File:Euclid.jpg Greek mathematician, 3rd century BC, as imagined by Raphael in this detail from [[The School of Athens]]No likeness or description of Euclids physical appearance made during his lifetime survived antiquity. Therefore, Euclids depiction in works of art depends on the artists imagination (see [[Euclid]].]] Mathematics is the study of quantity structure space and calculus Mathematician seek out patterns Lynn Steen (April 29, 1988). [[The Science of Patterns.]]Science (journal) 240: 611–616. and summarized at http://www.ascd.org/portal/site/ascd/template.chapter/menuitem.1889bf0176da7573127855b3e3108a0c/?chapterMgmtIdf97433df69abb010VgnVCM1000003d01a8c0RCRD Association for Supervision and Curriculum Development.], ascd.orgKeith Devlin Mathematics: The Science of Patterns: The Search for Order in Life, Mind and the Universe(Scientific American Paperback Library) 1996, ISBN 978-0-7167-5047-5 formulate new conjecture , and establish truth by Rigour#Mathematical rigour deductive reasoning from appropriately chosen axiom and definition .Jourdain. There is debate over whether mathematical objects such as number and points exist naturally or are human creations. The mathematician Benjamin Peirce called mathematics "the science that draws necessary conclusions".Peirce, p. 97. 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." Through the use of abstraction (mathematics) and logic l reasoning mathematics evolved from counting calculation measurement and the systematic study of the shape and motion (physics) of physical objects. Practical mathematics has been a human activity for as far back as History of Mathematics exist. Logic first appeared in Greek mathematics most notably in Euclid s [[Euclids Elements|Elements]]. Mathematics continued to develop, for example in China in 300 BC, in India in AD 100, and in the Muslim world in AD 800, until the Renaissance when mathematical innovations interacting with new timeline of scientific discoveries led to a rapid increase in the rate of mathematical discovery that continues to the present day.Eves 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.Peterson

Etymology

The word "mathematics" comes from the ancient Greek language μάθημα (máthēma, which means learning study science and additionally came to have the narrower and more technical meaning "mathematical study", even in Classical times.Both senses can be found in Plato. Liddell and Scott, s.voceαθηματικός Its adjective is (mathēmatikós, related to learning or studious which likewise further came to mean mathematical In particular, (mathēmatikḗ tékhnē, meant the mathematical art The apparent plural form in English, like the French plural form (and the less commonly used singular derivative , goes back to the Latin neuter plural (Cicero , based on the Greek plural 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 mathematicsanew, after the pattern of physics and metaphysics which were inherited from the Greek.[[The Oxford Dictionary of English Etymology]] [[Oxford English Dictionary]] sub"mathematics", "mathematic", "mathematics" In English, the noun mathematicstakes singular verb forms. It is often shortened to mathsor, in English-speaking North America, math

History

Image:Kapitolinischer Pythagoras adjusted.jpg (c.570-c.495 BC) has commonly been given credit for discovering the Pythagorean theorem Well-known figures in Greek mathematics also include Euclid Archimedes and Thales ]] The evolution of mathematics might be seen as an ever-increasing series of abstraction (mathematics) or alternatively an expansion of subject matter. The first abstraction, which is shared by many animals,lt;/ref> was probably that of number : 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 counting physicalobjects, Prehistory peoples also recognized how to count abstractquantities, like time – days, season , years.See, for example, Raymond L. Wilder, Evolution of Mathematical Concepts; an Elementary Study passimlt;/ref> Elementary arithmetic (addition subtraction multiplication and division (mathematics) naturally followed. Since numeracy pre-dated writing further steps were needed for recording numbers such as Tally sticks or the knotted strings called quipu used by the Inca to store numerical data.Numeral system have been many and diverse, with the first known written numerals created by Ancient Egypt in Middle Kingdom of Egypt texts such as the Rhind Mathematical Papyrus File:maya.svg ] The earliest uses of mathematics were in Trade land measurement painting and weaving patterns and the recording of time. More complex mathematics did not appear until around 3000 BC, when the Babylonian and Ancient Egypt began using arithmetic, algebra and geometry for taxation and other financial calculations, for building and construction, and for astronomy Kline 1990, Chapter 1. The systematic study of mathematics in its own right began with the Ancient Greeks between 600 and 300 BC."[http://books.google.com/books?id=drnY3Vjix3kC&pg=PA1&dq&hl=en#v=onepage&q=&f=false A History of Greek Mathematics: From Thales to Euclid]. Thomas Little Heath (1981). ISBN 0-486-24073-8 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 theorem and their mathematical proof "Sevryuk

