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Areas Of Mathematics

Areas of mathematics

The aim of this page is to list all areas of modern mathematics, with a brief explanation about their scope and links to other parts of Wikipedia, set out in a systematic way. The way research-level mathematics is internally organised is mostly determined by practitioners, and does change over time; this is in contrast with the apparently timeless syllabus divisions used in mathematics education, where calculus can seem to be much the same over a time scale of a century. Calculus itself does not appear as a major heading — most of the traditional material would be divided amongst topics under analysis. This illustrates, in part, the difficulty of communicating the principles of any large-scale organisation. The research on most calculus topics was carried out in the eighteenth century, and has long been assimilated. The story of why fields exist as specialties involves in most cases quite a long intellectual history (and sometimes institutional history). The American Mathematical Society's [http://www.ams.org/msc/ Mathematics Subject Classification (2000 edition)] has been used as a starting point to ensure all areas are covered, and related areas are close together. However, the MSC aims to classify mathematical papers, not mathematics itself, so additional categories have been used. See also the list of lists of mathematical topics (not to be confused with the far longer list of mathematical topics).

Foundations / general

- 00: General
- 01: History and biography
- 03: Mathematical logic and foundations
- 97: Mathematics education
- 00: Recreational mathematics

Algebra

The study of structure starting with numbers, first the familiar natural numbers and integers and their arithmetical operations, which are recorded in elementary algebra. The deeper properties of whole numbers are studied in number theory. The investigation of methods to solve equations leads to the field of abstract algebra, which, among other things, studies rings and fields, structures that generalize the properties possessed by everyday numbers. Long standing questions about ruler-and-compass constructions were finally settled by Galois theory. The physically important concept of vectors, generalized to vector spaces is studied in linear algebra. ; Combinatorics (MSC 05) : Studies finite collections of objects that satisfy specified criteria. In particular, it is concerned with "counting" the objects in those collections (enumerative combinatorics) and with deciding whether certain "optimal" objects exist (extremal combinatorics). It includes graph theory, used to describe inter-connected objects (a graph in this sense is a collection of connected points). See also the list of combinatorics topics, list of graph theory topics and glossary of graph theory. ; Order theory (MSC 06) : With any set of real numbers, it is possible to write them out in ascending order. Order Theory extends this idea to sets in general. It includes notions like lattices and ordered algebraic structures. See also the order theory glossary and the list of order topics. ; General algebraic systems (MSC 08) : Given a set, ways of combining or relating members of that set can be defined. If these obey certain rules, then a particular algebraic structure is formed. Universal algebra is the more formal study of these structures and systems. ; Number theory (MSC 11) : Number theory is traditionally concerned with the properties of integers. More recently, it has come to be concerned with wider classes of problems that have arisen naturally from the study of integers. It can be divided into elementary number theory (where the integers are studied without use of techniques from other mathematical fields); analytic number theory (where calculus and complex analysis are used as tools); algebraic number theory (which studies the algebraic numbers - the roots of polynomials with integer coefficients); Geometric number theory; combinatorial number theory and computational number theory. See also the list of number theory topics
- 12: Field theory and polynomials
- 13: Commutative rings and algebras
- 15: Linear and multilinear algebra; matrix theory
- 16: Associative rings and associative algebras
- 17: Non-associative rings and non-associative algebras
- 18: Category theory; homological algebra
- 19: K-theory
- 20: Group theory and generalizations
- 22: Topological groups, Lie groups, and analysis upon them (Also transformation groups, abstract harmonic analysis)

Analysis

Analysis is primarily concerned with change. Rates of change, accumulated change, multiple things changing relative to (or independently of) one another, etc.
- 26: Real functions, including derivatives and integrals
- 28: Measure and integration
- 30: Complex functions, including approximation theory in the complex domain
- 31: Potential theory
- 32: Several complex variables and analytic spaces
- 33: Special functions
- 34: Ordinary differential equations
- 35: Partial differential equations
- 37: Dynamical systems and ergodic theory
- 39: Difference equations and functional equations
- 40: Sequences, series, summability
- 41: Approximations and expansions
- 42: Fourier analysis, including Fourier transforms, trigonometric approximation, trigonometric interpolation, and orthogonal functions
- 43: Abstract harmonic analysis
- 44: Integral transforms, operational calculus
- 45: Integral equations
- 46: Functional analysis, including infinite-dimensional holomorphy, integral transforms in distribution spaces
- 47: Operator theory
- 49: Calculus of variations and optimal control; optimization (including geometric integration theory)
- 58: Global analysis, analysis on manifolds (including infinite-dimensional holomorphy) (Also: probabilistic potential theory, numerical approximation, representation theory, analysis on manifolds)

Geometry

Geometry (MSC 51) deals with spatial relationships, using fundamental qualities or axioms. Such axioms can be used in conjunction with mathematical definitions for points, straight lines, curves, surfaces, and solids to draw logical conclusions. ;Convex geometry (MSC 52) ;Discrete or combinatorial geometry (MSC 52): may be loosely defined as study of geometrical objects and properties that are discrete or combinatorial, either by their nature or by their representation. It includes the study of shapes such as the Platonic solids and the notion of tessellation. ;Differential geometry (MSC 53): is the study of geometry using calculus, and is very closely related to differential topology. Covers such areas as Riemannian geometry, curvature and differential geometry of curves. See also the glossary of differential geometry and topology. ;Algebraic geometry (MSC 14): Given a polynomial of two real variables, then the points on a plane where that function is zero will form a curve. An algebraic curve extends this notion to polynomials over a field in a given number of variables. Algebraic geometry may be viewed the study of these curves. See also the list of algebraic geometry topics and list of algebraic surfaces. ;Topology: Deals with the properties of a figure that do not change when the figure is continuously deformed. The main areas are point set topology (or general topology), algebraic topology, and the topology of manifolds, defined below. ;General topology (MSC 54): Also called point set topology. Properties of topological spaces. Includes such notions as open and closed sets, compact spaces, continuous functions, convergence, separation axioms, metric spaces, dimension theory. See also the glossary of general topology and the list of general topology topics. ;Algebraic topology (MSC 55): Properties of algebraic objects associated with a topological space and how these algebraic objects capture properties of such spaces. Contains areas like homology theory, cohomology theory, homotopy theory, and homological algebra. The latter deals with homotopy groups (including the fundamental group) as well as simplicial complexes and CW complexes (also called cell complexes). See also the list of algebraic topology topics. ;Manifolds (MSC 57): A manifold can be thought of as an n-dimensional generalization of a surface in the usual 3-dimensional Euclidean space. The study of manifolds includes differential topology, which looks at the properties of differentiable functions defined over a manifold. See also complex manifolds.

