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Euclid, Greek mathematician, 3rd century BC, as imagined by Raphael in this detail from The School of Athens.[1]
Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns,[2][3] formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions.[4]
There is debate over whether mathematical objects such as numbers and points exist naturally or are human creations. The mathematician Benjamin Peirce called mathematics "the science that draws necessary conclusions".[5] Albert Einstein, on the other hand, stated that "as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."[6]
Through the use of abstraction and logical reasoning, mathematics evolved from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity for as far back as written records exist. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Mathematics continued to develop, for example in China in 300 BCE, in India in 100 CE, and in Arabia in 800 CE, until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that continues to the present day.[7]
Mathematics is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, and the social sciences. Applied mathematics, the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries and sometimes leads to the development of entirely new mathematical disciplines, such as statistics and game theory. Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind, although practical applications for what began as pure mathematics are often discovered.[8]

Contents

Etymology

The word "mathematics" comes from the Greek μάθημα (máthēma), which means learning, study, science, and additionally came to have the narrower and more technical meaning "mathematical study", even in Classical times.[9] Its adjective is μαθηματικός (mathēmatikós), related to learning, or studious, which likewise further came to mean mathematical. In particular, μαθηματικὴ τέχνη (mathēmatikḗ tékhnē), Latin: ars mathematica, meant the mathematical art.
The apparent plural form in English, like the French plural form les mathématiques (and the less commonly used singular derivative la mathématique), goes back to the Latin neuter plural mathematica (Cicero), based on the Greek plural τα μαθηματικά (ta mathēmatiká), used by Aristotle, and meaning roughly "all things mathematical"; although it is plausible that English borrowed only the adjective mathematic(al) and formed the noun mathematics anew, after the pattern of physics and metaphysics, which were inherited from the Greek.[10] In English, the noun mathematics takes singular verb forms. It is often shortened to maths or, in English-speaking North America, math.

History

A quipu, used by the Inca to record numbers.
The evolution of mathematics might be seen as an ever-increasing series of abstractions, or alternatively an expansion of subject matter. The first abstraction, which is shared by many animals,[11] was probably that of numbers: the realization that a collection of two apples and a collection two oranges (for example) have something in common, namely quantity of their members.
In addition to recognizing how to count physical objects, prehistoric peoples also recognized how to count abstract quantities, like time – days, seasons, years.[12] Elementary arithmetic (addition, subtraction, multiplication and division) naturally followed.
Further steps needed writing or some other system for recording numbers such as tallies or the knotted strings called quipu used by the Inca to store numerical data.[citation needed] Numeral systems have been many and diverse, with the first known written numerals created by Egyptians in Middle Kingdom texts such as the Rhind Mathematical Papyrus.[citation needed]
The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns and the recording of time. More complex mathematics did not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic, algebra and geometry for taxation and other financial calculations, for building and construction, and for astronomy.[13] The systematic study of mathematics in its own right began with the Ancient Greeks between 600 and 300 BC.
Mathematics has since been greatly extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today. According to Mikhail B. Sevryuk, in the January 2006 issue of the Bulletin of the American Mathematical Society, "The number of papers and books included in the Mathematical Reviews database since 1940 (the first year of operation of MR) is now more than 1.9 million, and more than 75 thousand items are added to the database each year. The overwhelming majority of works in this ocean contain new mathematical theorems and their proofs."[14]

Inspiration, pure and applied mathematics, and aesthetics

Mathematics arises from many different kinds of problems. At first these were found in commerce, land measurement, architecture and later astronomy; nowadays, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. For example, the physicist Richard Feynman invented the path integral formulation of quantum mechanics using a combination of mathematical reasoning and physical insight, and today's string theory, a still-developing scientific theory which attempts to unify the four fundamental forces of nature, continues to inspire new mathematics.[15] Some mathematics is only relevant in the area that inspired it, and is applied to solve further problems in that area. But often mathematics inspired by one area proves useful in many areas, and joins the general stock of mathematical concepts. A distinction is often made between pure mathematics and applied mathematics. However pure mathematics topics often turn out to have applications, e.g. number theory in cryptography. This remarkable fact that even the "purest" mathematics often turns out to have practical applications is what Eugene Wigner has called "the unreasonable effectiveness of mathematics."[16] As in most areas of study, the explosion of knowledge in the scientific age has led to specialization: there are now hundreds of specialized areas in mathematics and the latest Mathematics Subject Classification runs to 46 pages.[17] Several areas of applied mathematics have merged with related traditions outside of mathematics and become disciplines in their own right, including statistics, operations research, and computer science.
For those who are mathematically inclined, there is often a definite aesthetic aspect to much of mathematics. Many mathematicians talk about the elegance of mathematics, its intrinsic aesthetics and inner beauty. Simplicity and generality are valued. There is beauty in a simple and elegant proof, such as Euclid's proof that there are infinitely many prime numbers, and in an elegant numerical method that speeds calculation, such as the fast Fourier transform. G. H. Hardy in A Mathematician's Apology expressed the belief that these aesthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. He identified criteria such as significance, unexpectedness, inevitability, and economy as factors that contribute to a mathematical aesthetic.[18] Mathematicians often strive to find proofs of theorems that are particularly elegant, a quest Paul Erdős often referred to as finding proofs from "The Book" in which God had written down his favorite proofs.[19][20] The popularity of recreational mathematics is another sign of the pleasure many find in solving mathematical questions.

Notation, language, and rigor

Leonhard Euler, who created and popularized much of the mathematical notation used today
Most of the mathematical notation in use today was not invented until the 16th century.[21] Before that, mathematics was written out in words, a painstaking process that limited mathematical discovery.[22] Euler (1707–1783) was responsible for many of the notations in use today. Modern notation makes mathematics much easier for the professional, but beginners often find it daunting. It is extremely compressed: a few symbols contain a great deal of information. Like musical notation, modern mathematical notation has a strict syntax (which to a limited extent varies from author to author and from discipline to discipline) and encodes information that would be difficult to write in any other way.
Mathematical language can also be hard for beginners. Words such as or and only have more precise meanings than in everyday speech. Moreover, words such as open and field have been given specialized mathematical meanings. Mathematical jargon includes technical terms such as homeomorphism and integrable. But there is a reason for special notation and technical jargon: mathematics requires more precision than everyday speech. Mathematicians refer to this precision of language and logic as "rigor".
The infinity symbol in several typefaces.
Mathematical proof is fundamentally a matter of rigor. Mathematicians want their theorems to follow from axioms by means of systematic reasoning. This is to avoid mistaken "theorems", based on fallible intuitions, of which many instances have occurred in the history of the subject.[23] The level of rigor expected in mathematics has varied over time: the Greeks expected detailed arguments, but at the time of Isaac Newton the methods employed were less rigorous. Problems inherent in the definitions used by Newton would lead to a resurgence of careful analysis and formal proof in the 19th century. Misunderstanding the rigor is a cause for some of the common misconceptions of mathematics. Today, mathematicians continue to argue among themselves about computer-assisted proofs. Since large computations are hard to verify, such proofs may not be sufficiently rigorous.[24]
Axioms in traditional thought were "self-evident truths", but that conception is problematic. At a formal level, an axiom is just a string of symbols, which has an intrinsic meaning only in the context of all derivable formulas of an axiomatic system. It was the goal of Hilbert's program to put all of mathematics on a firm axiomatic basis, but according to Gödel's incompleteness theorem every (sufficiently powerful) axiomatic system has undecidable formulas; and so a final axiomatization of mathematics is impossible. Nonetheless mathematics is often imagined to be (as far as its formal content) nothing but set theory in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory.[25]