Inspiration, pure and applied mathematics, and aesthetics

File:GodfreyKneller-IsaacNewton-1689.jpg (1643-1727), an inventor of calculus ]] 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 todays string theory a still-developing scientific theory which attempts to unify the four Fundamental interaction continues to inspire new mathematics.lt;/ref> 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 in the Natural Sciences .Eugene Wigner 1960, "http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html The Unreasonable Effectiveness of Mathematics in the Natural Sciences,]" [[Communications on Pure and Applied Mathematics]]13 1): 1–14. 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.http://www.ams.org/mathscinet/msc/pdfs/classification2010.pdf Mathematics Subject Classification 2010] 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 eleganceof mathematics, its intrinsic aesthetics and inner beauty Simplicity and generality are valued. There is beauty in a simple and elegant proof (mathematics) such as Euclid s proof that there are infinitely many prime number , and in an elegant numerical method that speeds calculation, such as the fast Fourier transform G. H. Hardy in [[A Mathematicians 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.lt;/ref> 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.lt;/ref>lt;/ref> The popularity of recreational mathematics is another sign of the pleasure many find in solving mathematical questions.

Notation, language, and rigor

File:Leonhard Euler 2.jpg 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.http://jeff560.tripod.com/mathsym.html Earliest Uses of Various Mathematical Symbols] (Contains many further references). Before that, mathematics was written out in words, a painstaking process that limited mathematical discovery.Kline, p. 140, on Diophantus p.261, on Franciscus Vieta Leonhard 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 orand onlyhave more precise meanings than in everyday speech. Moreover, words such as [[open set|open]]and [[field (mathematics)|field]]have been given specialized mathematical meanings. Mathematical jargon includes technical terms such as [[homeomorphism]]and [[Integral|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". File:Infinity symbol.svg 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 "theorem ", based on fallible intuitions, of which many instances have occurred in the history of the subject.See [[false proof]]for simple examples of what can go wrong in a formal proof. The Four color theorem#History contains examples of false proofs accidentally accepted by other mathematicians at the time. 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 proof . Since large computations are hard to verify, such proofs may not be sufficiently rigorous.Ivars Peterson, The Mathematical Tourist Freeman, 1988, ISBN 0-7167-1953-3. p. 4 "A few complain that the computer program cant be verified properly", (in reference to the Haken-Apple proof of the Four Color Theorem). Axiom in traditional thought were "self-evident truths", but that conception is problematic. At a formal level, an axiom is just a string of symbolic logic 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 independence (mathematical logic) 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.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."

Mathematics as science

File:Carl Friedrich Gauss.jpg himself known as the "prince of mathematicians",lt;/ref> referred to mathematics as "the Queen of the Sciences".]] Carl Friedrich Gauss referred to mathematics as "the Queen of the Sciences".Waltershausen In the original Latin Regina Scientiarum as well as in German language Königin der Wissenschaften the word corresponding to sciencemeans (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 naturalscience 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.Einstein, p. 28. The quote is Einsteins 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]] Many philosophers believe that mathematics is not experimentally falsifiability and thus not a science according to the definition of Karl Popper lt;/ref> However, in the 1930s important work in mathematical logic convinced many mathematicians that mathematics cannot be reduced to logic alone, and Karl Popper concluded that "most mathematical theories are, like those of physics and biology hypothesis deductive pure mathematics therefore turns out to be much closer to the natural sciences whose hypotheses are conjectures, than it seemed even recently."Popper 1995, p. 56 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 knowledgeand thus includes mathematics.Ziman In any case, mathematics shares much in common with many fields in the physical sciences, notably the deductive reasoning of assumptions. intuition (knowledge) and experiment tion also play a role in the formulation of conjecture 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 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 mathematiciansfeel 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 othersfeel 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 university 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 Mathematical awards are generally kept separate from their equivalents in science. The most prestigious award in mathematics is the "The Fields Medal is now indisputably the best known and most influential award in mathematics. MonastyrskyRiehm established in 1936 and now awarded every 4 years. It is often considered the equivalent of sciences Nobel Prize . 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 problem , 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 Hilberts problems.