Applied mathematics

Probability and statistics

;Probability theory (MSC 60) : the study of how likely a given event is to occur.
- 60G/H: Stochastic processes (including probabilistic potential theory) ;Statistics (MSC 62): Analysis of data, and how representative it is. See also the list of statistical topics.

Computational sciences

- 65: Numerical analysis, including numerical methods
- 68: Computer science

Physical sciences

;Mechanics: addresses what happens when a real physical object is subjected to forces. This divides naturally into the study of rigid solids, deformable solids, and fluids, detailed below. ;Particle mechanics (MSC 70): In mathematics, a particle is a point-like, perfectly rigid, solid object. Particle mechanics deals with the results of subjecting particles to forces. It includes celestial mechanics — the study of the motion of celestial objects. ;Mechanics of deformable solids (MSC 74) : Most real-world objects are not point-like nor perfectly rigid. More importantly, objects change shape when subjected to forces. This subject has a very strong overlap with continuum mechanics, which is concerned with continuous matter. It deals with such notions as stress, strain and elasticity. See also continuum mechanics. ;Fluid mechanics (MSC 76): Fluids in this sense includes not just liquids, but flowing gases, and even solids under certain situations. (For example, dry sand can behave like a fluid). It includes such notions as viscosity, turbulent flow and laminar flow (its opposite). See also fluid dynamics.
- 78: Optics, electromagnetic theory
- 80: Classical thermodynamics, heat transfer
- 81: Quantum theory, including quantum optics
- 82: Statistical mechanics, structure of matter
- 83: Relativity and gravitational theory, including relativistic mechanics
- 85: Astronomy and astrophysics
- 86: Geophysics

Non-physical sciences

- 90: Operations research, mathematical programming
- 91: Game theory, economics, social and behavioral sciences
- 92: Biology (see also mathematical biology) and other natural sciences
- 93: Systems theory; control, including optimal control
- 94: Information and communication, circuits
- 97: Mathematics education Category:Mathematics th:สาขาของคณิตศาสตร์

Mathematics

Mathematics is often defined as the study of topics such as quantity, structure, space, and change. Another view, held by many mathematicians, is that mathematics is the body of knowledge justified by deductive reasoning, starting from axioms and definitions. Practical mathematics, in nearly every society, is used for such purposes as accounting, measuring land, or predicting astronomical events. Mathematical discovery or research often involves discovering and cataloging patterns, without regard for application. The remarkable fact that the "purest" mathematics often turns out to have practical applications is what Eugene Wigner has called "the unreasonable effectiveness of mathematics." Today, the natural sciences, engineering, economics, and medicine depend heavily on new mathematical discoveries. The word "mathematics" comes from the Greek μάθημα (máthema) meaning "science, knowledge, or learning" and μαθηματικός (mathematikós) meaning "fond of learning". It is often abbreviated maths in Commonwealth English and math in North American English.

History

:Main article: History of mathematics The evolution of mathematics might be seen to be an ever-increasing series of abstractions, or alternatively an expansion of subject matter. The first abstraction was probably that of numbers. The realization that two apples and two oranges do have something in common, namely that they fill the hands of exactly one person, was a breakthrough in human thought. In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. Arithmetic (e.g. addition, subtraction, multiplication and division), naturally followed. Monolithic monuments testify to a knowledge of geometry. Further steps need writing or some other system for recording numbers such as tallies or the knotted strings called khipu used by the Inca empire to store numerical data. Numeral systems have been many and diverse. Historically, the major disciplines within mathematics arose, from the start of recorded history, out of the need to do calculations on taxation and commerce, to understand the relationships among numbers, to measure land, and to predict astronomical events. These needs can be roughly related to the broad subdivision of mathematics, into the studies of quantity, structure, space, and change. Mathematics since has been much extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries have been made throughout history and continue to be made today.

Inspiration, pure and applied mathematics, and aesthetics

Mathematics arises wherever there are difficult problems that involve quantity, structure, space, or change. At first these were found in commerce, land measurement and later astronomy; nowadays, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. Newton invented infinitesimal calculus and Feynman his Feynman path integral using a combination of reasoning and physical insight, and today's string theory also inspires new mathematics. 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. As in most areas of study, the explosion of knowledge in the scientific age has led to specialization in mathematics. One major distinction is between pure mathematics and applied mathematics. Within applied mathematics, two major areas have split off and become disciplines in their own right, statistics and computer science. Many mathematicians talk about the elegance of mathematics, its intrinsic aesthetics and inner beauty. Simplicity and generality are valued. There is beauty also in a clever proof, such as Euclid's proof that there are infinitely many prime numbers, and in a numerical method that speeds calculation, such as the fast Fourier transform. G. H. Hardy in "A Mathematicians Apology" expressed the belief that these esthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. Main article: Mathematical beauty.