Mathematics as science

Carl Friedrich Gauss, himself known as the "prince of mathematicians",[26] referred to mathematics as "the Queen of the Sciences".
Carl Friedrich Gauss referred to mathematics as "the Queen of the Sciences".[27] In the original Latin Regina Scientiarum, as well as in German Königin der Wissenschaften, the word corresponding to science means (field of) knowledge. Indeed, this is also the original meaning in English, and there is no doubt that mathematics is in this sense a science. The specialization restricting the meaning to natural science is of later date. If one considers science to be strictly about the physical world, then mathematics, or at least pure mathematics, is not a science. Albert Einstein stated that "as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."[6]
Many philosophers believe that mathematics is not experimentally falsifiable, and thus not a science according to the definition of Karl Popper.[28] However, in the 1930s important work in mathematical logic showed that mathematics cannot be reduced to logic, and Karl Popper concluded that "most mathematical theories are, like those of physics and biology, hypothetico-deductive: pure mathematics therefore turns out to be much closer to the natural sciences whose hypotheses are conjectures, than it seemed even recently."[29] Other thinkers, notably Imre Lakatos, have applied a version of falsificationism to mathematics itself.
An alternative view is that certain scientific fields (such as theoretical physics) are mathematics with axioms that are intended to correspond to reality. In fact, the theoretical physicist, J. M. Ziman, proposed that science is public knowledge and thus includes mathematics.[30] In any case, mathematics shares much in common with many fields in the physical sciences, notably the exploration of the logical consequences of assumptions. Intuition and experimentation also play a role in the formulation of conjectures in both mathematics and the (other) sciences. Experimental mathematics continues to grow in importance within mathematics, and computation and simulation are playing an increasing role in both the sciences and mathematics, weakening the objection that mathematics does not use the scientific method.[citation needed] In his 2002 book A New Kind of Science, Stephen Wolfram argues that computational mathematics deserves to be explored empirically as a scientific field in its own right.
The opinions of mathematicians on this matter are varied. Many mathematicians feel that to call their area a science is to downplay the importance of its aesthetic side, and its history in the traditional seven liberal arts; others feel that to ignore its connection to the sciences is to turn a blind eye to the fact that the interface between mathematics and its applications in science and engineering has driven much development in mathematics. One way this difference of viewpoint plays out is in the philosophical debate as to whether mathematics is created (as in art) or discovered (as in science). It is common to see universities divided into sections that include a division of Science and Mathematics, indicating that the fields are seen as being allied but that they do not coincide. In practice, mathematicians are typically grouped with scientists at the gross level but separated at finer levels. This is one of many issues considered in the philosophy of mathematics.[citation needed]
Mathematical awards are generally kept separate from their equivalents in science. The most prestigious award in mathematics is the Fields Medal,[31][32] established in 1936 and now awarded every 4 years. It is often considered the equivalent of science's Nobel Prizes. The Wolf Prize in Mathematics, instituted in 1978, recognizes lifetime achievement, and another major international award, the Abel Prize, was introduced in 2003. These are awarded for a particular body of work, which may be innovation, or resolution of an outstanding problem in an established field. A famous list of 23 such open problems, called "Hilbert's problems", was compiled in 1900 by German mathematician David Hilbert. This list achieved great celebrity among mathematicians, and at least nine of the problems have now been solved. A new list of seven important problems, titled the "Millennium Prize Problems", was published in 2000. Solution of each of these problems carries a $1 million reward, and only one (the Riemann hypothesis) is duplicated in Hilbert's problems.

Fields of mathematics

An abacus, a simple calculating tool used since ancient times.
Mathematics can, broadly speaking, be subdivided into the study of quantity, structure, space, and change (i.e. arithmetic, algebra, geometry, and analysis). In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic, to set theory (foundations), to the empirical mathematics of the various sciences (applied mathematics), and more recently to the rigorous study of uncertainty.

Quantity

The study of quantity starts with numbers, first the familiar natural numbers and integers ("whole numbers") and arithmetical operations on them, which are characterized in arithmetic. The deeper properties of integers are studied in number theory, from which come such popular results as Fermat's Last Theorem. Number theory also holds two problems widely considered to be unsolved: the twin prime conjecture and Goldbach's conjecture.
As the number system is further developed, the integers are recognized as a subset of the rational numbers ("fractions"). These, in turn, are contained within the real numbers, which are used to represent continuous quantities. Real numbers are generalized to complex numbers. These are the first steps of a hierarchy of numbers that goes on to include quarternions and octonions. Consideration of the natural numbers also leads to the transfinite numbers, which formalize the concept of "infinity". Another area of study is size, which leads to the cardinal numbers and then to another conception of infinity: the aleph numbers, which allow meaningful comparison of the size of infinitely large sets.
1, 2, 3\,...\! ...-2, -1, 0, 1, 2\,...\!  -2, \frac{2}{3}, 1.21\,\! -e, \sqrt{2}, 3, \pi\,\! 2, i, -2+3i, 2e^{i\frac{4\pi}{3}}\,\!
Natural numbers Integers Rational numbers Real numbers Complex numbers

Structure

Many mathematical objects, such as sets of numbers and functions, exhibit internal structure as a consequence of operations or relations that are defined on the set. Mathematics then studies properties of those sets that can be expressed in terms of that structure; for instance number theory studies properties of the set of integers that can be expressed in terms of arithmetic operations. Moreover, it frequently happens that different such structured sets (or structures) exhibit similar properties, which makes it possible, by a further step of abstraction, to state axioms for a class of structures, and then study at once the whole class of structures satisfying these axioms. Thus one can study groups, rings, fields and other abstract systems; together such studies (for structures defined by algebraic operations) constitute the domain of abstract algebra. By its great generality, abstract algebra can often be applied to seemingly unrelated problems; for instance a number of ancient problems concerning compass and straightedge constructions were finally solved using Galois theory, which involves field theory and group theory. Another example of an algebraic theory is linear algebra, which is the general study of vector spaces, whose elements called vectors have both quantity and direction, and can be used to model (relations between) points in space. This is one example of the phenomenon that the originally unrelated areas of geometry and algebra have very strong interactions in modern mathematics.
Elliptic curve simple.svg Rubik's cube.svg Group diagdram D6.svg Lattice of the divisibility of 60.svg
Number theory Group theory Graph theory Order theory