Fields of mathematics

File:Abacus 6.png 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 mathematical analysis . In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to mathematical logic to set theory (foundations of mathematics , to the empirical mathematics of the various sciences (applied mathematics , and more recently to the rigorous study of uncertainty

Quantity

lt;!-- This section is linked from List of basic mathematics topics --> The study of quantity starts with number , first the familiar natural number and integer ("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 number ("Fraction (mathematics) ). These, in turn, are contained within the real number , which are used to represent Continuous function quantities. Real numbers are generalized to complex number . These are the first steps of a hierarchy of numbers that goes on to include quarternion and octonion . Consideration of the natural numbers also leads to the transfinite number , which formalize the concept of "infinity . Another area of study is size, which leads to the cardinal number and then to another conception of infinity: the aleph number , which allow meaningful comparison of the size of infinitely large sets. :| style"border:1px solid #ddd; text-align:center; margin:auto" cellspacing"20" | $1,\; 2,\; 3\backslash ,...\backslash !$ || $...-2,\; -1,\; 0,\; 1,\; 2\backslash ,...\backslash !$ || $-2,\; \backslash frac2\}3\},\; 1.21\backslash ,\backslash !$ || $-e,\; \backslash sqrt2\},\; 3,\; \backslash pi\backslash ,\backslash !$ || $2,\; i,\; -2+3i,\; 2e^i\backslash frac4\backslash pi\}3\}\}\backslash ,\backslash !$ |- | Natural number || Integer || Rational number || Real number || Complex number |}

Structure

lt;!-- This section is linked from List of basic mathematics topics --> Many mathematical objects, such as set (mathematics) of numbers and function (mathematics) exhibit internal structure as a consequence of operation (mathematics) or relation (mathematics) 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 integer that can be expressed in terms of arithmetic operations. Moreover, it frequently happens that different such structured sets (or Mathematical structure exhibit similar properties, which makes it possible, by a further step of abstraction to state axiom for a class of structures, and then study at once the whole class of structures satisfying these axioms. Thus one can study group (mathematics) ring (mathematics) field (mathematics) 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 space , whose elements called vector (geometric) 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. Combinatorics studies ways of enumerating the number of objects that fit a given structure. :| style"border:1px solid #ddd; text-align:center; margin:auto" cellspacing"15" | || File:Elliptic curve simple.svg || File:Rubik's cube.svg || File:Group diagdram D6.svg || File:Lattice of the divisibility of 60.svg |- | Combinatorics || Number theory || Group theory || Graph theory || Order theory |}

Space

lt;!-- This section is linked from List of basic mathematics topics --> 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 geometry (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 manifold , in particular, Vector calculus 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 group 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. :| style"border:1px solid #ddd; text-align:center; margin:auto" cellspacing"15" | File:Illustration to Euclid's proof of the Pythagorean theorem.svg || File:Sine cosine plot.svg || File:Hyperbolic triangle.svg || File:Torus.png || File:Mandel zoom 07 satellite.jpg || File:Measure illustration.png |- |Geometry || Trigonometry || Differential geometry || Topology || Fractal || Measure Theory |}