Notation, language, and rigor

Mathematical writing is not easily accessible to the layperson. A Brief History of Time, Stephen Hawking's 1988 bestseller, contained a single mathematical equation. This was the author's compromise with the publisher's advice, that each equation would halve the sales. The reasons for the inaccessibility even of carefully-expressed mathematics can be partially explained. Contemporary mathematicians strive to be as clear as possible in the things they say and especially in the things they write (this they have in common with lawyers). They refer to rigor. To accomplish rigor, mathematicians have extended natural language. There is precisely-defined vocabulary for referring to mathematical objects, and stating certain common relations. There is an accompanying mathematical notation which, like musical notation, has a definite content and also has a strict grammar (under the influence of computer science, more often now called syntax). Some of the terms used in mathematics are also common outside mathematics, such as ring, group and category; but are not such that one can infer the meanings. Some are specific to mathematics, such as homotopy and Hilbert space. It was said that Henri Poincaré was only elected to the Académie Française so that he could tell them how to define automorphe in their dictionary. Rigor is fundamentally a matter of mathematical proof. Mathematicians want their theorems to follow mechanically from axioms by means of formal axiomatic reasoning. This is to avoid mistaken 'theorems', based on fallible intuitions; of which plenty of examples have occurred in the history of the subject (for example, in mathematical analysis). Axioms in traditional thought were 'self-evident truths', but that conception turns out not to be workable in pushing the mathematical boundaries. 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 (strong enough) axiom system has undecidable formulas; and so a final axiomatization of mathematics is unavailable. 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.

Is mathematics a science?

Carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. The mathematician-physicist Leon M. Lederman has quipped: "The physicists defer only to mathematicians, and the mathematicians defer only to God (though you may be hard pressed to find a mathematician that modest)." If one considers science to be strictly about the physical world, then mathematics, or at least pure mathematics, is not a science. 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. [http://info.med.yale.edu/therarad/summers/ziman.htm] 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.

Overview of fields of mathematics

As noted above, the major disciplines within mathematics first arose out of the need to do calculations in commerce, to understand the relationships between numbers, to measure land, and to predict astronomical events. These four needs can be roughly related to the broad subdivision of mathematics 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) and to the empirical mathematics of the various sciences (applied mathematics). The study of quantity starts with numbers, first the familiar natural numbers and integers and their arithmetical operations, which are characterized in arithmetic. The deeper properties of whole numbers are studied in number theory. The study of structure began with investigations of Pythagorean triples. Neolithic monuments on the British Isles are constructed using Pythagorean triples. Eventually, this led to the invention of more abstract numbers, such as the square root of two. The deeper structural properties of numbers are studied in abstract algebra and the investigation of groups, rings, fields and other abstract number systems. Included is the important concept of vectors, generalized to vector spaces and studied in linear algebra. The study of vectors combines three of the fundamental areas of mathematics, quantity, structure, and space. The study of space originates with geometry, beginning with Euclidean geometry. Trigonometry combines space and number. 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, calculus on manifolds. Within algebraic geometry is the description of geometric objects as solution sets of polynomal 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 be the greatest growth area in 20th century mathematics. Understanding and describing change is a common theme in the natural sciences, and calculus was developed as a most useful tool. The central concept used to describe a changing quantity is that of a function. Many problems lead quite naturally to relations between a quantity and its rate of change, and the methods of differential equations. The numbers used to represent continuous quantities are the real numbers, and the detailed study of their properties and the properties of real-valued functions is known as real analysis. These have been generalized, with the inclusion of the square root of negative one, to the complex numbers, which are studied in complex analysis. Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions. One of many applications of functional analysis is quantum mechanics. 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. Beyond quantity, structure, space, and change are areas of pure mathematics that can be approached only by deductive reasoning. In order to clarify the foundations of mathematics, the fields of mathematical logic and set theory were developed. Mathematical logic, which divides into recursion theory, model theory, and proof theory, is now closely linked to computer science. When electronic computers were first conceived, several essential theoretical concepts in computer science were shaped by mathematicians, leading to the fields of computability theory, computational complexity theory, and information theory. Many of those topics are now investigated in theoretical computer science. Discrete mathematics is the common name for the fields of mathematics most generally useful in computer science. An important field in applied mathematics is statistics, which uses probability theory as a tool and allows the description, analysis, and prediction of phenomena where chance plays a part. It is used in all the sciences. Numerical analysis investigates methods for using computers to efficiently solve a broad range of mathematical problems that are typically beyond human capacity, and taking rounding errors or other sources of error into account to obtain credible answers.

Major themes in mathematics

An alphabetical and subclassified list of mathematical topics is available. The following list of themes and links gives just one possible view. For a fuller treatment, see Areas of mathematics or the list of lists of mathematical topics.

Change

:Ways to express and handle change in mathematical functions, and changes between numbers. : :Arithmetic – Calculus – Vector calculus – Analysis – Differential equations – Dynamical systems – Chaos theory – List of functions

Foundations and methods

:Approaches to understanding the nature of mathematics. :philosophy of mathematics – mathematical intuitionism – mathematical constructivism – foundations of mathematics – set theory – symbolic logic – model theory – category theory – Logic – reverse mathematics – table of mathematical symbols

Discrete mathematics

:Discrete mathematics involves techniques that apply to objects that can only take on specific, separated values. : :Combinatorics – Naive set theory – Theory of computation– Cryptography – Graph theory

Applied mathematics

:Applied mathematics uses the full knowledge of mathematics to solve real-world problems. :Mathematical physics – Mechanics – Fluid mechanics – Numerical analysis – Optimization – Probability – Statistics – Mathematical economics – Financial mathematics – Game theory – Mathematical biology – Cryptography – Information theory

Important theorems

:These theorems have interested mathematicians and non-mathematicians alike. :See list of theorems for more :Pythagorean theorem – Fermat's last theorem – Gödel's incompleteness theorems – Fundamental theorem of arithmetic – Fundamental theorem of algebra – Fundamental theorem of calculus – Cantor's diagonal argument – Four color theorem – Zorn's lemma – Euler's identity – classification theorems of surfaces – Gauss-Bonnet theorem – Quadratic reciprocity – Riemann-Roch theorem.

Important conjectures

See list of conjectures for more :These are some of the major unsolved problems in mathematics. :Goldbach's conjecture – Twin Prime Conjecture – Riemann hypothesis – Poincaré conjecture – Collatz conjecture – P=NP? – open Hilbert problems.