Space

The study of space originates with geometry – in particular, Euclidean geometry. Trigonometry is the branch of mathematics that deals with relationships between the sides and the angles of triangles and with the trigonometric functions; it combines space and numbers, and encompasses the well-known Pythagorean theorem. The modern study of space generalizes these ideas to include higher-dimensional geometry, non-Euclidean geometries (which play a central role in general relativity) and topology. Quantity and space both play a role in analytic geometry, differential geometry, and algebraic geometry. Within differential geometry are the concepts of fiber bundles and calculus on manifolds, in particular, vector and tensor calculus. Within algebraic geometry is the description of geometric objects as solution sets of polynomial equations, combining the concepts of quantity and space, and also the study of topological groups, which combine structure and space. Lie groups are used to study space, structure, and change. Topology in all its many ramifications may have been the greatest growth area in 20th century mathematics; it includes point-set topology, set-theoretic topology, algebraic topology and differential topology. In particular, instances of modern day topology are metrizability theory, axiomatic set theory, homotopy theory, and Morse theory. Topology also includes the now solved Poincaré conjecture and the controversial four color theorem, whose only proof, by computer, has never been verified by a human.
Illustration to Euclid's proof of the Pythagorean theorem.svg Sine cosine plot.svg Hyperbolic triangle.svg Torus.png Mandel zoom 07 satellite.jpg Measure illustration.png
Geometry Trigonometry Differential geometry Topology Fractal geometry Measure Theory

Change

Understanding and describing change is a common theme in the natural sciences, and calculus was developed as a powerful tool to investigate it. Functions arise here, as a central concept describing a changing quantity. The rigorous study of real numbers and functions of a real variable is known as real analysis, with complex analysis the equivalent field for the complex numbers. Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions. One of many applications of functional analysis is quantum mechanics. Many problems lead naturally to relationships between a quantity and its rate of change, and these are studied as differential equations. Many phenomena in nature can be described by dynamical systems; chaos theory makes precise the ways in which many of these systems exhibit unpredictable yet still deterministic behavior.
Integral as region under curve.svg Vector field.svg Airflow-Obstructed-Duct.png Limitcycle.jpg Lorenz attractor.svg Princ argument ex1.png
Calculus Vector calculus Differential equations Dynamical systems Chaos theory Complex analysis

Foundations and philosophy

In order to clarify the foundations of mathematics, the fields of mathematical logic and set theory were developed. Mathematical logic includes the mathematical study of logic and the applications of formal logic to other areas of mathematics; set theory is the branch of mathematics that studies sets or collections of objects. Category theory, which deals in an abstract way with mathematical structures and relationships between them, is still in development. The phrase "crisis of foundations" describes the search for a rigorous foundation for mathematics that took place from approximately 1900 to 1930.[33] Some disagreement about the foundations of mathematics continues to present day. The crisis of foundations was stimulated by a number of controversies at the time, including the controversy over Cantor's set theory and the Brouwer-Hilbert controversy.
Mathematical logic is concerned with setting mathematics on a rigorous axiomatic framework, and studying the results of such a framework. As such, it is home to Gödel's second incompleteness theorem, perhaps the most widely celebrated result in logic, which (informally) implies that any formal system that contains basic arithmetic, if sound (meaning that all theorems that can be proven are true), is necessarily incomplete (meaning that there are true theorems which cannot be proved in that system). Gödel showed how to construct, whatever the given collection of number-theoretical axioms, a formal statement in the logic that is a true number-theoretical fact, but which does not follow from those axioms. Therefore no formal system is a true axiomatization of full number theory.[citation needed] Modern logic is divided into recursion theory, model theory, and proof theory, and is closely linked to theoretical computer science.
 p \Rightarrow q \, Venn A intersect B.svg Commutative diagram for morphism.svg
Mathematical logic Set theory Category theory

Discrete mathematics

Discrete mathematics is the common name for the fields of mathematics most generally useful in theoretical computer science. This includes, on the computer science side, computability theory, computational complexity theory, and information theory. Computability theory examines the limitations of various theoretical models of the computer, including the most powerful known model – the Turing machine. Complexity theory is the study of tractability by computer; some problems, although theoretically solvable by computer, are so expensive in terms of time or space that solving them is likely to remain practically unfeasible, even with rapid advance of computer hardware. Finally, information theory is concerned with the amount of data that can be stored on a given medium, and hence deals with concepts such as compression and entropy.
On the purely mathematical side, this field includes combinatorics and graph theory.
As a relatively new field, discrete mathematics has a number of fundamental open problems. The most famous of these is the "P=NP?" problem, one of the Millennium Prize Problems.[34]
\begin{matrix} (1,2,3) & (1,3,2) \\ (2,1,3) & (2,3,1) \\ (3,1,2) & (3,2,1) \end{matrix} DFAexample.svg Caesar3.svg 6n-graf.svg
Combinatorics Theory of computation Cryptography Graph theory

Applied mathematics

Applied mathematics considers the use of abstract mathematical tools in solving concrete problems in the sciences, business, and other areas.
Applied mathematics has significant overlap with the discipline of statistics, whose theory is formulated mathematically, especially with probability theory. Statisticians (working as part of a research project) "create data that makes sense" with random sampling and with randomized experiments; the design of a statistical sample or experiment specifies the analysis of the data (before the data be available). When reconsidering data from experiments and samples or when analyzing data from observational studies, statisticians "make sense of the data" using the art of modelling and the theory of inference – with model selection and estimation; the estimated models and consequential predictions should be tested on new data.[35]
Computational mathematics proposes and studies methods for solving mathematical problems that are typically too large for human numerical capacity. Numerical analysis studies methods for problems in analysis using ideas of functional analysis and techniques of approximation theory; numerical analysis includes the study of approximation and discretization broadly with special concern for rounding errors. Other areas of computational mathematics include computer algebra and symbolic computation.