Change

lt;!-- This section is linked from List of basic mathematics topics --> Understanding and describing change is a common theme in the natural science , and calculus was developed as a powerful tool to investigate it. function (mathematics) arise here, as a central concept describing a changing quantity. The rigorous study of real number and functions of a real variable is known as real analysis with complex analysis the equivalent field for the complex number . Functional analysis focuses attention on (typically infinite-dimensional) space#Mathematics 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 equation . Many phenomena in nature can be described by dynamical system ; chaos theory makes precise the ways in which many of these systems exhibit unpredictable yet still deterministic system (mathematics) behavior. | style"border:1px solid #ddd; text-align:center; margin:auto" cellspacing"20" | File:Integral as region under curve.svg || File:Vector field.svg || File:Airflow-Obstructed-Duct.png || File:Limitcycle.jpg || File:Lorenz attractor.svg || File:Princ argument ex1.png |- | Calculus || Vector calculus | Differential equation || Dynamical system || 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 Set (mathematics) or collections of objects. Category theory which deals in an abstract way with mathematical structure 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.Luke Howard Hodgkin & Luke Hodgkin, A History of Mathematics Oxford University Press, 2005. 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 theory and the Brouwer-Hilbert controversy Mathematical logic is concerned with setting mathematics within a rigorous axiom tic framework, and studying the implications of such a framework. As such, it is home to Gödel's incompleteness theorems which (informally) imply 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. Whatever finite collection of number-theoretical axioms is taken as a foundation, Gödel showed how to construct a formal statement that is a true number-theoretical fact, but which does not follow from those axioms. Therefore no formal system is a complete axiomatization of full number theory.Modern logic is divided into recursion theory model theory and proof theory and is closely linked to theoretical computer science computer science :| style"border:1px solid #ddd; text-align:center; margin:auto" cellspacing"15" | $p\; \backslash Rightarrow\; q\; \backslash ,$|| File:Venn A intersect B.svg || File:Commutative diagram for morphism.svg |- | Mathematical logic || Set theory || Category theory || |}

Theoretical computer science

Theoretical computer science includes computability theory (computation) 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. A famous problem is the "P = NP problem problem, one of the Millennium Prize Problems http://www.claymath.org/millennium/P_vs_NP/ Clay Mathematics Institute], PNP, claymath.org 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 data compression and Entropy (information theory) :| style"border:1px solid #ddd; text-align:center; margin:auto" cellspacing"15" | File:DFAexample.svg || File:Caesar3.svg |- | Theory of computation || Cryptography |}

Applied mathematics

Applied mathematics considers the use of abstract mathematical tools in solving concrete problems in the science , 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 design of 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 study statisticians "make sense of the data" using the art of statistical model and the theory of statistical inference – with statistical model model selection and estimation the estimated models and consequential Scientific method#Predictions from the hypothesis should be statistical hypothesis testing on Scientific method#Evaluation and improvement 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. 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 (mathematics) 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 error . Other areas of computational mathematics include computer algebra and symbolic computation

Image:Gravitation space source.png | Mathematical physics lt;/center> Image:BernoullisLawDerivationDiagram.svg | Fluid dynamics lt;/center> Image:Composite trapezoidal rule illustration small.svg | Numerical analysis lt;/center> Image:Maximum boxed.png | Optimization (mathematics) lt;/center> Image:Two red dice 01.svg | Probability theory lt;/center> Image:Oldfaithful3.png | Statistics lt;/center> Image:Market Data Index NYA on 20050726 202628 UTC.png | Financial mathematics lt;/center> Image:Arbitrary-gametree-solved.svg | Game theory lt;/center> Image:Signal transduction v1.png | Mathematical biology lt;/center> Image:Ch4-structure.png | Mathematical chemistry lt;/center> Image:GDP PPP Per Capita IMF 2008.png | Mathematical economics lt;/center> Image:Simple feedback control loop2.png | Control theory lt;/center>

See also

* Definitions of mathematics * Dyscalculia * Iatromathematicians * Logics * Mathematical anxiety * Mathematical game * Mathematical model * Mathematical problem * Mathematical structure * Mathematics and art * Mathematics competitions * Mathematics education * Portal:Mathematics * Pattern * Philosophy of mathematics * Pseudomathematics

