However Pythagoras believed in the absoluteness of numbers, and could not accept the existence of irrational numbers. He could not disprove their existence through logic, but his beliefs would not accept the existence of irrational numbers and so he sentenced Hippasus to death by drowning.

The sixteenth century saw the final acceptance by Europeans of negative, integral and fractional numbers. The seventeenth century saw decimal fractions with the modern notation quite generally used by mathematicians. But it was not until the nineteenth century that the irrationals were separated into algebraic and transcendental parts, and a scientific study of theory of irrationals was taken once more. It had remained almost dormant since Euclid. The year 1872 saw the publication of the theories of Karl Weierstrass (by his pupil Kossak), Heine (Crelle, 74), Georg Cantor (Annalen, 5), and Richard Dedekind. Méray had taken in 1869 the same point of departure as Heine, but the theory is generally referred to the year 1872. Weierstrass's method has been completely set forth by Salvatore Pincherle (1880), and Dedekind's has received additional prominence through the author's later work (1888) and the recent endorsement by Paul Tannery (1894). Weierstrass, Cantor, and Heine base their theories on infinite series, while Dedekind founds his on the idea of a cut (Schnitt) in the system of real numbers, separating all rational numbers into two groups having certain characteristic properties. The subject has received later contributions at the hands of Weierstrass, Kronecker (Crelle, 101), and Méray.

Continued fractions, closely related to irrational numbers (and due to Cataldi, 1613), received attention at the hands of Euler, and at the opening of the nineteenth century were brought into prominence through the writings of Joseph Louis Lagrange. Other noteworthy contributions have been made by Druckenmüller (1837), Kunze (1857), Lemke (1870), and Günther (1872). Ramus (1855) first connected the subject with determinants, resulting, with the subsequent contributions of Heine, Möbius, and Günther, in the theory of Kettenbruchdeterminanten. Dirichlet also added to the general theory, as have numerous contributors to the applications of the subject.

#### Transcendental numbers and reals

The first results concerning transcendental numbers were Lambert's 1761 proof that π cannot be rational, and also that en is irrational if n is rational (unless n = 0). (The constant e was first referred to in Napier's 1618 work on logarithms.) Legendre extended this proof to show that π is not the square root of a rational number. The search for roots of quintic and higher degree equations was an important development, the Abel–Ruffini theorem (Ruffini 1799, Abel 1824) showed that they could not be solved by radicals (formula involving only arithmetical operations and roots). Hence it was necessary to consider the wider set of algebraic numbers (all solutions to polynomial equations). Galois (1832) linked polynomial equations to group theory giving rise to the field of Galois theory.

Even the set of algebraic numbers was not sufficient and the full set of real number includes transcendental numbers.[citation needed] The existence of which was first established by Liouville (1844, 1851). Hermite proved in 1873 that e is transcendental and Lindemann proved in 1882 that π is transcendental. Finally Cantor shows that the set of all real numbers is uncountably infinite but the set of all algebraic numbers is countably infinite, so there is an uncountably infinite number of transcendental numbers.

### Infinity

The earliest known conception of mathematical infinity appears in the Yajur Veda - an ancient script in India, which at one point states "if you remove a part from infinity or add a part to infinity, still what remains is infinity". Infinity was a popular topic of philosophical study among the Jain mathematicians circa 400 BC. They distinguished between five types of infinity: infinite in one and two directions, infinite in area, infinite everywhere, and infinite perpetually.

In the West, the traditional notion of mathematical infinity was defined by Aristotle, who distinguished between actual infinity and potential infinity; the general consensus being that only the latter had true value. Galileo's Two New Sciences discussed the idea of one-to-one correspondences between infinite sets. But the next major advance in the theory was made by Georg Cantor; in 1895 he published a book about his new set theory, introducing, among other things, transfinite numbers and formulating the continuum hypothesis. This was the first mathematical model that represented infinity by numbers and gave rules for operating with these infinite numbers.