History and the world of mathematicians

Mathematics and other fields

:Mathematics and architecture – Mathematics and education – Mathematics of musical scales

Common misconceptions

Mathematics is not a closed intellectual system, in which everything has already been worked out. There is no shortage of open problems. Pseudomathematics is a form of mathematics-like activity undertaken outside academia, and occasionally by mathematicians themselves. It often consists of determined attacks on famous questions, consisting of proof-attempts made in an isolated way (that is, long papers not supported by previously published theory). The relationship to generally-accepted mathematics is similar to that between pseudoscience and real science. The misconceptions involved are normally based on:
- misunderstanding of the implications of mathematical rigour;
- attempts to circumvent the usual criteria for publication of mathematical papers in a learned journal after peer review, with assumptions of bias;
- lack of familiarity with, and therefore underestimation of, the existing literature. The case of Kurt Heegner's work shows that the mathematical establishment is neither infallible, nor unwilling to admit error in assessing 'amateur' work. And like astronomy, mathematics owes much to amateur contributors such as Fermat and Mersenne. Mathematics is not accountancy. Although arithmetic computation is crucial to accountants, their main concern is to verify that computations are correct through a system of doublechecks. Advances in abstract mathematics are mostly irrelevant to the efficiency of concrete bookkeeping, but the use of computers clearly does matter. Mathematics is not numerology. Numerology uses modular arithmetic to reduce names and dates down to numbers, but assigns emotions or traits to these numbers intuitively or on the basis of traditions. Mathematical concepts and theorems need not correspond to anything in the physical world. In the case of geometry, for example, it is not relevant to mathematics to know whether points and lines exist in any physical sense, as geometry starts from axioms and postulates about abstract entities called "points" and "lines" that we feed into the system. While these axioms are derived from our perceptions and experience, they are not dependent on them. And yet, mathematics is extremely useful for solving real-world problems. It is this fact that led Eugene Wigner to write an essay on The Unreasonable Effectiveness of Mathematics in the Natural Sciences. Mathematics is not about unrestricted theorem proving, any more than literature is about the construction of grammatically correct sentences. However, theorems are elements of formal theories, and in some cases computers can generate proofs of these theorems more or less automatically, by means of automated theorem provers. These techniques have proven useful in formal verification of programs and hardware designs. However, they are unlikely to generate (in the near term, at least) mathematics with any widely recognized aesthetic value.

Bibliography

- Benson, Donald C., The Moment Of Proof: Mathematical Epiphanies (1999).
- Courant, R. and H. Robbins, What Is Mathematics? (1941);
- Davis, Philip J. and Hersh, Reuben, The Mathematical Experience. Birkhäuser, Boston, Mass., 1980. A gentle introduction to the world of mathematics.
- Boyer, Carl B., History of Mathematics, Wiley, 2nd edition 1998 available, 1st edition 1968 . A concise history of mathematics from the Concept of Number to contemporary Mathematics.
- Gullberg, Jan, Mathematics--From the Birth of Numbers. W.W. Norton, 1996. 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 math encyclopedia, in ten (expensive) volumes, the most complete and authoritative work available. Also in paperback and on CD-ROM.
- Kline, M., Mathematical Thought from Ancient to Modern Times (1973).
- Pappas, Theoni, The Joy Of Mathematics (1989).

External links

- [http://www.cut-the-knot.org/ Interactive Mathematics Miscellany and Puzzles] — A collection of articles on various math topics, with interactive Java illustrations at cut-the-knot
- Rusin, Dave: [http://www.math-atlas.org/ The Mathematical Atlas]. A guided tour through the various branches of modern mathematics.
- Stefanov, Alexandre: [http://us.geocities.com/alex_stef/mylist.html Textbooks in Mathematics]. A list of free online textbooks and lecture notes in mathematics.
- Weisstein, Eric et al.: [http://www.mathworld.com/ MathWorld: World of Mathematics]. An online encyclopedia of mathematics.
- 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.
- A mathematical thesaurus maintained by the [http://nrich.maths.org/ NRICH] project at the University of Cambridge (UK), [http://thesaurus.maths.org/ Connecting Mathematics]
- [http://planetmath.org/ Planet Math]. An online math encyclopedia under construction, focusing on modern mathematics. Uses the GFDL, allowing article exchange with Wikipedia. Uses TeX markup.
- [http://www.mathforge.net/ Mathforge]. A news-blog with topics ranging from popular mathematics to popular physics to computer science and education.
- [http://www.youngmath.net/concerns Young Mathematicians Network (YMN)]. A math-blog "Serving the Community of Young Mathematicians". Topics include: Math News, Grad and Undergrad Life, Job Search, Career, Work & Family, Teaching, Research, Misc...
- [http://metamath.org/ Metamath]. A site and a language, that formalize math from its foundations.
- [http://world.std.com/~reinhold/dir/mathmovies.html Math in the Movies]. A guide to major motion pictures with scenes of real mathematics
- [http://math.cofc.edu/faculty/kasman/MATHFICT/default.html Mathematics in fiction]. Links to works of fiction that refer to mathematics or mathematicians.
- [http://www.mathhelpforum.com/math-help Math Help Forum]. A forum, for math help, math discussion and debate.
- [http://www.sosmath.com/CBB S.O.S. Mathematics Cyberboard] a math help forum which incorporates a LaTeX extension, making it easier for members to write and display math formulae.
- [http://www-history.mcs.st-and.ac.uk/~history/ Mathematician Bibliography]. Extensive history and quotes from all famous mathematicians.
- [http://www.physicsmathforums.com/ Physics Math Forums]
- Category:School subjects fiu-vro:Matõmaatiga zh-min-nan:Sò·-ha̍k ko:수학 ms:Matematik ja:数学 simple:Mathematics th:คณิตศาสตร์

Mathematics education

] Mathematics education is the study of practices and methods of both the teaching and learning of mathematics. Furthermore, mathematics educators are concerned with the development of tools that facilitate practice and/or the study of practice. Mathematics education has been a hotly debated subject in modern society. There is an ambiguity in the term for it refers both to these practices in classrooms around the world, but also to an emergent discipline with its own journals, conferences, etc. The main international body involved is the [http://www.kurims.kyoto-u.ac.jp/IMU/ICMI/ International Commission on Mathematical Instruction].