See also

Notes

  1. ^ No likeness or description of Euclid's physical appearance made during his lifetime survived antiquity. Therefore, Euclid's depiction in works of art depends on the artist's imagination (see Euclid).
  2. ^ Steen, L.A. (April 29, 1988). The Science of Patterns. Science, 240: 611–616. and summarized at Association for Supervision and Curriculum Development., ascd.org
  3. ^ Devlin, Keith, Mathematics: The Science of Patterns: The Search for Order in Life, Mind and the Universe (Scientific American Paperback Library) 1996, ISBN 9780716750475
  4. ^ Jourdain.
  5. ^ Peirce, p. 97.
  6. ^ a b Einstein, p. 28. The quote is Einstein's answer to the question: "how can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?" He, too, is concerned with The Unreasonable Effectiveness of Mathematics in the Natural Sciences.
  7. ^ Eves
  8. ^ Peterson
  9. ^ Both senses can be found in Plato. Liddell and Scott, s.voceμαθηματικός
  10. ^ The Oxford Dictionary of English Etymology, Oxford English Dictionary, sub "mathematics", "mathematic", "mathematics"
  11. ^ S. Dehaene; G. Dehaene-Lambertz; L. Cohen (Aug 1998). "Abstract representations of numbers in the animal and human brain". Trends in Neuroscience 21 (8): pp. 355–361. doi:10.1016/S0166-2236(98)01263-6. 
  12. ^ See, for example, Raymond L. Wilder, Evolution of Mathematical Concepts; an Elementary Study, passim
  13. ^ Kline 1990, Chapter 1.
  14. ^ Sevryuk
  15. ^ Johnson, Gerald W.; Lapidus, Michel L. (2002). The Feynman Integral and Feynman's Operational Calculus. Oxford University Press. 
  16. ^ Eugene Wigner, 1960, "The Unreasonable Effectiveness of Mathematics in the Natural Sciences," Communications on Pure and Applied Mathematics 13(1): 1–14.
  17. ^ Mathematics Subject Classification 2010
  18. ^ Hardy, G. H. (1940). A Mathematician's Apology. Cambridge University Press. 
  19. ^ Gold, Bonnie; Simons, Rogers A. (2008). Proof and Other Dilemmas: Mathematics and Philosophy. MAA. 
  20. ^ Aigner, Martin; Ziegler, Gunter M. (2001). Proofs from the Book. Springer. 
  21. ^ Earliest Uses of Various Mathematical Symbols (Contains many further references).
  22. ^ Kline, p. 140, on Diophantus; p.261, on Vieta.
  23. ^ See false proof for simple examples of what can go wrong in a formal proof. The history of the Four Color Theorem contains examples of false proofs accidentally accepted by other mathematicians at the time.
  24. ^ Ivars Peterson, The Mathematical Tourist, Freeman, 1988, ISBN 0-7167-1953-3. p. 4 "A few complain that the computer program can't be verified properly", (in reference to the Haken-Apple proof of the Four Color Theorem).
  25. ^ Patrick Suppes, Axiomatic Set Theory, Dover, 1972, ISBN 0-486-61630-4. p. 1, "Among the many branches of modern mathematics set theory occupies a unique place: with a few rare exceptions the entities which are studied and analyzed in mathematics may be regarded as certain particular sets or classes of objects."
  26. ^ Zeidler, Eberhard (2004). Oxford User's Guide to Mathematics. Oxford, UK: Oxford University Press. p. 1188. ISBN 0198507631. 
  27. ^ Waltershausen
  28. ^ Shasha, Dennis Elliot; Lazere, Cathy A. (1998). Out of Their Minds: The Lives and Discoveries of 15 Great Computer Scientists. Springer. p. 228. 
  29. ^ Popper 1995, p. 56
  30. ^ Ziman
  31. ^ "The Fields Medal is now indisputably the best known and most influential award in mathematics." Monastyrsky
  32. ^ Riehm
  33. ^ Luke Howard Hodgkin & Luke Hodgkin, A History of Mathematics, Oxford University Press, 2005.
  34. ^ Clay Mathematics Institute, P=NP, claymath.org
  35. ^ Like other mathematical sciences such as physics and computer science, statistics is an autonomous discipline rather than a branch of applied mathematics. Like research physicists and computer scientists, research statisticians are mathematical scientists. Many statisticians have a degree in mathematics, and some statisticians are also mathematicians.

References

External links

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Quotes

Up to date as of January 14, 2010

From Wikiquote

Mathematics is the body of knowledge centered on concepts such as quantity, structure, space, and change, and the academic discipline which studies them. It evolved, through the use of abstraction and logical reasoning, from counting, calculation, measurement, and the study of the shapes and motions of physical objects. Mathematicians explore such concepts, aiming to formulate new conjectures and establish their truth by rigorous deduction from appropriately chosen axioms and definitions.

Contents

Sourced

  • Any author who uses mathematics should always express in ordinary language the meaning of the assumptions he admits, as well as the significance of the results obtained. The more abstract his theory, the more imperative this obligation. In fact, mathematics are and can only be a tool to explore reality. In this exploration, mathematics do not constitute an end in itself, they are and can only be a means.
  • Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where the explorers often get lost. Rigour should be a signal to the historian that the maps have been made, and the real explorers have gone elsewhere.
    • W.S. Anglin, in Mathematics and History, elucidating the symmetry between the creative and logical aspects of mathematics.
  • If in other sciences we should arrive at certainty without doubt and truth without error, it behooves us to place the foundations of knowledge in mathematics.
  • Great fleas have little fleas upon their backs to bite 'em,
    And little fleas have lesser fleas, and so ad infinitum,
    And the great fleas themselves, in turn, have greater fleas to go on,
    While these again have greater still, and greater still, and so on.
  • I shall here present the view that numbers, even whole numbers, are words, parts of speech, and that mathematics is their grammar. Numbers were therefore invented by people in the same sense that language, both written and spoken, was invented. Grammar is also an invention. Words and numbers have no existence separate from the people who use them. Knowledge of mathematics is transmitted from one generation to another, and it changes in the same slow way that language changes. Continuity is provided by the process of oral or written transmission.
    • Carl Eckart, Our Modern Idol: Mathematical Science (1984), p. 95
  • Mathematics has the dubious honor of being the least popular subject in the curriculum ... Future teachers pass through the elementary schools learning to detest mathematics. They drop it in high school as early as possible. They avoid it in teachers colleges because it is not required. They return to the elementary school to teach a new generation to detest it.
    • Report of the Educational Testing Service, Princeton, N. J., as quoted in TIME magazine (18 June 1956), cited by George Pólya, How to Solve It, Page ix in Expanded Princeton Science Library Edition (2004), ISBN 0-691-11966-X
  • Numbers exist only in our minds. There is no physical entity that is number 1. If there were, 1 would be in a place of honor in some great museum of science, and past it would file a steady stream of mathematicians gazing at 1 in wonder and awe.
    • Linear Algebra by Fraleigh/Beauregard
  • A man with all the algebra in the world is often only an ass when he knows nothing else. Perhaps in ten years society may derive advantage from the curves which these visionary algebraists will have laboriously squared. I congratulate posterity beforehand. But to tell you the truth I see nothing but a scientific extravagance in all these calculations. That which is neither useful nor agreeable is worthless. And as for useful things, they have all been discovered; and to those which are agreeable, I hope that good taste will not admit algebra among them.
  • As to your Newton, I confess I do not understand his void and his gravity; I admit he has demonstrated the movement of the heavenly bodies with more exactitude than his forerunners; but you will admit it is an absurdity to to maintain the existence of Nothing.
    • Frederick the Great, Letters of Voltaire and Frederick the Great (New York: Brentano's, 1927), transl. Richard Aldington, letter 221 from Frederick to Voltaire, 25 November 1777.
  • Euler calculated the force of the wheels necessary to raise the water in a reservoir … My mill was carried out geometrically and could not raise a drop of water fifty yards from the reservoir. Vanity of vanities! Vanity of geometry!
    • Frederick the Great, Letters of Voltaire and Frederick the Great (New York: Brentano's, 1927), transl. Richard Aldington, letter 221 from Frederick to Voltaire, 25 November 1777.
  • The idea that theorems follow from the postulates does not correspond to simple observation. If the Pythagorean theorem were found to not follow from the postulates, we would again search for a way to alter the postulates until it was true. Euclid's postulates came from the Pythagorean theorem, not the other way around.
    • Richard Hamming, "The Unreasonable Effectiveness of Mathematics", The American Mathematical Monthly 87 (2), February 1980, pp. 81-90
  • ... from the intrinsic evidence of his creation, the Great Architect of the Universe now begins to appear as a pure mathematician.
    • Sir James Jeans, The Mysterious Universe, pg. 165.
  • Mathematics, rightly viewed, possesses not only truth, but supreme beauty —a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as poetry.
  • 1050 is a long way from infinity.
    • Daniel Shanks, Solved and Unsolved Problems in Number Theory, 3rd edition, chapter IV, page 217.
    • Computer calculation even up to a big number can't really say much about asymptotic behaviour.
  • So, nat'ralists observe, a flea
    Hath smaller fleas that on him prey,
    And these have smaller still to bite 'em
    And so proceed ad infinitum.
  • The Koch Curve - Triangles outside triangles outside triangles ad infinitum the Koch curve goes, it's infinitely infinitesimal, this self-similarity shows. A length too long to measure, an area too small to see, what else can this contradiction be, behold fractal geometry.
    • Bernt Wahl, in The Adventures of the Fractal Explorer, about the beauty of Fractal Geometry