In the 1960s, Abraham Robinson showed how infinitely large and infinitesimal numbers can be rigorously defined and used to develop the field of nonstandard analysis. The system of hyperreal numbers represents a rigorous method of treating the ideas about infinite and infinitesimal numbers that had been used casually by mathematicians, scientists, and engineers ever since the invention of calculus by Newton and Leibniz.

A modern geometrical version of infinity is given by projective geometry, which introduces "ideal points at infinity," one for each spatial direction. Each family of parallel lines in a given direction is postulated to converge to the corresponding ideal point. This is closely related to the idea of vanishing points in perspective drawing.

### Complex numbers

The earliest fleeting reference to square roots of negative numbers occurred in the work of the mathematician and inventor Heron of Alexandria in the 1st century AD, when he considered the volume of an impossible frustum of a pyramid. They became more prominent when in the 16th century closed formulas for the roots of third and fourth degree polynomials were discovered by Italian mathematicians (see Niccolo Fontana Tartaglia, Gerolamo Cardano). It was soon realized that these formulas, even if one was only interested in real solutions, sometimes required the manipulation of square roots of negative numbers.

This was doubly unsettling since they did not even consider negative numbers to be on firm ground at the time. The term "imaginary" for these quantities was coined by René Descartes in 1637 and was meant to be derogatory (see imaginary number for a discussion of the "reality" of complex numbers). A further source of confusion was that the equation

$\left \left( \sqrt\left\{-1\right\}\right \right)^2 =\sqrt\left\{-1\right\}\sqrt\left\{-1\right\}=-1$

seemed to be capriciously inconsistent with the algebraic identity

$\sqrt\left\{a\right\}\sqrt\left\{b\right\}=\sqrt\left\{ab\right\},$

which is valid for positive real numbers a and b, and which was also used in complex number calculations with one of a, b positive and the other negative. The incorrect use of this identity, and the related identity

$\frac\left\{1\right\}\left\{\sqrt\left\{a\right\}\right\}=\sqrt\left\{\frac\left\{1\right\}\left\{a\right\}\right\}$

in the case when both a and b are negative even bedeviled Euler. This difficulty eventually led him to the convention of using the special symbol i in place of √−1 to guard against this mistake.

The 18th century saw the labors of Abraham de Moivre and Leonhard Euler. To De Moivre is due (1730) the well-known formula which bears his name, de Moivre's formula:

$\left(\cos \theta + i\sin \theta\right)^\left\{n\right\} = \cos n \theta + i\sin n \theta \,$

and to Euler (1748) Euler's formula of complex analysis:

$\cos \theta + i\sin \theta = e ^\left\{i\theta \right\}. \,$

The existence of complex numbers was not completely accepted until the geometrical interpretation had been described by Caspar Wessel in 1799; it was rediscovered several years later and popularized by Carl Friedrich Gauss, and as a result the theory of complex numbers received a notable expansion. The idea of the graphic representation of complex numbers had appeared, however, as early as 1685, in Wallis's De Algebra tractatus.

Also in 1799, Gauss provided the first generally accepted proof of the fundamental theorem of algebra, showing that every polynomial over the complex numbers has a full set of solutions in that realm. The general acceptance of the theory of complex numbers is not a little due to the labors of Augustin Louis Cauchy and Niels Henrik Abel, and especially the latter, who was the first to boldly use complex numbers with a success that is well known.

Gauss studied complex numbers of the form a + bi, where a and b are integral, or rational (and i is one of the two roots of x2 + 1 = 0). His student, Ferdinand Eisenstein, studied the type a + , where ω is a complex root of x3 − 1 = 0. Other such classes (called cyclotomic fields) of complex numbers are derived from the roots of unity xk − 1 = 0 for higher values of k. This generalization is largely due to Ernst Kummer, who also invented ideal numbers, which were expressed as geometrical entities by Felix Klein in 1893. The general theory of fields was created by Évariste Galois, who studied the fields generated by the roots of any polynomial equation F(x) = 0.

In 1850 Victor Alexandre Puiseux took the key step of distinguishing between poles and branch points, and introduced the concept of essential singular points; this would eventually lead to the concept of the extended complex plane.