Notes

References

* Benson, Donald C., The Moment of Proof: Mathematical Epiphanies Oxford University Press USA; New Ed edition (December 14, 2000). ISBN 0-19-513919-4. * Carl B. Boyer A History of Mathematics Wiley; 2 edition (March 6, 1991). ISBN 0-471-54397-7. — A concise history of mathematics from the Concept of Number to contemporary Mathematics. * Courant, R. and H. Robbins, What Is Mathematics? : An Elementary Approach to Ideas and Methods Oxford University Press, USA; 2 edition (July 18, 1996). ISBN 0-19-510519-2. * Philip J. Davis and Reuben Hersh [[The Mathematical Experience]] Mariner Books; Reprint edition (January 14, 1999). ISBN 0-395-92968-7. — A gentle introduction to the world of mathematics. * * Eves, Howard, An Introduction to the History of Mathematics Sixth Edition, Saunders, 1990, ISBN 0-03-029558-0. * Jan Gullberg Mathematics — From the Birth of Numbers W. W. Norton & Company 1st edition (October 1997). ISBN 0-393-04002-X. — An encyclopedic overview of mathematics presented in clear, simple language. * Hazewinkel, Michiel (ed.), [[Encyclopaedia of Mathematics]] Kluwer Academic Publishers 2000. — A translated and expanded version of a Soviet mathematics encyclopedia, in ten (expensive) volumes, the most complete and authoritative work available. Also in paperback and on CD-ROM, and http://eom.springer.de/default.htm online]. * Jourdain, Philip E. B., The Nature of Mathematics in The World of Mathematics James R. Newman, editor, Dover Publications 2003, ISBN 0-486-43268-8. * Morris Kline Mathematical Thought from Ancient to Modern Times Oxford University Press, USA; Paperback edition (March 1, 1990). ISBN 0-19-506135-7. * * Oxford English Dictionary second edition, ed. John Simpson and Edmund Weiner, Clarendon Press 1989, ISBN 0-19-861186-2. * [[The Oxford Dictionary of English Etymology]] 1983 reprint. ISBN 0-19-861112-9. * Pappas, Theoni, The Joy Of Mathematics Wide World Publishing; Revised edition (June 1989). ISBN 0-933174-65-9. * * Peterson, Ivars, Mathematical Tourist, New and Updated Snapshots of Modern Mathematics Owl Books, 2001, ISBN 0-8050-7159-8. * * * * * *