Unsourced

  • Mathematics is music for the mind; Music is mathematics for the soul.
    • Stanley Gudder
  • Mathematics is man's attempt on understanding nature.
  • A mathematician is a blind man in a dark room looking for a black cat which isn't there.
  • Mathematics seems to endow one with something like a new sense.
  • Anyone who cannot cope with mathematics is not fully human. At best he is a tolerable sub-human who has learned to wear shoes, bathe, and not make messes in the house.
  • As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.
  • Do not worry about your problems with mathematics, I assure you mine are far greater.
  • God does not care about our mathematical difficulties. He integrates empirically.
  • I don't believe in mathematics.
  • Since the mathematicians have invaded the theory of relativity, I do not understand it myself anymore.
  • As long as algebra is taught in school, there will be prayer in school.
  • He who can properly define and divide is to be considered a god.
  • The knowledge of which geometry aims is the knowledge of the eternal.
  • Mathematics is like checkers in being suitable for the young, not too difficult, amusing, and without peril to the state.
  • I admit that mathematical science is a good thing. But excessive devotion to it is a bad thing.
  • I am accustomed, as a professional mathematician, to living in a sort of vacuum, surrounded by people who declare with an odd sort of pride that they are mathematically illiterate.
    • David Mumford
  • I have no faith in political arithmetic.
  • If I feel unhappy, I do mathematics to become happy. If I am happy, I do mathematics to keep happy.
    • Alfréd Rényi
  • In great mathematics there is a very high degree of unexpectedness, combined with inevitability and economy.
  • In mathematics you don't understand things. You just get used to them.
  • It is easy to lie with statistics. It is hard to tell the truth without it.
  • Life is good for only two things, discovering mathematics and teaching mathematics.
    • Siméon Poisson
  • Mathematical proofs, like diamonds, are hard and clear, and will be touched with nothing but strict reasoning.
  • Mathematicians are like lovers. Grant a mathematician the least principle, and he will draw from it a consequence which you must also grant him, and from this consequence another.
    • Bernard Le Bouyer de Fontenelle
  • [Mathematics] is an independent world created out of pure intelligence.
  • Mathematics is not yet capable of coping with the naïveté of the mathematician himself.
    • Abraham Kaplan
  • Mathematics is the only instructional material that can be presented in an entirely undogmatic way.
    • Max Dehn
  • Mathematics is the science of what is clear by itself.
  • Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true. (referring to the axiomatic method, where certain properties of an (otherwise unknown) structure are assumed and consequences thereof are then logically derived)
  • Mathematics takes us into the region of absolute necessity, to which not only the actual word, but every possible word, must conform.
  • How dare we speak of the laws of chance? Is not chance the antithesis of all law?
  • Measure what is measurable, and make measurable what is not so.
  • No human investigation can be called real science if it cannot be demonstrated mathematically.
  • Now I feel as if I should succeed in doing something in mathematics, although I cannot see why it is so very important...The knowledge doesn't make life any sweeter or happier, does it?
  • One cannot escape the feeling that these mathematical formulas have an independent existence and an intelligence of their own, that they are wiser than we are, wiser even than their discoverers, that we get more out of them than was originally put into them.
    • Heinrich Rudolf Hertz
  • Sex is the mathematics urge sublimated.
    • M. C. Reed
  • The art of doing mathematics consists in finding that special case which contains all the germs of generality.
  • The infinite! No other question has ever moved so profoundly the spirit of man.
  • Mathematics is a game played according to certain simple rules with meaningless marks on paper.
  • The imaginary number is a fine and wonderful recourse of the divine spirit, almost an amphibian between being and not being.
  • The mathematician has reached the highest rung on the ladder of human thought.
  • The science of mathematics presents the most brilliant example of how pure reason may successfully enlarge its domain without the aid of experience.
  • The simplest schoolboy is now familiar with facts for which Archimedes would have sacrificed his life.
  • There is no branch of mathematics, however abstract, which may not some day be applied to phenomena of the real world.
  • There is something I don't understand about algebra: It has been around for thousands of years, yet no one has ever found out what the value of "x" or "y" really is.
    • Richard van der Merwe

Misattributed

  • The good Christian should beware of mathematicians, and all those who make empty prophesies. The danger already exists that the mathematicians have made a covenant with the devil to darken the spirit and to confine man in the bonds of Hell.
    • Misattributed to St. Augustine. This is a very bad mistranslation of De genesim ad litteram libri XII, book 2, 17.37. 'Mathematici' in Latin means astrologers, not mathematicians, and the book makes repeated attacks on astrology. The text really reads: For which reason both astrologers and those impiously making divinings, as the truth says emphatically, must be avoided by the good Christian, lest after making a pact of agreement they entangle their soul in a hidden partnership with demons.

See also

External links

Wikipedia
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Look up mathematics in Wiktionary, the free dictionary

Study guide

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(Redirected to Portal:Mathematics article)

From Wikiversity

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The Mathematics Portal

Welcome to the Mathematics Portal! .This page connects visitors to the learning resources that have been developed by various Wikiversity content development projects.^ These cross-connections enable insights to be developed into the various parts; together, they strengthen belief in the correctness and underlying unity of the whole structure.
  • Chapter 2: The Nature of Mathematics 28 January 2010 0:26 UTC www.project2061.org [Source type: FILTERED WITH BAYES]

^ "I wanted to literally make algebra child's play," explains Borenson, who developed his system in working with children of various abilities, including children with learning difficulties.
  • Hands On Equations 10 February 2010 11:011 UTC www.borenson.com [Source type: General]

^ In order to develop a Christian approach to an exact science such as Statistics, it is useful to view such a science within the various broader contexts with which it is connected.
  • Mathematics and Christianity 28 January 2010 0:26 UTC www.asa3.org [Source type: Academic]

.Wikiversity participants who are interested in mathematics are invited to create and develop learning projects and learning resources and help organize them by developing this portal.^ The goal of the partnership is to create opportunities for the fellows to communicate their research to a variety of audiences and to enhance the mathematics taught in the K-12 learning environments.
  • University of Hawaii Mathematics Department 2 February 2010 15:42 UTC www.math.hawaii.edu [Source type: FILTERED WITH BAYES]

^ My skills are for hire in tutoring, teaching; helping people at work with simple formulas or with research projects for profit or not; and helping math teachers learn the maths they are suppose to teach.
  • Arithmetic Video Lessons, Notes and Exercises 3 February 2010 14:24 UTC whyslopes.com [Source type: FILTERED WITH BAYES]

^ It addresses all aspects of an academic career: improving the teaching and learning of mathematics, engaging in research and scholarship, and participating in professional activities.
  •  Department of Mathematics and Statistics 28 January 2010 0:26 UTC math.eku.edu [Source type: FILTERED WITH BAYES]

.View a list of existing learning projects for mathematics: Mathematics learning projects.^ Talking, Writing, and Mathematical Thinking Click the “View a Sample Chapter/Article” link to download the first chapter, which discusses how students’ writing can lead to deeper learning.