### Prime numbers

Prime numbers have been studied throughout recorded history. Euclid devoted one book of the Elements to the theory of primes; in it he proved the infinitude of the primes and the fundamental theorem of arithmetic, and presented the Euclidean algorithm for finding the greatest common divisor of two numbers.

In 240 BC, Eratosthenes used the Sieve of Eratosthenes to quickly isolate prime numbers. But most further development of the theory of primes in Europe dates to the Renaissance and later eras.

In 1796, Adrien-Marie Legendre conjectured the prime number theorem, describing the asymptotic distribution of primes. Other results concerning the distribution of the primes include Euler's proof that the sum of the reciprocals of the primes diverges, and the Goldbach conjecture which claims that any sufficiently large even number is the sum of two primes. Yet another conjecture related to the distribution of prime numbers is the Riemann hypothesis, formulated by Bernhard Riemann in 1859. The prime number theorem was finally proved by Jacques Hadamard and Charles de la Vallée-Poussin in 1896. The conjectures of Goldbach and Riemann yet remain to be proved or refuted.

## Word alternatives

Some numbers traditionally have alternative words to express them, including the following:

## References

1. ^ "Introduction to Mathematical Philosophy" by Bertrand Russell, page 56

Template:· Transcendental numbersTemplate:· Quaternions ($\scriptstyle\mathbb\left\{H\right\}$)Template:· Octonions ($\scriptstyle\mathbb\left\{O\right\}$)Template:· Sedenions ($\scriptstyle\mathbb\left\{S\right\}$)Template:· Cayley–Dickson constructionTemplate:· Split-complex numbers

|group2= Complex
extensions | list2= Bicomplex numbersTemplate:· BiquaternionsTemplate:· Split-quaternionsTemplate:· TessarinesTemplate:· Hypercomplex numbersTemplate:· Musean hypernumbersTemplate:· Superreal numbersTemplate:· Hyperreal numbersTemplate:· Supernatural numbersTemplate:· Surreal numbers

|group3=Other extensions | list3= Dual numbersTemplate:· Transfinite numbersTemplate:· Extended real numbersTemplate:· Cardinal numbersTemplate:· Ordinal numbersTemplate:· p-adic numbers }}

# Wiktionary

Up to date as of January 15, 2010

### Definition from Wiktionary, a free dictionary

Wikipedia has an article on:

## English

Most common English words: city « held « help « #342: number » alone » body » point

### Etymology 1

Anglo-Norman noumbre, from Old French nombre, from Latin numerus, from Proto-Indo-European *nem- (to divide).

#### Pronunciation

• (UK) enPR: nŭm'bə(r), IPA: /ˈnʌm.bə(ɹ)/, SAMPA: /"nVm.b@(r\)/
• (US) IPA: /ˈnʌm.bɚ/, SAMPA: /"nVm.b3/
• help, file
• Hyphenation: num‧ber

#### Noun

Wikipedia has an article on:
 Singular number Plural numbers
number (plural numbers)
.
1. (countable) An abstract entity used to describe quantity.^ Moles and their uses The mole (abbreviated mol) is the the SI measure of quantity of a "chemical entity", which can be an atom, molecule, formula unit, electron or photon.
• All about Avogadro's number and the mole 10 January 2010 11:011 UTC www.chem1.com [Source type: Reference]

Zero, one, -1, 2.5, and pi are all numbers.
2. (countable) A numeral: a symbol for a non-negative integer
The number 8 is usually made with a single stroke.
.
3. (countable, mathematics) A member of one of several classes: natural numbers,integers, rational numbers, real numbers, complex numbers, quaternions.^ Complex numbers are pairs c = (x, y) of two real numbers.
• What's a number? from Interactive Mathematics Miscellany and Puzzles 10 January 2010 11:011 UTC www.cut-the-knot.org [Source type: Reference]

^ Rational numbers and integers are all algebraic.
• What's a number? from Interactive Mathematics Miscellany and Puzzles 10 January 2010 11:011 UTC www.cut-the-knot.org [Source type: Reference]

^ I will not give a mathematical definition of “real number”.

