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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]



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.


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.


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


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


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


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


  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.,
  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,
  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.


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Welcome to the School of Mathematics!

Since you're here, you either are someone wishing to share your knowledge of Mathematics, or you are someone who wishes to gain knowledge about Mathematics. If you are the first kind of person, have a look around, and see how you can contribute. If you are of the second kind, read on!

Mathematics has many facets. Though it has a wealth of applications, Mathematics is also a science, and an art, in its own right. Like other sciences, Mathematics is useful, but, just like other forms of art, Mathematics is beautiful, in its own unique way. You might not see this beauty at first, but the main goal of the Wikiversity School of Mathematics is to let you see this beauty, through inquiry and learning. To achieve this goal, we want to inspire you, since inspiration is essential to learning, discovery, and ultimately, seeing the beauty of Mathematics. A little bit of inspiration can put the beauty of Mathematics, and all its facets, before your eyes.

The School of Mathematics is a work in progress. We are trying to organize our material into a logical order, create new material, and revise existing material. Although we try to keep clarity in mind, if any of the material is confusing, you don't know where to start, or you have other questions, do not hesitate to pose your question at our help desk. There is no such thing as a "dumb question"; learning is driven by questions!

The School of Mathematics wishes you a very warm welcome. We hope that you will benefit from what we have to offer, and we hope that we will benefit from what you have to offer, either now or in the future.


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The Lorenz attractor is a chaotic map, noted for its butterfly shape. The map shows how the state of a dynamical system evolves over time in a complex, non-repeating pattern. The attractor itself, and the equations from which it is derived, were introduced by Edward Lorenz in 1963, who derived it from the simplified equations of convection rolls arising in the equations of the atmosphere.

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HSC Extension 1 and 2 Mathematics Development stage: 00% (as of Mar 12, 2007) - A-level Mathematics Development stage: 25% (as of Jan 1, 2000) - Abstract algebra Development stage: 50% (as of Jan 1, 2000) - Algebra Development stage: 50% (as of Jan 1, 2000) - Applicable Mathematics Development stage: 25% (as of Jan 1, 2000) - Applied Math Basics Development stage: 50% (as of Jan 1, 2000) - Beginning Mathematics Development stage: 25% (as of Jan 1, 2000) - Calculus Development stage: 50% (as of Jan 1, 2000) - Complex Analysis Development stage: 25% (as of Jan 1, 2000) - Conic Sections Development stage: 00% (as of Jan 1, 2000) - Differential Geometry Development stage: 00% (as of Dec 11, 2007) - Discrete mathematics Development stage: 25% (as of Jan 1, 2000) - Introduction to Game Theory Development stage: 25% (as of Jan 1, 2000) - Finite Model TheoryDevelopment stage: 50% (as of May 1, 2007) - Functional AnalysisDevelopment stage: 25% (as of March 25, 2009) - Geometry Development stage: 25% (as of Jan 23, 2005) - Geometry for elementary school Development stage: 50% (as of Jan 1, 2000) - Handbook of Descriptive Statistics Development stage: 25% (as of Jan 1, 2000) - High-School Mathematics Extensions Development stage: 50% (as of Jan 1, 2000) - Linear algebra Development stage: 50% (as of Jan 1, 2000) - Mathematical Proof Development stage: 00% (as of Jan 1, 2000) - Numerical Methods Development stage: 25% (as of Jan 1, 2000) - Pre-Algebra Development stage: 00% (as of Mar 15, 2006) - Probability Development stage: 25% (as of Jan 1, 2000) - Real Analysis Development stage: 75% (as of May 23, 2008) - Set Theory Development stage: 25% (as of Mar 19, 2005) - Statistics Development stage: 25% (as of Jan 1, 2000) - Topology Development stage: 50% (as of Apr 4, 2000) - Trigonometry Development stage: 75% (as of Feb 4, 2006) - The Book of Mathematical Proofs Development stage: 00% (as of Sep 27, 2007)(Sep 27, 2007) - Vectors Development stage: 00% (as of Feb 5, 2009)(Feb 5, 2009) - Graph Theory

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Up to date as of January 23, 2010
(Redirected to Basic Math article)

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Basic Math
Box artwork for Basic Math.
Developer(s) Atari
Publisher(s) Atari
Release date(s)
Genre(s) Education
System(s) Atari 2600
Players 1-2

Basic Math was one of the nine Atari 2600 titles that launched with the system in 1977. In an effort to show the public that Atari was about more than just fun and games, and that the Atari 2600 could also be used as a tool for learning, Atari developed an educational title as part of their initial line up of available cartridges. The first one deals with math, as the name implies.

