Mathematics

Ancient Greek mathematician Euclid (holding calipers), 3rd century BC, as imagined by Raphael in this detail from the School of Athens (1509–1511)

Math is the study of numbers, shapes, and patterns. The word comes from the Greek μάθημα (máthema), meaning "science, knowledge, or learning", and is sometimes shortened to math or maths.

It is the study of:

Numbers: including how things can be counted.
Structure: including how things are organized, but also how they can be or could have been. This subfield is usually called algebra.
Place: where things are, and spatial arrangement, including arrangements of spaces themselves. This subfield is usually called geometry.
Change: how things become different. This subfield is usually called analysis.

Applied math is useful for solving problems in the real world. People working in business, science, engineering, and construction use math.^[1]^[2]

Problem-solving in mathchange source

Math solves problems by using logic. One of the main tools of logic used by mathematicians is deduction. Deduction is a special way of thinking to discover and prove new truths using old truths. To a mathematician, the reason something is true (called a proof) is just as important as the fact that it is true, and this reason is often found using deduction. Using deduction is what makes mathematical thinking different from other kinds of scientific thinking, which might rely on experiments or on interviews.^[3]

Logic and reasoning are used by math people to create general rules, which are an important part of math. These rules leave out information that is not important so that a single rule can cover many situations. By finding general rules, mathematics solves many problems at the same time as these rules can be used on other problems.^[4] These rules can be called theorems (if they have been proven) or conjectures (if it is not known if they are true yet).^[5] Most math people use non-logical and creative reasoning in order to find a logical proof.^[6]

Sometimes, math people finds and studies rules or ideas that we don't understand yet. Often in mathematics, ideas and rules are chosen because they are considered simple or neat. On the other hand, sometimes these ideas and rules are found in the real world after they are studied in mathematics; this has happened many times in the past. In general, studying the rules and ideas of mathematics can help us understand the world better. Some examples of math problems are addition, subtraction, multiplication, division, calculus, fractions and decimals. Algebra problems are solved by evaluating certain variables. A calculator answers every math problem in the four basic arithmetic operations.

Areas of study in mathchange source

Numberchange source

Math includes the study of numbers and quantities. It is a branch of science that deals with the logic of shape, quantity, and arrangement. Most of the areas listed below are studied in many different fields of mathematics, including set theory and mathematical logic. The study of number theory usually focuses more on the structure and behavior of the integers rather than on the actual foundations of numbers themselves, and so is not listed in this given subsection.

$1,2,3,\ldots$	$\ldots ,-1,0,1,\ldots$	${\frac {1}{2}},{\frac {2}{3}},0.125,\ldots$	$\tau ,\pi ,e,{\sqrt {2}},\ldots$	$1+i,2e^{i\pi /3},\ldots$
Natural numbers	Integers	Rational numbers	Real numbers	Complex numbers
$0,1,\ldots ,\omega ,\omega +1,\ldots ,2\omega ,\ldots$	$\aleph _{0},\aleph _{1},\ldots$	$+,-,\times ,/$	$<,\leq ,=,\geq ,>$	$f(x)={\sqrt {x}}$
Ordinal numbers	Cardinal numbers	Arithmetic operations	Arithmetic relations	Functions, see also special functions

Structurechange source

Structural math studies objects' and constructions' shape and integrity. These are areas of algebra and calculus.


Number theory	Abstract algebra	Linear algebra	Order theory	Graph theory

Shapechange source

Some areas of math study the shapes of things. Most of these areas are part of the study of geometry.


Topology	Geometry	Trigonometry	Differential geometry	Fractal geometry

Changechange source

Some areas of math study the way things change. Most of these areas are part of the study of analysis.


Calculus	Vector calculus	Analysis

Differential equations	Dynamical systems	Chaos theory

Applied mathematicschange source

Applied math uses symbolic logic to solve problems in areas like engineering and physics.

Numerical analysis – Optimization – Probability theory – Statistics – Mathematical finance – Game theory – Mathematical physics – Fluid dynamics - Computational algorithms

Famous theoremschange source

These theorems and conjectures have interested mathematicians and amateurs alike:

Pythagorean theorem – FLT – Goldbach's conjecture – Twin Prime Conjecture – Gödel's incompleteness theorems – Poincaré conjecture – Cantor's diagonal argument – Four color theorem – Zorn's lemma – Euler's Identity – Church-Turing thesis

These theorems and hypotheses have greatly changed mathematics:

Central limit theorem classification theorems of surfaces – Continuum hypothesis – Fourier Theorem – Fundamental theorem of calculus – Fundamental theorem of algebra – Fundamental theorem of arithmetic – Fundamental theorem of projective geometry – Gauss-Bonnet theorem - Kantorovich theorem – P Versus NP – Pythagorean theorem – Riemann hypothesis

These are a few conjectures that have been called "revolutionary":

Beal Conjecture (a generalization of FLT) – Birch and Swinnerton-Dyer Conjecture – Collatz Conjecture – Goldbach's Conjecture –Hodge Conjecture – Poincaré Conjecture

Foundations and methodschange source

Set theory – Symbolic logic – Model theory – Category theory – Logic – Table of mathematical symbols

History and the world of mathematicianschange source

History of mathematics – Timeline of mathematics – Mathematicians – Fields Medal – Abel Prize – Millennium Prize Problems (Clay Math Prize) – International Mathematical Union – Mathematics competitions – Lateral thinking – Mathematics and gender

Awards in mathematicschange source

There is no Nobel Prize in mathematics. Mathematicians can receive the Abel Prize and the Fields Medal for important works.^[7]^[8]

The Clay Mathematics Institute has said it will give one million dollars to anyone who solves one of the Millennium Prize Problems.

Mathematical toolschange source

There are many tools used to do math or find answers to math problems.

Older tools

Newer tools

Graphing calculators, scientific calculators and more complex computer visual tools
Programming languages
Computer algebra systems (listing) and automated matrix analysis
statistics software (for example SPSS)
SAS (programming language)
R (programming language)

Referenceschange source

Related pageschange source

Other websiteschange source

Mathematics Citizendium
Mathematics at the Open Directory Project

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v t e Mathematics (areas of mathematics)
Foundations	Category theory Information theory Mathematical logic Philosophy of mathematics Set theory Type theory
Algebra	Abstract Elementary Group theory Linear Commutative Multilinear Universal Homological
Analysis	Calculus Real analysis Complex analysis Differential equations Functional analysis Harmonic analysis Measure theory
Discrete	Combinatorics Graph theory Order theory Game theory
Geometry	Algebraic Analytic Differential Euclidean Discrete Finite
Number theory	Arithmetic Algebraic number theory Analytic number theory Diophantine geometry
Topology	Algebraic Geometric General Differential Homotopy theory
Applied	Control theory Mathematical physics Probability Statistics Engineering mathematics Mathematical biology Mathematical chemistry Mathematical economics Mathematical finance Mathematical psychology Mathematical sociology Mathematical statistics Operations research
Computational	Computer science Theory of computation Computational complexity theory Numerical analysis Optimization Computer algebra
Related topics	History of mathematics Recreational mathematics Mathematics and art Mathematics education