^ Canadian Mathematics Society's List of Projects for Science Fairs To date there have not been a lot of mathematics projects in the science fairs and we believe that one reason for this might be that it is not at all clear what a mathematics project might involve.
  • Mathematics Archives - K12 Internet Sites 2 February 2010 15:42 UTC archives.math.utk.edu [Source type: Academic]

^ Also, participants will build classroom vocabulary lists, learn methods on how to speak and write mathematics, and explore vocabulary strategies, such as personalized word walls and mathematical reflections.

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Featured learning resource

.Exercises at Introduction to differentiation lead participants through calculations of compound interest that introduce e (2.7182818284...^ Simple Interest Vs Compound Interest Understanding the basic difference between simple interest vs compound interest is of prime importance, in order to calculate the amount you have to pay or deposit in a bank.
  • Mathematics - Math Articles 28 January 2010 0:26 UTC www.buzzle.com [Source type: FILTERED WITH BAYES]

^ A study of polynomials and rational functions leads to the introduction of the basic ideas of differential and integral calculus.

^ How to Calculate Compound Interest - Calculating Compound Interest Calculation of compound interest is one of the basic and widely used mathematical calculation.
  • Mathematics - Math Articles 28 January 2010 0:26 UTC www.buzzle.com [Source type: FILTERED WITH BAYES]

) and derivatives of functions.
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Selected picture

Solutions of the the Bessel differential equation.
.Bessel functions arise in many mathematical models such as those for vibrating surfaces.^ Many leading historians of mathematics have contributed to the perception by either omitting any mention of Arabic/Islamic mathematics in the historical development of the subject or with statements such as that made by Duhem in [ 3 ]:- ...

^ Incorporating elements of logic and abstraction, mathematics is found in many career fields such as the natural sciences, the medical field, engineering and even music.
  • History of Mathematics: Video Series | eHow Videos 28 January 2010 0:26 UTC www.ehow.com [Source type: FILTERED WITH BAYES]

^ Many of the logistic ideas are similar to those of the formalists, but the latter group does not believe that mathematics can be deduced from logic alone.

.When mathematically analyzing a vibrating drum the boundary conditions lead to solutions that are harmonic functions.^ Develops tools to analyze phenomena in the physical and life sciences, from cell aggregation to vibrating drums to traffic jams.

^ Just as physical constants provide "boundary conditions" for the physical universe, mathematical constants somehow characterize the structure of mathematics.
  • Frank Potter's Science Gems - Mathematics 28 January 2010 0:26 UTC www.sciencegems.com [Source type: Academic]

^ Under linear and time-harmonic conditions, a set of periodic Green's functions is derived to combine the interactions of an infinite number of identical equispaced floating bodies.
  • Alltop - Top Math News 2 February 2010 15:42 UTC math.alltop.com [Source type: General]

.When using cylindrical coordinates, the solutions are sines, cosines or Bessel functions (in the radial direction).^ The chain rule and inverse function theorems for several variables with applications to maxima and minima, integration in polar, cylindrical, and spherical coordinate systems.
  • UMES - Math and Computer Science - Undergraduate Course Description 28 January 2010 0:26 UTC www.umes.edu [Source type: Academic]

^ It is very useful to be able to find the roots of a general function F(x), which are the solutions of the equation F(x) = 0, whose degree is the largest power of x involved.
  • Algebra 10 February 2010 11:011 UTC www.du.edu [Source type: Academic]

^ Calculus tutorial Karl's calculus tutorial, limits, continuity, derivatives, applications of derivatives, exponentials and logarithms, trig functions (sine, cosine, etc.
  • The educational encyclopedia, mathematics 28 January 2010 0:26 UTC www.educypedia.be [Source type: Academic]

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Selected research

Mersenne primes.
.Marin Mersenne's name is widely known because of his interest in prime numbers that are one less than a power of two.^ Multiplying Two Numbers Close to but less than 100 .
  • Dividing a Whole Number by a Fraction Whose Numerator is 1 10 February 2010 11:011 UTC www.algebra1help.com [Source type: FILTERED WITH BAYES]

^ Multiplication - Numbers less than 1000 .
  • 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]

^ Division - Numbers less than 1000 .
  • 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]

.Mersenne also did research in the area of the theory of music and musical instruments (see Wikipedia).^ A professor at UCLA since 1987, Popa is a world-leading researcher in the areas of functional analysis, operator algebras, subfactor theory, and ergodic theory.
  • UCLA Department of Mathematics 2 February 2010 15:42 UTC www.math.ucla.edu [Source type: Academic]

^ After seeing this movie, I did some research on Drancy and was shocked to learn it was the French, with the approval of the Nazi's, who did this.
  • Emotional Arithmetic (2007) 3 February 2010 14:24 UTC www.imdb.com [Source type: FILTERED WITH BAYES]

^ Current research interest areas of the faculty include applied probability, nonparametric statistics, and statistical reliability theory and applications.
  • NJIT - Graduate Programs: Mathematics 28 January 2010 0:26 UTC catalog.njit.edu [Source type: Academic]

.Participants at Mersenne primes are encouraged to join the distributed computing project that allows the computing power of personal computers to help search for Mersenne prime numbers.^ Divisibility properties of the integers, prime and composite numbers, modular arithmetic, congruence equations, Diophantine equations, the distribution of primes and discussion of some famous unsolved problems.

^ LCM, GCD Factor two numbers into primes, then use a Venn diagram to compute their Least Common Multiple (LCM) and Greatest Common Divisor (GCD).
  • Math Graphic Organizer Printouts - EnchantedLearning.com 2 February 2010 15:42 UTC www.enchantedlearning.com [Source type: General]

^ Numbers, constants and computation π, e, log 2 and other constants, binary splitting, Newton's iteration and other algorithms, counting primes, and more...

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Did you know...

The red curve is a cycloid.
.The Euler-Lagrange equation of classical mechanics was discovered during attempts to find a curve for which the time taken by a frictionless particle sliding down it under uniform gravity to its lowest point is independent of its starting point.^ Much of classical algebra is concerned with finding solutions to equations or systems of equations, i.e., finding the roots , or values of the unknowns, that upon substitution into the original equation will make it a numerical identity.
  • algebra Facts, information, pictures | Encyclopedia.com articles about algebra 10 February 2010 11:011 UTC www.encyclopedia.com [Source type: Academic]

^ Algebra may divided into "classical algebra" (equation solving or "find the unknown number" problems) and "abstract algebra", also called "modern algebra" (the study of groups, rings, and fields).
  • Highlights in the History of Algebra 10 February 2010 11:011 UTC www.ucs.louisiana.edu [Source type: Reference]

^ 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]


.Did you know that Wikiversity can never have enough examples and solved problems?^ For example, do you know what "the akashic records" are?
  • Brain Exercises - Mental Maths - Mental Math - Mental Arithmetic. 3 February 2010 14:24 UTC www.mental-workout.com [Source type: General]

^ Now this problem is for you to solve!