The equation eiπ + 1 = 0 includes the most important numbers: 1, 0, π, i, and e.
.
4. (Followed by a numeral; used attributively) Indicating the position of something in a list or sequence.^ I use numbered lists to help people follow along, but I also add statistics whenever and whereever I can.

^ The use of three dots at the end of the list is a common mathematical notation to indicate that the list keeps going forever.
• The Real Number System 10 January 2010 11:011 UTC www.jamesbrennan.org [Source type: FILTERED WITH BAYES]

^ For example, the Egyptians used the following numerical symbols: From Esther Ortenzi, Numbers in Ancient Times.

Abbreviations: No or No., no or no. (in each case, sometimes written with a superscript "o", like Nº or №). The symbol "#" is also used in this manner.
Horse number 5 won the race.
5. Quantity.
Any number of people can be reading from a given repository at a time
His army is vast in number.
.
6. (grammar) Of a word or phrase, the state of being singular, dual or plural, shown by inflection.^ Another option is to read the Hebrew word for thousand with a dual meaning of chief and 1,000, with the chiefs numbering one less than the stated figure.
• Book of the Bible, Numbers :: NIV Study Bible 10 January 2010 11:011 UTC biblica.com [Source type: FILTERED WITH BAYES]

Adjectives and nouns should agree in gender, number, and case.
7. (countable) A performance; especially, a single song or song and dance routine within a larger show.
For his second number, he sang "The Moon Shines Bright".
##### Synonyms
• (mathematical number): scalar
##### Derived terms
• Look at pages starting with number.

#### Verb

 Infinitive to number Third person singular numbers Simple past numbered Past participle numbered Present participle numbering
to number (third-person singular simple present numbers, present participle numbering, simple past and past participle numbered)
1. (transitive) To label (items) with numbers; to assign numbers to (items).
Number the baskets so that we can find them easily.
2. (intransitive) To total or count; to amount to.
I don’t know how many books are in the library, but they must number in the thousands.

### Etymology 2

From numb + -er.

#### Pronunciation

• (RP) enPR: nŭm'ə, IPA: /ˈnʌmə/, SAMPA: /"nVm@(r)/
• (US): enPR: nŭm'ər, IPA: /'nʌmɚ/, SAMPA: /"nVm3/
• help, file
• Hyphenation: num‧ber

number
1. Comparative form of numb: more numb.

# Simple English

For the book in the Bible, see Numbers (Bible).

A number is a concept from mathematics, used to count or measure. Depending on the field of mathematics, where numbers are used, there are different definitions:

Numbers are also used for other things besides counting. Numbers are used when things are measured. Numbers are used to study how the world works. Mathematics is a way to use numbers to learn about the world and make things. The study of the rules of the natural world is called science. The work that uses numbers to make things is called engineering

## Numbering methods

### Numbers for people

There are different ways of giving symbols to numbers. These methods are called number systems. The most common number system that people use is the base ten number system. The base ten number system is also called the decimal number system. The base ten number system is common because people have ten fingers and ten toes. There are 10 different symbols {0,1,2,3,4,5,6,7,8,9} used in the base ten number system. These ten symbols are called digits.[1]

A symbol for a number is made up of these ten digits. The position of the digits shows how big the number is. For example, the number 23 in the decimal number system really means 2 times 10 plus 3, and 101 means 1 times a hundred (=100) plus 0 times 10 (=0) plus 1 times 1 (=1).

### Numbers for machines

Another number system is more common for machines. The machine number system is called the binary number system. The binary number system is also called the base two number system. There are two different symbols (0,1) used in the base two number system. These two symbols are called bits.[2]

A symbol for a binary number is made up of these two bit symbols. The position of the bit symbols shows how big the number is. For example, the number 10 in the binary number system really means 1 times 2 plus 0, and 101 means 1 times four (=4) plus 0 times two (=0) plus 1 times 1 (=1). The binary number 10 is the same as the decimal number 2. The binary number 101 is the same as the decimal number 5.