Basic Math can be used to teach young children simple arithmetic. It provides examples of all four basic arithmetic operations (addition, subtraction, multiplication, and division), and the choice of whether to present all problems that can be found in a particular column of a table, or whether to present problem completely at random. Time limits can be set for providing the answer by using the difficulty switches. 10 problems are presented in every set.

As can be expected, Basic Math was not in high demand by Atari 2600 owners, but parents were happy to buy it for their children in an age where the benefits of computers on education were only beginning to come to light and be understood. Sometime around 1980, Atari decided to give the game a "peppier" name and changed it to Fun with Numbers. The cartoon illustration that often accompanied the game's description in catalogs was of a little Einstein inside the TV set (which was not included in the game). It was released as Math under the Sears Telegames label.




  • Color/BW: Switch between color display and black & white display. (This feature made the game look better on black & white TVs that were still prominent at the time of the game's release.)
  • Right Difficulty Switch: The right difficulty switch is used to set whether there is a time limit for answering problems or not. Set the right difficulty switch to A to enable time limits, and set the time limit with the left difficulty switch.
  • Left Difficulty Switch: When the right difficulty switch is set to A, the left difficulty switch determines how much of a time limit the player gets to answer a problem, depending on which game variation is being played.
    • Games 1-4: When set to B, the player gets 24 seconds to answer a problem. When set to A, the player only gets 12 seconds.
    • Games 5-8: When set to B, the player gets 12 seconds to answer a series of one digit problems. When set to A, the player gets 24 seconds to answer a set of two digit problems.
  • Game Select: Select a game variation. The variations cycle from 1 to 8 and start back over at 1. See the Game Variation section below.
  • Game Reset: Starts a new game in whatever game variation is currently selected. The player will get ten new problems, and the tally of correct answers is reset to 0.
  • Joystick: Use the joystick to enter in your answers. Press the joystick up and down to cycle through the digits 0 through 9, and back to 0 again, or in the other direction. Press left or right to move the answer bar to another place in the answer, such as the tens digit, or the remainder digit.
  • Button: Press the button when you are done changing the numbers and you want to submit the current value as your answer.

How to play

After starting a game, the player is presented with their first problem. Use the joystick to change the value shown beneath the answer bar. Press the joystick up or down to cycle through all ten digits, and press the joystick left or right to line up the answer bar with the digit that you would like to change. Press the button when you have completed selecting your answer. You will hear a tone and then a happy musical selection if you got the answer right, or a sad musical selection if you got the answer wrong. If you got the wrong answer, your answer will be replaced with the correct answer before moving on. After all ten problems have been presented, the player is presented with the number of problems he or she answered correctly on the left side out of the number of problems (10) that they were given on the right side.

As an example, if the problem in question was 6 × 5, the player would start by pushing up on the joystick until the number 0 appeared in the ones digit. Then the player would push the joystick to the left, and line the answer bar up with the tens digit position, and then push up on the joystick until the number 3 appeared. Then the player would push the button. In another example, if the problem in question was 5 ÷ 3, the player would start by pushing up on the joystick until the number 1 appeared in the ones digit. Now, in order to indicate the remainder, the player must push the joystick to the right, not once, but twice. With a space between the ones and the remainder digit, the player would push up on the joystick until a 2 appeared, and push the button to submit his answer. If the 2 appeared only one space to the right of the 1, the answer would be considered incorrect.

Game Variations

Table Problems


Game variations 1 through 4 are Table Problem variations. Each variation is a different operation; 1 is addition, 2 is subtraction, 3 is multiplication, and 4 is division. The player begins each game by selecting which top number they want to work with. The game then proceeds to randomly choose the numbers that appear on the bottom of the problem. If the game runs out of problems for a particular number, the game advances the player's selection by one. For example, if the player selects game variation 1 (addition) and chooses to work with the number 5, the game will ask the player to add 5 to every number from 1 to 9. Since there are ten problem, the player will get one additional problem adding the number 6 to a randomly chosen number. In another example, the player selects game variation 4 (division) and chooses to work with the number 6. The game will ask the player to divide 6 by every number from 1 to 6. When those numbers are exhausted, the game will present four more problems, asking the player to divide 7 by four numbers between 1 and 7.

Random Problems


Game variations 5 through 8 are Random Problem variations. Each variation is a different operation; 5 is addition, 6 is subtraction, 7 is multiplication, and 8 is division. The player does not choose what number to work with, as both the top number and the bottom number are chosen at random. The version provides players with a more difficult challenge, since the sets of problems are unknown until they are presented to the player, and tend to be more complicated.


Up to date as of February 01, 2010

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Developer(s) Sears
Publisher(s) Sears
Release date
Genre Edutainment
Mode(s) Single player
Age rating(s) N/A
Atari 2600
Platform(s) Atari 2600
Input Atari 2600 Joystick
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