^ Then you are given a similar problem to solve.
  • Algebra Games - The fun way to learn Algebra 10 February 2010 11:011 UTC www.squidoo.com [Source type: General]

Wikiversity does not always have to re-create the wheel. .Do you know a good website with mathematics example problems?^ For example, do you know what "the akashic records" are?
  • Brain Exercises - Mental Maths - Mental Math - Mental Arithmetic. 3 February 2010 14:24 UTC www.mental-workout.com [Source type: General]

^ This website assumes you know the basics, i.e.

^ If you are interested in helping to develop the "User Contributed" section of the website (not yet publicly available), please let me know.
  • Free Math Worksheets 2 February 2010 15:42 UTC math-drills.com [Source type: FILTERED WITH BAYES]

.Wikiversity needs branches of the Hunter-gatherers project for each academic subject area.^ In the first stage of its work, Project 2061 commissioned panels of scientists, mathematicians, and technologists to identify the knowledge and skills students should have in five subject areas.
  • Mathematics Archives - K12 Internet Sites 2 February 2010 15:42 UTC archives.math.utk.edu [Source type: Academic]

^ Is the project devoted only to mathematics (or a single subject area), or is there a link to other curricular areas?
  • Math Projects 2 February 2010 15:42 UTC www.ct4me.net [Source type: FILTERED WITH BAYES]

^ Project-based online conversations typically range in length from 6 weeks to a full academic year, as students' needs and interests dictate.
  • Math Projects 2 February 2010 15:42 UTC www.ct4me.net [Source type: FILTERED WITH BAYES]

.Please make note of good online mathematics resources at Hunter-gatherers/Mathematics and link to these resources from the appropriate mathematics pages of Wikiversity.^ To find these resources, scroll to "Mathematics".

^ Many of these resources are accessible on this link.
  • Mathematics Archives - K12 Internet Sites 2 February 2010 15:42 UTC archives.math.utk.edu [Source type: Academic]

^ Link to other online Math resources.

Example. .Did you know that there is a wiki for mathematics example problems?^ For example, do you know what "the akashic records" are?
  • Brain Exercises - Mental Maths - Mental Math - Mental Arithmetic. 3 February 2010 14:24 UTC www.mental-workout.com [Source type: General]

^ Problems can be defined so that you'll know...
  • Download Arithmetic Problems Software: Math Quiz Creator, EMSolution Arithmetic, Calculation Made Easy, ... 3 February 2010 14:24 UTC www.filebuzz.com [Source type: FILTERED WITH BAYES]

^ New Did you know?
  • Algebra Games - The fun way to learn Algebra 10 February 2010 11:011 UTC www.squidoo.com [Source type: General]

.See www.exampleproblems.com.^ More information is available at: http://www.lorealusa.com/?uid=forwomeninscience See also Programs for Students See also Prizes for Students and Educators Grants in the Mathematical Sciences .
  • AWM Career Resources 28 January 2010 0:26 UTC www.awm-math.org [Source type: Academic]

^ Roman Numeral Conversion http://www.ivtechstudios.com/roman/ Enter the number you want to convert (Arabic or Roman) and instantly see the equivalent!
  • Math Portal: 9 - 12 2 February 2010 15:42 UTC fcit.usf.edu [Source type: FILTERED WITH BAYES]

^ For more information, see www.careercast.com/jobs/jobsRated .
  •  Department of Mathematics and Statistics 28 January 2010 0:26 UTC math.eku.edu [Source type: FILTERED WITH BAYES]

.How do participants in the mathematics area of Wikipedia feel about linking to .com websites?^ Talking, Writing, and Mathematical Thinking Click the “View a Sample Chapter/Article” link to download the first chapter, which discusses how students’ writing can lead to deeper learning.

^ "Not a book about mathematics itself, but rather about how the brain deals with numbers".

^ An extensive source of information about the crisis in mathematics education is the website of Mathematically Correct .
  • Illinois Loop: Mathematics 28 January 2010 0:26 UTC www.illinoisloop.org [Source type: FILTERED WITH BAYES]

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Related portals

Many mathematics-related learning resources are for specific scientific sub-disciplines and can be found with the aid of other Wikiversity portals: Engineering and Technology - Life Sciences - Physical Sciences - Social Sciences - - General Science Portal
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Content development projects

.Mathematics-related content development projects exist on pages in "School:" or "Topic:" namespaces.^ Miscellaneous Web Pages related to Mathematics .
  • A Catalog of Mathematics Resources on WWW and the Internet 2 February 2010 15:42 UTC mthwww.uwc.edu [Source type: Academic]

^ It gave mathematics a whole new development path so much broader in concept to that which had existed before, and provided a vehicle for future development of the subject.

^ The library of over 300 mathematics assessment tasks developed during the project remains freely available through this web site.
  • NH Department of Education - Math Curriculum 28 January 2010 0:26 UTC www.ed.state.nh.us [Source type: Academic]



Schools: Mathematics - Computer Science - Economics - Olympiads
Topics: (Edit within the MathDeps Template)
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Categories

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Things you can do

Stock post message.svg To-do list for Portal:Mathematics: edit · history · watch · refresh

.Here are some tasks you can do:
  • Page requests: There should be a system that orders each learning resource according to a sequence from easy to advanced.
  • Verify: Sources need to be cited.^ There is no implied ordering to the sequence; duplicates need not be adjacent.
    • SPARQL Algebra (Draft) 10 February 2010 11:011 UTC www.w3.org [Source type: Academic]

    ^ Here you can take a self-directed course to learn about techniques for collecting and analyzing data that does not require complex mathematics.
    • A Math Guide to FCIT 2 February 2010 15:42 UTC fcit.usf.edu [Source type: FILTERED WITH BAYES]

    ^ Learn the Ham Sandwich Theorem, and how fast you should run in the rain to stay the driest.

    Content that is not significantly different from existing Wikipedia or Wikibooks content needs to be modified, hopefully by inclusion of learning activities for learners.
  • Proof reading: Page name cleanup
  • Other: Wikiversity open tasks
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Mathematics news

Graphical representation of a dynamical system.
."Stephen Smale awarded Wolf Prize in mathematics" by Robert Sanders
Stephen Smale, known for his work on Topology, the study of dynamical systems and a list of 18 problems in mathematics to be solved in the 21st century, known as Smale's problems.
^ Notably absent from this list is an award for Mathematics.
  • Math Jokes and Archimedes - Jokes and Science 2 February 2010 15:42 UTC www.juliantrubin.com [Source type: Original source]

^ A bachelor's degree in mathematics is a key that unlocks hundreds of different doors, ranging from law school to systems analysis to a career in business to graduate study in mathematics.

^ A short introduction to those aspects of C and C++ essential for mathematics, followed by extensive work with mathematics problems in which computation plays an important role.