## Names of numbers

English has special names for some of the numbers in the decimal number system that are 'powers of ten'. All of these power of ten numbers in the decimal number system use just the symbol 1 and the symbol zero. For example, ten tens is the same as ten times ten, or one hundred. In symbols, this is "10 × 10 = 100". Also, ten hundreds is the same as ten times one hundred, or one thousand. In symbols, this is "10 × 100 = 10 × 10 × 10 = 1000". Some other power of ten numbers also have special names:

When dealing with larger numbers than this, there are two different ways of naming the numbers in English. Under the 'long scale' a new name is given every time the number is a million times larger than the last named number. It is also called the 'British Standard'. This scale used to be common in Britain but is not often used in English speaking countries today. It is still used in some other European nations. Another scale is the 'short scale' under which a new name is given every time a number is a thousand times larger than the last named number. This scale is a lot more common in most English speaking nations today.

• 1 000 000 000 - One Billion (Short Scale), One Milliard (Long Scale).
• 1 000 000 000 000 - One Trillion (Short Scale), One Billion (Long Scale)
• 1 000 000 000 000 000 - One Quadrillion (Short Scale), One Billiard (Long Scale)

## Types of numbers

### Natural numbers

Natural numbers are the numbers which we normally use for counting, 1,2,3,4,5,6,7,8,9,10 etc. Some people say that 0 is a natural number, too.

Another name for these numbers is positive numbers. These numbers are sometimes written as +1 to show that they are different from the negative numbers. But not all positive numbers are natural (for example $\frac\left\{1\right\}\left\{2\right\}$ is positive, but not natural).

Natural number has two forms. Some people call it counting numbers whereas some people call it whole numbers. If you are confused, just remember that the whole number can be including 0 but the counting number does not include 0. The natural number can be named as whole number or counting number.

### Negative numbers

Negative numbers are numbers less than zero.

One way to think of negative numbers is using a number line. We call one point on this line zero. Then we will label (write the name of) every position on the line by how far to the right of the zero point it is, for example the point one is one centimeter to the right, the point two is two centimeters to the right.

Now think about a point which is one centimeter to the left of the zero point. We cannot call this point one, as there is already a point called one. We therefore call this point minus 1 (-1) (as it is one centimeter away, but in the opposite direction).

A drawing of a number line is below.

[[File:]]

All the normal operations of mathematics can be done with negative numbers:

If people add a negative number to another this is the same as taking away the positive number with the same numerals. For example 5 + (-3) is the same as 5 - 3, and equals 2.

If they take away a negative number from another this is the same as adding the positive number with the same numerals. For example 5 - (-3) is the same as 5 + 3, and equals 8.

If they multiply two negative numbers together they get a positive number. For example -5 times -3 is 15.

If they multiply a negative number by a positive number, or multiply a positive number by a negative number, they get a negative result. For example 5 times -3 is -15.

### Integers

Integers are whole numbers. This is all the natural numbers, all their opposites, and the number zero. Decimal numbers and fractions are not integers.

### Rational numbers

Rational numbers are numbers which can be written as fractions. This means that they can be written as a divided by b, where the numbers a and b are integers, and b is not equal to 0.

Some rational numbers, such as 1/10, need a finite number of digits after the decimal point to write them in decimal form. The number one tenth is written in decimal form as 0.1. Numbers written with a finite decimal form are rational. Some rational numbers, such as 1/11, need an infinite number of digits after the decimal point to write them in decimal form. There is a repeating pattern to the digits following the decimal point. The number one eleventh is written in decimal form as 0.0909090909....

A percentage is not totally a rational number but it has relationship with fraction and decimal. Sometimes, ratio considered as a rational number.

### Irrational numbers

Irrational numbers are numbers which cannot be written as a fraction, but do not have imaginary parts.

File:Square root of 2
√2 is irrational

Irrational numbers often occur in geometry. For instance if we have a square which has sides of 1 meter, the distance between opposite corners is the square root of two. This is an irrational number. In decimal for it is written as 1.414213... Mathematicians have proved that the square root of every natural number is either an integer or an irrational number.

One well known irrational number is pi. This is the circumference of a circle divided by its diameter. This number is the same for every circle. The number pi is approximately 3.1415926359... .

An irrational number cannot be fully written down in decimal form. It would have an infinite number of digits after the decimal point. These digits would also not repeat.