History

Elementary mathematics was part of the education system in most ancient civilisations, including Ancient Greece, the Roman empire, Vedic society and ancient Egypt. In most cases, a formal education was only available to male children with a sufficiently high status, wealth or caste. In Plato's division of the liberal arts into the trivium and the quadrivium, the quadrivium included the mathematical fields of arithmetic and geometry. This structure was continued in the structure of classical education that was developed in medieval Europe. Teaching of geometry was almost universally based on Euclid's Elements. Apprentices to trades such as masons, merchants and money-lenders could expect to learn such practical mathematics as was relevant to their profession. The first mathematics textbooks to be written in English were published by Robert Recorde, beginning with The Grounde of Artes in 1540. In the Renaissance the academic status of mathematics declined, because it was strongly associated with trade and commerce. Although it continued to be taught in European universities, it was seen as subservient to the study of Natural, Metaphysical and Moral Philosophy. This trend was somewhat reversed in the seventeenth century, with the University of Aberdeen creating a Mathematics Chair in 1613, followed by the Chair in Geometry set up in University of Oxford in 1619 and the Lucasian Chair of Mathematics, established by the University of Cambridge in 1662. However, it was uncommon for mathematics to be taught outside of the universities. Isaac Newton, for example, received no formal mathematics teaching until he joined Trinity College, Cambridge in 1661. In the eighteenth and nineteenth centuries the industrial revolution led to an enormous increase in urban populations. Basic numeracy skills, such as the ability to tell the time, count money and carry out simple arithmetic, became essential in this new urban lifestyle. Within the new public education systems, mathematics became a central part of the curriculum from an early age. By the twentieth century mathematics was part of the core curriculum in all developed countries. However, diverse and changing ideas about the purpose of mathematical education led to little overall consistency in the content or methods that were adopted.

Objectives

At different times and in different cultures and countries, mathematics education has attempted to achieve a variety of different objectives. These objectives have included:
- The teaching of basic numeracy skills to all pupils
- The teaching of practical mathematics (arithmetic, elementary algebra, plane and solid geometry, trigonometry) to most pupils, to equip them to follow a trade or craft
- The teaching of abstract mathematical concepts (such as set and function) at an early age
- The teaching of selected areas of mathematics (such as Euclidean geometry) as an example of an axiomatic system and a model of deductive reasoning
- The teaching of selected areas of mathematics (such as calculus) as an example of the intellectual achievements of the modern world
- The teaching of advanced mathematics to those pupils who wish to follow a career in science
- The teaching of heuristics and other problem-solving strategies to solve nonroutine problems. Methods of teaching mathematics have varied in line with changing objectives.

Standards

Throughout most of history, standards for mathematics education were set locally, by individual schools or teachers, depending on the levels of achievement that were relevant to and realistic for their pupils. In modern times there has been a move towards regional or national standards, usually under the umbrella of a wider standard school curriculum. In England, for example, standards for mathematics education are set as part of the [http://www.nc.uk.net/webdav/servlet/XRM?Page/@id=6004&Subject/@id=22 National Curriculum for England]. In the USA, the National Council of Teachers of Mathematics [http://nctm.org] has produced a series of documents, most recently Principles and Standards for School Mathematics [http://standards.nctm.org/], setting forth a consensus on broad goals for school mathematics. More specific education standards are generally set at a state level - in California, for example, the California State Board of Education sets standards for mathematics education [http://www.cde.ca.gov/be/st/ss/mthmain.asp].

Levels

Different levels of mathematics are taught at different ages. Sometimes a class may be taught at an earlier age as a special or "honors" class. A rough guide to the ages at which the sub-topics of arithmetics and algebra are taught is as follows:
- Addition : ages 5-7; more digits ages 8-9
- Subtraction : ages 5-7; more digits ages 8-9
- Multiplication : ages 7-8; more digits ages 9-10
- Division : age 8; more digits ages 9-10
- Pre-Algebra : ages 11-12
- Algebra : ages 13+
- Geometry : ages 14-15+
- Precalculus : ages 16+
- Calculus : ages 18+

Methods

The method or methods used in any particular context are largely determined by the objectives that the relevant educational system is trying to achieve. Methods of teaching mathematics include the following:
- Classical education - the teaching of mathematics within the classical education syllabus of the middle ages was typically based on Euclid's Elements, which was taught as a paradigm of deductive reasoning
- Rote learning - the teaching of mathematical results, definitions and concepts by repetition and memorisation. Typically used to teach multiplication tables.
- Exercises - the teaching of mathematical skills by completing large numbers of exercises of a similar type, such as adding vulgar fractions or solving quadratic equations. For example, Cuisenaire rods are used as a method of teaching fractions.
- Problem solving - the cultivation of mathematical ingenuity, creativity and heuristic thinking by setting students open-ended, unusual, and sometimes insoluble problems. The problems can range from simple word problems to problems from international mathematics competitions such as the International Mathematical Olympiad.
- New math - a method of teaching mathematics which focuses on abstract concepts such as set theory rather than practical applications.
- Historical method - teaching the development of mathematics within an historical, social and cultural context. Provides more human interest than a purely abstract approach.
- Reform - based on the Constructivism (learning theory). Students usually work on contextual problems in small groups, with the problems themselves acting as vehicles for mathematics content. The use of calculator and manipulatives are encouraged, but algebra skills and rote memorization are deemphasized. These methods are not exclusive, and any given system of mathematical education will probably combine several different methods.