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Quotes

."A man provided with paper, pencil, and rubber, and subject to strict discipline, is in effect a universal Turing Machine."^ Moreover, the EDVAC is a model for a ``universal computing machine'' in the sense of Turing [ T ].
  • Teaching Discrete Mathematics 28 January 2010 0:26 UTC www.math.nmsu.edu [Source type: Academic]

-Alan Turing
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Wikibooks

Up to date as of January 23, 2010
(Redirected to Wikibooks:Mathematics bookshelf article)

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Bookshelves
.Biology | Computer Science | Computer Software | Education | Health science | History | Humanities | Language and Literature | Languages | Law | Mathematics | Natural Sciences | Physics | Programming Languages | Social Sciences | Study Guides | Misc.^ Throughout human history mathematics has been used as a language of science of patterns, models, and relationships to describe phenomena in the physical, social, and natural environment.
  • Kansas Wesleyan University :: Academics :: Mathematics :: Home 28 January 2010 0:26 UTC www.kwu.edu [Source type: Academic]

^ New Jersey: Core Curriculum Content Standards - Adopted in 1996 and subject to revision every five years, the NJ standards deal with the arts, career education, ELA, FACS, health, math, PE, science, social studies, technological literacy, and world languages.
  • Developing Educational Standards - Math 2 February 2010 15:42 UTC edstandards.org [Source type: Academic]

^ Genesee Area Mathematics/Science/Technology Center The Genesee Area Mathematics/Science/Technology Center seeks to harness community resources for supporting the dedicated parents, students, and educators.
  • Mathematics Archives - K12 Internet Sites 2 February 2010 15:42 UTC archives.math.utk.edu [Source type: Academic]

| Wikibooks Help
Mathematics
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All Mathematics books.
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Contents

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Introductory Mathematics

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Basic Math

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Algebra

Elementary

Intermediate

Geometry

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Trigonometry

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Discrete Mathematics

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Calculus

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Statistics and Probability

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Higher Mathematics

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Foundations

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Analysis

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Linear Algebra

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Advanced Algebra

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Geometry and Topology

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Differential Equations

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Applied Mathematics

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Other Math Books

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Program Specific Math

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Misc.

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Simple English

File:Egyptian A'h-mosè or Rhind Papyrus (1065x1330).png
The Papyrus Rhind is the source of most of modern knowledge about mathematics in Ancient Egypt

Mathematics (sometimes shortened as "maths" or "math"), is the study of numbers, shapes and patterns. Mathematicians are people whose job is to learn and discover such things in mathematics. Mathematics is useful for solving problems that occur in the real world, so many people, besides mathematicians, study and use mathematics. Today, mathematics is needed in many jobs. Business, science, engineering, and construction need some knowledge of mathematics.

Mathematicians solve problems by using logic. Mathematicians often use deduction. Deduction is a special way of thinking to discover and prove new truths using old truths. To a mathematician, the reason something is true is just as important as the fact that it is true. Using deduction is what makes mathematical thinking different from other kinds of thinking.

Contents

About

Mathematics includes the study of:

  • Numbers (example 2+2=4)
  • Structure: how things are organized.
  • Place: where things are and their arrangement.
  • Change: how things become different over time.

Mathematics uses logic to study these things and to create general rules, which are an important part of mathematics. These rules leave out information that is not important so that a single rule can cover many situations. By finding general rules, mathematics solves many problems at the same time.

A proof gives a reason why a rule in mathematics is correct. This is done by using certain other rules that everyone agrees are correct, which are called axioms. A rule that has a proof is sometimes called a theorem. Experts in mathematics perform research to create new theorems. Sometimes experts find an idea that they think is a theorem but can not find a proof for it. That idea is called a conjecture until they find a proof.

Sometimes, mathematics finds and studies rules or ideas that have not yet been found in the real world. Often in mathematics, ideas and rules are chosen because they are considered simple or beautiful. Sometimes these ideas and rules are found in the real world after they are studied in mathematics. This has happened many times in the past. This means that studying the rules and ideas of mathematics can help us know the world better.

Number

Mathematics includes the study of number, or quantity.
1, 2, 3, \ldots \ldots, -1, 0, 1, \ldots \frac{1}{2}, \frac{2}{3}, 0.125,\ldots \pi, e, \sqrt{2},\ldots 1+i, 2e^{i\pi/3},\ldots
Natural numbers Integers Rational numbers Real numbers Complex numbers
\omega, \omega + 1, \ldots, 2\omega, \ldots \aleph_0, \aleph_1, \ldots +,-,\times,\div >,\ge, =, \le, < f(x) = \sqrt x
Ordinal numbers Cardinal numbers Arithmetic operations Arithmetic relations Functions

Structure

Some areas of mathematics study the structure that an object has.
File:Elliptic curve File:Free File:Eigenvectoren.pdf File:Lattice of the divisibility of [[File:|128px]]
Number theory Abstract algebra Linear algebra Order theory Graph theory

Shape

Some areas of mathematics study the shapes of things.
[[File:|128px]] [[File:|128px]] [[File:|160px]] File:Osculating File:Koch
Topology Geometry Trigonometry Differential geometry Fractal geometry

Change

Some areas of mathematics study the way things change
File:Integral as region under File:Vector [[File:|128px]]
Calculus Vector calculus Analysis
File:Damping [[File:|128px]] [[File:|128px]]
Differential equations Dynamical systems Chaos theory

Applied mathematics

Applied mathematics uses mathematics to solve problems of other areas such as engineering , physics, and computing.
Numerical analysis – Optimization – Probability theoryStatistics – Mathematical finance – Game theoryMathematical physicsFluid dynamics - computational algorithms

Famous theorems

These theorems have interested mathematicians and people who are not mathematicians.

Pythagorean theoremFermat's last theoremGoldbach's conjectureTwin Prime Conjecture – Gödel's incompleteness theorems – Poincaré conjecture – Cantor's diagonal argument – Four color theorem – Zorn's lemma – Euler's IdentityChurch-Turing thesis

These are theorems and conjectures that have greatly changed mathematics.

Riemann hypothesisContinuum hypothesis – P=NP – Pythagorean theorem – Central limit theorem – Fundamental theorem of calculusFundamental theorem of algebraFundamental theorem of arithmetic – Fundamental theorem of projective geometry – classification theorems of surfaces – Gauss-Bonnet theorem – Fermat's last theorem

Foundations and methods

Progress in understanding the nature of mathematics also influences the way mathematicians study their subject.

Philosophy of mathematics – Mathematical intuitionism – Mathematical constructivism – Foundations of mathematics – Set theory – Symbolic logic – Model theory – Category theoryLogic – Reverse Mathematics – Table of mathematical symbols

History and the world of mathematicians

Mathematics in history, and the history of mathematics.

History of mathematics – Timeline of mathematics – MathematiciansFields medal – Abel Prize – Millennium Prize Problems (Clay Math Prize) – International Mathematical Union – Mathematics competitions – Lateral thinking – Maths and gender

Name

The word "mathematics" comes from the Greek word "μάθημα" (máthema). The Greek word "μάθημα" means "science, knowledge, or learning".

Often, the word "mathematics" is made shorter into maths (in British English) or math (in American English). The short words math or maths are often used for arithmetic, geometry or simple algebra by young students and their schools.

Awards in mathematics

There is no Nobel prize in mathematics. Mathematicians can receive the Abel prize and the Fields Medal for important works.

The Clay Mathematics Institute have said they will give one million dollars to anyone who solves one of the Millennium Prize Problems

Mathematical tools

Tools that are used to do mathematics or find answers to mathematics problems.

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Citable sentences

Up to date as of December 30, 2010

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








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