### Real numbers

Real numbers is a name for all the sets of numbers listed above

• The rational numbers, including integers
• The irrational numbers

This is all numbers but for imaginary numbers.

### Imaginary numbers

Imaginary numbers are formed by real numbers multiplied by the number i. This number is the square root of minus one (-1).

There is no number in the real numbers which when squared makes the number -1. Therefore mathematicians invented a number. They called this number i, or the imaginary unit.

Imaginary numbers operate under the same rules as real numbers:

• The sum of two imaginary numbers is found by pulling out (factoring out) the i. For example 2i + 3i = (2 + 3)i = 5i.
• The difference of two imaginary numbers is found similarly. For example 5i - 3i = (5 - 3)i = 2i.
• When multiplying two imaginary numbers, remember that i × i (i2) is -1. For example 5i × 3i = ( 5 × 3 ) × ( i × i ) = 15 × (-1) = -15

Imaginary numbers were called imaginary because when they were first found many mathematicians did not think they existed.[needs proof] The person who discovered imaginary numbers was Gerolamo Cardano in the 1500s. The first to use the word imaginary number was René Descartes. The first people to use these numbers were Leonard Euler and Carl Friedrich Gauss. Both lived in the 18th century.

### Complex numbers

Complex numbers are numbers which have two parts; a real part and an imaginary part. Every type of number written above is also a complex number.

Complex numbers are a more general form of numbers. Every equation can be solved using only complex numbers.

The complex numbers can be drawn on a number plane. This is composed of a real number line, and an imaginary number line.

           3i|_
|
|
2i|_          . 2+2i
|
|
i|_
|
|
|_____|_____|_____|_____|_____|_____|_____|_____|
-2    -1     0     1     2     3     4     5     6
|
-i|_                .3-i
|
|
.-2-2i   -2i|_
|
|
-3i|_
|


All of normal mathematics can be done with complex numbers:

• To sum two complex numbers they sum the real and imaginary parts separately. For example (2 + 3i) + (3 + 2i) = (2 + 3) + (3 + 2)i= 5 + 5i.
• If they subtract one complex number from another they subtract the real and imaginary parts separately. For example (7 + 5i) - (3 + 3i) = (7 - 3) + (5 - 3)i = 4 + 2i.

To multiply two complex numbers is complicated. It is easiest to describe in general terms, with two complex numbers a + bi and c + di.

$\left( a + b \mathrm\left\{i\right\} \right) \times \left( c + d\mathrm\left\{i\right\} \right) = a \times c + a \times d\mathrm\left\{i\right\} + b\mathrm\left\{i\right\} \times c + b\mathrm\left\{i\right\} \times d\mathrm\left\{i\right\} = ac + ad\mathrm\left\{i\right\} + bc\mathrm\left\{i\right\} -bd = \left( ac - bd \right) + \left( ad + bc \right)\mathrm\left\{i\right\}$

For example (4 + 5i) × (3 + 2i) = (4 × 3 - 5 × 2) + (4 × 2 + 5 × 3)i = (12 - 10) + (8 + 15)i = 2 + 23i.

### Transcendental numbers

A real or complex number is called Transcendental number if it cannot be obtained as a result of an algebraic equation with integer coefficients.

$a_\left\{n\right\}x^\left\{n\right\} + \dots + a_\left\{2\right\}x^2 + a_\left\{1\right\}x + a_\left\{0\right\} = 0$

Proving that a certain number is transcendental can be extremely difficult. Each transcendental number is also an irrational number. The first people to see that there were transcendental numbers were Gottfried Wilhelm Leibniz and Leonhard Euler. The first to actually prove there were transcendental numbers was Joseph Liouville. He did this in 1844.

Well known transcendental numbers:

• e
• π
• ea for algebraic a 0
• $2^\left\{\sqrt\left\{2\right\}\right\}$

## Notes

1. A finger or a toe is also called a digit
2. A bit is a short form of the words "binary digit".
bjn:Wilangan

# Citable sentences

Up to date as of December 30, 2010

Here are sentences from other pages on Number, which are similar to those in the above article.