Mathematics teachers

The following people all taught mathematics at some stage in their lives, although they are better known for other things:
- Lewis Carroll, pen name of British author Charles Dodgson, lectured in mathematics at Christ Church, Oxford
- John Dalton, British chemist and physicist, taught mathematics in schools and colleges in Manchester, Oxford and York
- Tom Lehrer, American songwriter and satirist, taught mathematics at Harvard and MIT
- Georg Joachim Rheticus, Austrian cartographer and disciple of Copernicus, taught mathematics at the University of Wittenberg
- Edmund Rich, Archbishop of Canterbury in the 13th century, lectured on mathematics at the universities of Oxford and Paris
- Archie Williams, American athlete and Olympic gold medalist, taught mathematics at high schools in California

External links

- [http://www-gap.dcs.st-and.ac.uk/~history/Education/index.html History of Mathematical Education]
- [http://igpme.org/ International Group for the Psychology of Mathematics Education]

Scholarly Journals: Print

- [http://springerlink.metapress.com/app/home/journal.asp?wasp=79421927ccf647fcb830fafe34a31b4a&referrer=parent&backto=linkingpublicationresults,1:102875,1 Educational Studies in Mathematics]
- [http://my.nctm.org/eresources/journal_home.asp?journal_id=1 Journal for Research in Mathematics Education]
- [http://flm.educ.ualberta.ca/ For the Learning of Mathematics]

Scholarly Journals: on-line

- [http://www.ex.ac.uk/~PErnest/ The Philosophy of Mathematics Education Journal homepage]
-

Calculus

:For other uses of the term calculus see calculus (disambiguation) Calculus is a central branch of mathematics, developed from algebra and geometry. The origin of the word stems from the nascent development of mathematics: the early Greeks used pebbles arranged in patterns to learn arithmetic and geometry, and the Latin word for "pebble" is "calculus," a diminutive of calx (genitive calcis) meaning "limestone." Calculus is built on two major complementary ideas. The first is differential calculus, which studies the rate of change in one quantity relative to the rate of change in another quantity. This can be illustrated by the slope of a line. The second is integral calculus, which studies the accumulation of quantities, such as areas under a curve, linear distance traveled, or volume displaced. These two operations are inverses of one another, as explained by the fundamental theorem of calculus. Examples of typical differential calculus problems include:
- finding the acceleration and speed of a free-falling body at a particular moment
- finding the optimal number of units a company should produce to maximize their profit. Examples of integral calculus problems include:
- finding the amount of water pumped by a pump with a set power input but varying conditions of pumping losses and pressure
- finding the amount of parking lot plowed by a snowplow of given power with varying rates of snowfall. Today, calculus is used in every branch of the physical sciences, in computer science, in statistics, and in engineering; in economics, business, and medicine; and as a general method whenever the goal is an optimum solution to a problem that can be given in mathematical form. From a mathematical standpoint, it is used in conjunction with limits which, roughly speaking, allow the control or accurate description of an otherwise uncontrollable output.

Differential calculus

The derivative measures the sensitivity of one variable to small changes in another variable. Consider the formula: :

\mathrm = \frac

for an object moving at constant speed. The speed of a car, as measured by the speedometer, is the derivative of the car's distance traveled, as measured by the odometer, as a function of time. Calculus is a mathematical tool for dealing with this complex but natural and familiar situation. Differential calculus can be used to determine the instantaneous speed at any given instant, while the formula speed = distance divided by time only gives the average speed. The formula cannot be applied to an instant in time because it then gives the meaningless quotient zero divided by zero. Calculus avoids division by zero using the limit which, roughly speaking, is a method of controling an otherwise uncontrolable output, such as division by zero or multiplication by infinity. More formally, differential calculus defines the instantaneous rate of change (the derivative) of a mathematical function's value, with respect to changes of the variable. The derivative is defined as a limit of a difference quotient. The derivative of a function, if it exists, gives information about its graph. It is useful for finding optimum solutions to problems, called maxima and minima of a function. It is proved mathematically that these optimum solutions exist either where the graph is flat, so that the slope is zero, or where the graph has a sharp turn (cusp where the derivative does not exist, or at the endpoints of the graph. Another application of differential calculus is Newton's method, a powerful equation solving algorithm Differential calculus has been applied to many questions that were first formulated in other areas, such as business or medicine. The derivative lies at the heart of the physical sciences. Newton's law of motion, Force = Mass × Acceleration, involves calculus because acceleration is the derivative of the velocity. (See Differential equation.) Maxwell's theory of electromagnetism and Einstein's theory of general relativity are also expressed in the language of differential calculus, as is the basic theory of electrical circuits and much of engineering. It is also applied to problems in biology, economics, and many other areas. The derivative of a function y with respect to x is usually expressed as either y′ (read "y-prime") or as :

\frac

Integral calculus

There are two types of integral in calculus, the indefinite and the definite. The indefinite integral is simply the antiderivative. That is, F is an antiderivative of f when f is a derivative of F. (This use of captial letters and lower case letters is common in calculus. The lower case letter represents the derivative of the capital letter.) The definite integral evaluates the cumulative effect of many small changes in a quantity. The simplest instance is the formula :

\mathrm = \mathrm \cdot \mathrm

for calculating the distance a car moves during a period of time when it is traveling at constant speed. The distance moved is the cumulative effect of the small distances moved in each instant. Calculus is also able to deal with the natural situation in which the car moves with changing speed. Integral calculus determines the exact distance traveled during an interval of time by creating a series of better and better approximations, called Riemann sums, that approach the exact distance as a limit. More formally, we say that the definite integral of a function on an interval is a limit of Riemann sum approximations. Applications of integral calculus arise whenever the problem is to compute a number that is in principle (approximately) equal to the sum of a large number of small quantities. The classic geometric application is to area computations. In principle, the area of a region can be approximated by chopping it up into many pieces (typically rectangles, or, in polar coordiantes, circular sectors), and then adding the areas of those pieces. The length of an arc, the area of a surface, and the volume of a solid can also be expressed as definite integrals. Probability, the basis for statistics, provides another important application of integral calculus. The symbol of integration is ∫, a stretched s (which stands for "sum"). The precise meanings of expressions involving integrals can be found in the main article Integral. The definite integral, written as: :

\int_a^b f(x)\, dx

is read "the integral from a to b of f(x) dx".

Foundations

The rigorous foundation of calculus is based on the notions of a function and of a limit; the latter has a theory ultimately depending on that of the real numbers as a continuum. Its tools include techniques associated with elementary algebra, and mathematical induction. The modern study of the foundations of calculus is known as real analysis. This includes full definitions and proofs of the theorems of calculus. It also provides generalisations such as measure theory and distribution theory.