External links

* http://freebookcentre.net/SpecialCat/Free-Mathematics-Books-Download.html Free Mathematics books] Free Mathematics books collection. * Encyclopaedia of Mathematics online encyclopaedia from http://eom.springer.de Springer], Graduate-level reference work with over 8,000 entries, illuminating nearly 50,000 notions in mathematics. * http://hyperphysics.phy-astr.gsu.edu/Hbase/hmat.html HyperMath site at Georgia State University] * http://www.freescience.info/mathematics.php FreeScience Library] The mathematics section of FreeScience library * Rusin, Dave: [http://www.math-atlas.org/ The Mathematical Atlas] A guided tour through the various branches of modern mathematics. (Can also be found at http://www.math.niu.edu/~rusin/known-math/index/index.html NIU.edu].) * Polyanin, Andrei: [http://eqworld.ipmnet.ru/ EqWorld: The World of Mathematical Equations] An online resource focusing on algebraic, ordinary differential, partial differential (mathematical physics , integral, and other mathematical equations. * Cain, George: http://www.math.gatech.edu/~cain/textbooks/onlinebooks.html Online Mathematics Textbooks] available free online. * http://www.tricki.org/ Tricki], Wiki-style site that is intended to develop into a large store of useful mathematical problem-solving techniques. * http://math.chapman.edu/cgi-bin/structures?HomePage Mathematical Structures], list information about classes of mathematical structures. * http://etext.lib.virginia.edu/DicHist/analytic/anaVII.html Math & Logic: The history of formal mathematical, logical, linguistic and methodological ideas.] In The Dictionary of the History of Ideas. * http://www-history.mcs.st-and.ac.uk/~history/ Mathematician Biographies]. The MacTutor History of Mathematics archive Extensive history and quotes from all famous mathematicians. * [http://metamath.org/ Metamath] A site and a language, that formalize mathematics from its foundations. * http://www.nrich.maths.org/public/index.php Nrich], a prize-winning site for students from age five from University of Cambridge * http://garden.irmacs.sfu.ca Open Problem Garden], a wiki of open problems in mathematics * [http://planetmath.org/ Planet Math] An online mathematics encyclopedia under construction, focusing on modern mathematics. Uses the CC BY SA license, allowing article exchange with Wikipedia. Uses TeX markup. * http://www-math.mit.edu/daimp Some mathematics applets, at MIT] * Weisstein, Eric et al.: [http://www.mathworld.com/ MathWorld: World of Mathematics] An online encyclopedia of mathematics. * Patrick Jones http://www.youtube.com/user/patrickJMT Video Tutorials] on Mathematics * http://en.citizendium.org/wiki/Theory_(mathematics) Citizendium: Theory (mathematics)]. Category:Mathematics Category:Mathematical sciences Category:Formal sciences Category:Greek loanwords af:Wiskunde als:Mathematik am:ትምህርተ ሂሳብ ang:Rīmcræft ar:رياضيات an:Matematicas roa-rup:Mathematicã as:গণিত ast:Matemátiques ay:Jakhu az:Riyaziyyat bn:গণিত zh-min-nan:Sò͘-ha̍k map-bms:Matematika ba:Математика be:Матэматыка be-x-old:Матэматыка bar:Mathematik bo:རྩིས་རིག bs:Matematika br:Matematik bg:Математика ca:Matemàtiques cv:Математика ceb:Matematika cs:Matematika ch:Matematika sn:Masvomhu co:Matematica cy:Mathemateg da:Matematik de:Mathematik dv:ރިޔާޟިއްޔާތު nv:Ałhíʼayiiltááh dsb:Matematika et:Matemaatika el:Μαθηματικά eml:Matemâtica myv:Математика es:Matemáticas eo:Matematiko ext:Matemáticas eu:Matematika fa:ریاضیات fo:Støddfrøði fr:Mathématiques frr:Matematiik fy:Wiskunde fur:Matematiche ga:Matamaitic gv:Maddaght gd:Matamataig gl:Matemáticas gan:數學 gu:ગણિત hak:Sṳ-ho̍k xal:Эсв ko:수학 haw:Makemakika hy:Մաթեմատիկա hi:गणित hr:Matematika io:Matematiko ig:Ọmúmú-ónúọgụgụ bpy:গণিত id:Matematika ia:Mathematica ie:Matematica os:Математикæ is:Stærðfræði it:Matematica he:מתמטיקה jv:Matématika kl:Matematikki kn:ಗಣಿತ krc:Математика ka:მათემატიკა ks:علم ریاضی csb:Matematika kk:Математика ky:Математика sw:Hisabati ht:Matematik ku:Matematîk lad:Matematika lo:ຄະນິດສາດ la:Mathematica lv:Matemātika lb:Mathematik lt:Matematika lij:Matematica li:Mathematiek jbo:cmaci lmo:Matemàtega hu:Matematika mk:Математика mg:Fanisana ml:ഗണിതം mt:Matematika mr:गणित arz:رياضيات ms:Matematik mwl:Matemática mdf:Математиксь mn:Математик my:သင်္ချာ nah:Tlapōhuayōtl nl:Wiskunde nds-nl:Wiskunde ne:गणित new:गणित ja:数学 no:Matematikk nn:Matematikk nrm:Caltchul nov:Matematike oc:Matematicas uz:Matematika pa:ਗਣਿਤ pag:Matematiks ps:شمېر پوهنه km:គណិតវិទ្យា pms:Matemàtica nds:Mathematik pl:Matematyka pt:Matemática crh:Riyaziyat ro:Matematică qu:Yupay yachay ru:Математика sah:Математика sm:Matematika sa:गणितं sc:Matemàtica sco:Mathematics stq:Mathematik sq:Matematika scn:Matimàtica si:ගණිතය simple:Mathematics ss:Tekubala sk:Matematika sl:Matematika szl:Matymatyka so:Xisaab ckb:بیرکاری srn:Sabi fu Teri sr:Математика sh:Matematika su:Matematika fi:Matematiikka sv:Matematik tl:Matematika ta:கணிதம் kab:Tusnakt tt:Математика te:గణితము tet:Matemátika th:คณิตศาสตร์ tg:Математика tr:Matematik tk:Matematika bug:Matematika uk:Математика ur:ریاضی za:Soqyoz vec:Matemàtega vi:Toán học vo:Matemat fiu-vro:Matõmaatiga zh-classical:數學 war:Matematika wo:Xayma wuu:数学 yi:מאטעמאטיק yo:Mathimátíkì zh-yue:數學 diq:Matematik bat-smg:Matematėka zh:数学