Fundamental theorem of calculus

The fundamental theorem of calculus states that differentiation and integration are, in a certain sense, inverse operations. More precisely, if one defines one function as the integral of another function, then differentiating the newly defined function returns the fuction you started with. Furthermore, if you want to find the value of a definite integral, you usually do so by evaluating an antiderivative. Here is the mathematical formulation of the Fundamental Theorem of Calculus: If a function f is continuous on the interval [a, b] and if F is a function whose derivative is f on the interval [a, b], then :

\int_^ f(x)\,dx = F(b) - F(a).

:Also, for every x in the interval [a, b], :

\frac\int_a^x f(t)\, dt = f(x).

This realization, made by both Newton and Leibniz, was key to the massive proliferation of analytic results after their work became known. The fundamental theorem provides an algebraic method of computing many definite integrals—without performing limit processes—by finding formulas for antiderivatives. It is also a prototype solution of a differential equation. Differential equations relate an unknown function to its derivatives, and are ubiquitous in the sciences.

Applications

The development and use of calculus has had wide reaching effects on nearly all areas of modern living. It underlies nearly all of the sciences, especially physics. Virtually all modern developments such as building techniques, aviation, and other technologies make fundamental use of calculus. Many algebraic formulas now used for ballistics, heating and cooling, and other practical sciences were worked out through the use of calculus. In a handbook, an algebraic formula based on calculus methods may be applied without knowing its origins. The success of calculus has been extended over time to differential equations, vector calculus, calculus of variations, complex analysis, and differential topology.

History

The origins of integral calculus are generally regarded as going back no farther than to the time of the ancient Greeks, circa 200 B.C., though there is some evidence that the ancient Egyptians may have had some hint of the idea at a much earlier date. (See Moscow Mathematical Papyrus.) The Hellenic mathematician Eudoxus is generally credited with the method of exhaustion, which made it possible to compute the areas of regions and the volumes of solids. Archimedes developed this method further, and invented heuristic methods which resemble modern calculus. Of all the mathematicians of the ancient world, he was the closest to discovering integral calculus, but never made the breakthrough, and after him study of calculus did not advance appreciably for more than a thousand years. An Indian mathematician, Bhaskara (1114-1185), developed a number of ideas that can now be seen to be forerunners of calculus, including the idea now known as "Rolle's theorem". He was the first to conceive of differential calculus. The 14th century Indian mathematician Madhava, along with other mathematicians of the Kerala school, studied infinite series, convergence, differentiation, and iterative methods for solution of non-linear equations. Jyestadeva of the Kerala school wrote the first calculus text, the Yuktibhasa, which explores methods and ideas of calculus repeated in Europe only by the seventeenth century. Calculus, towards the end of the early modern period and into the first years of the eighteenth century, was a time of major innovation in Europe, making accessible answers to old questions, and providing a new method in mathematical physics. Several mathematicians contributed to this breakthrough, notably John Wallis and Isaac Barrow. James Gregory proved a result equivalent to the Fundamental Theorem of Calculus in 1668. Leibniz and Newton pulled these ideas together into a coherent whole and they are usually credited with the independent and nearly simultaneous creation of calculus. Newton was the first to apply calculus to physics and Leibniz developed much of the notation used in calculus today; he often spent days determining appropriate symbols for concepts. It was generations after Newton and Leibniz that Cauchy and other mathematicians finally put calculus on a rigorous basis, with the definition of the limit. There has been considerable debate about whether Newton or Leibniz was first to come up with the important concepts of calculus. The truth of the matter is that the ideas of calculus were a part of the mathematical knowledge of their day, and they independently put those pieces together in different but coherent ways. The mathematical proofs of much of what they did came later, with Cauchy and others. This controversy between Leibniz and Newton was unfortunate in that it divided English-speaking mathematicians from those in Europe for many years, setting back British analysis (calculus-based mathematics) for a long time. Newton's terminology and notation was retained in British usage until the early 19th century, long after it had been replaced by Leibniz's notation everywhere else. The work of the Analytical Society successfully saw the introduction of Leibniz's notation in Great Britain. It is now thought that Newton had discovered several ideas related to calculus earlier than Leibniz; but Leibniz published first. Today, both are given equal credit. Lesser credit for ideas that led to the development of calculus is given to Descartes, Barrow, de Fermat, Huygens, and Wallis.

External links

- [http://www.math.uiowa.edu/~stroyan/InfsmlCalculus/InfsmlCalc.htm A Brief Introduction to Infinitesimal Calculus] by Keith Duncan Stroyan of the University of Iowa.
- [http://www.math.wisc.edu/~keisler/calc.html Elementary Calculus: An Approach Using Infinitesimals] by H. Jerome Keisler, an out-of-print book available on the web.
- [http://mathworld.wolfram.com/Calculus.html MathWorld general article on calculus]
- [http://www-groups.dcs.st-and.ac.uk/~history/Projects/Pearce/Chapters/Ch9_3.html Madhava of Sangamagramma ]
- [http://integrals.wolfram.com/ Online Integrator by Mathematica]
- [http://www.ericdigests.org/pre-9217/calculus.htm The Role of Calculus in College Mathematics]
- [http://www-groups.dcs.st-and.ac.uk/~history/Projects/Pearce/Chapters/Ch8_5.html Work of Bhaskaracharya II]
- ja:微分積分学 ko:미적분학 simple:Calculus th:แคลคูลัส

Eighteenth Century

As a means of recording the passage of time, the 18th century refers to the century that lasted from 1701 through 1800 in the Gregorian calendar. European history scholars will sometimes specifically refer to the 18th century as 1715-1789, denoting the period of time between the death of Louis XIV of France and the start of the French Revolution.

Events

- 1701-14: War of the Spanish Succession
- 1703: Saint Petersburg founded by Peter the Great. Russian capital until 1918.
- 1707: Act of Union passed merging the Scottish and the English Parliaments, thus establishing The Kingdom of Great Britain.
- 1707: After Aurangzeb's death, the Mughal Empire enters a long decline.
- 1715: Louis XIV dies
- 1718: City of New Orleans founded by the French in North America
- 1720: The South Sea Bubble
- 1721: Robert Walpole becomes the first Prime Minister of Great Britain (de facto).
- 1721: Treaty of Nystad signed, ending the Great Northern War.
- 1722: Afghans conquer Iran, ending the Safavid dynasty.
- 1722: Kangxi Emperor of China dies.
- 1733-38: War of the Polish Succession
- 1735-99: The Qianlong Emperor of China oversees a huge expansion in territory.
- 1736: Nadir Shah assumes title of Shah of Persia and founds the Afsharid dynasty. Rules until his death in 1747.
- 1739: Nadir Shah defeats the Mughals and sacks Delhi.
- 1740: Frederick the Great crowned King of Prussia.
- 1740-48: War of the Austrian Succession
- 1741: Russians begin settling the Aleutian Islands.
- 1747: Ahmad Shah founds the Durrani Empire in modern day Afghanistan.
- 1750: peak of the Little Ice Age
- 1755: The Lisbon earthquake
- 1756-63: Seven Years' War fought among European powers in various theaters around the world.
- 1757: Battle of Plassey signals the beginning of British rule in India.
- 1760: George III becomes King of Britain.
- 1762-96: Reign of Catherine the Great of Russia.
- 1763-66: Pontiac's Rebellion in North America
- 1766-99: Anglo-Mysore Wars
- 1767: Burmese conquer the Ayutthaya kingdom.
- 1768: Gurkhas conquer Nepal.
- 1768-1774: Russo-Turkish War
- 1769: Spanish missionaries establish the first of 21 missions in California.
- 1772-95: The Partitions of Poland end the Polish-Lithuanian Commonwealth and erase Poland from the map for 123 years.
- 1775-82: First Anglo-Maratha War
- 1775-83: American Revolution
- 1779-1879: Cape Frontier Wars between British and Boer settlers and the Xhosas in South Africa
- 1785-95: Northwest Indian War between the United States and Native Americans
- 1787: Freed slaves from London found Freetown in present-day Sierra Leone.
- 1788: First European settlement established in Australia at Sydney.
- 1789: George Washington elected President of the United States. Serves until 1797.
- 1789-99: The French Revolution
- 1791-1804: The Haitian Revolution
- 1792-1815: The Great French War starts as the French Revolutionary Wars which lead into the Napoleonic Wars.
- 1792: New York Stock & Exchange Board founded.
- 1793: Upper Canada bans slavery.
- 1795: Pinckney's Treaty between the United States and Spain grants the Mississippi Territory to the US.
- 1796: British eject Dutch from Ceylon.
- 1796-1804: White Lotus Rebellion in China.
- 1797: Napoleon's invasion and partition of the Republic of Venice ends over 1,000 years of independence for the Serene Republic.
- 1798: Irish Rebellion against British Rule
- 1798-1800: Quasi-War between the United States and France.
- 1799: Napoleon stages a coup d'état and becomes dictator of France.
- 1799: Dutch East India Company is dissolved.

Significant people

- Ueda Akinari (Japanese writer)
- Queen Anne (British monarch)
- Marie Antoinette (French royalty and symbol of anti-Revolutionary ire)
- Benedict Arnold, considered a traitor by many people on both sides (United States and Britain) of the American Revolutionary War.
- Johann Sebastian Bach (composer)
- Pierre Beaumarchais (French writer)
- Jeremy Bentham (English jurist, philosopher, and legal and social reformer)
- Napoleon Bonaparte (general and first consul of France)
- François Boucher (French painter)
- Edmund Burke (British statesman and philosopher who supported the American Revolution)
- Robert Burns (Scottish poet)
- Catherine the Great (Russian Tsaritsa)
- James Cook (British navigator)
- Denis Diderot (French writer and philosopher)
- Leonhard Euler (mathematician)
- Jean-Honoré Fragonard (French painter)
- Benjamin Franklin (American revolutionary, inventor, printer, and diplomat)
- Frederick the Great (Prussian monarch)
- Thomas Gainsborough (painter)
- King George III (British monarch)
- Christoph Willibald Gluck (German composer)
- Johann Wolfgang von Goethe (German writer)
- Thomas Gray (British writer)
- George Frideric Handel (German composer)
- Alexander Hamilton (American revolutionary, lawyer, and statesman)
- Joseph Haydn (Austrian composer)
- William Hogarth (painter and engraver)
- David Hume (philosopher)
- Thomas Jefferson (American revolutionary, philosopher, and statesman)
- Samuel Johnson (British writer and literary critic)
- Immanuel Kant (philosopher)
- Wolfgang von Kempelen (Hungarian scientist, pioneer in experimental phonetics)
- John Law (Scottish economist)
- Louis XIV of France (monarch)
- Louis XV of France (monarch)
- Louis XVI of France (monarch)
- James Madison (American revolutionary, writer, and statesman)
- Maria Theresa of Austria (Holy Roman Empress, Queen of Hungary and Bohemia)
- Michikinikwa (Miami tribe chief and war leader)
- Wolfgang Amadeus Mozart (composer)
- Thomas Paine (British intellectual and philosopher who advocated for the American Revolution)
- Philip II, Duke of Orléans (Regent of France)
- Alexander Pope (British poet)
- Francis II Rákóczi (prince of Hungary and Transylvania, leader of the Hungarian freedom war)
- Jean-Philippe Rameau (French composer and music theorist)
- Sir Joshua Reynolds (painter)
- Maximilien Robespierre (French Revolutionary leader and dictator)
- Jean-Jacques Rousseau (French writer and philosopher)
-

Areas of mathematics

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Mathematics

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Inspiration, pure and applied mathematics, and aesthetics

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Overview of fields of mathematics

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Mathematics education

History

Objectives

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Mathematics teachers

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Scholarly Journals: Print

Scholarly Journals: on-line

Calculus

Differential calculus

Integral calculus

Foundations

Fundamental theorem of calculus

Applications

History

See also

Further reading

External links

Eighteenth Century

Events

Significant people