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-{{more sources|date=July 2018}}
-Mathematics is the study of [[number]]s, [[shape]]s and [[pattern]]s. The word comes from the [[Greek language|Greek]] word "μάθημα" (máthema), meaning "[[science]], [[knowledge]], or [[learning]]", and is sometimes shortened to '''maths''' (in [[England]], [[Australia]], [[Ireland]], and [[New Zealand]]) or '''math''' (in the [[United States]] and [[Canada]]).<ref>{{Cite news|url=http://www.slate.com/blogs/lexicon_valley/2014/12/08/math_versus_maths_how_americans_and_brits_deploy_the_collective_noun.html|title=Why Do Brits Say Maths and Americans Say Math?|last=Waldman|first=Katy|date=2014-12-08|work=Slate|access-date=2018-07-30|language=en-US|issn=1091-2339}}</ref> The short words are often used for [[arithmetic]], [[geometry]] or simple [[algebra]] by [[student]]s and their [[schools]].
-Mathematics includes the [[Education|study]] of:
-* Numbers: how things can be [[Counting|counted]].
-*[[Structure]]: how things are [[organisation|organized]]. This subfield is usually called [[algebra]].
-* Place: where things are and their arrangement. This subfield is usually called [[geometry]].
-* Change: how things become different. This subfield is usually called [[Mathematical analysis|analysis]].
-Mathematics is useful for solving [[problem]]s that occur in the real world, so many people besides [[mathematician]]s study and use mathematics. Today, some mathematics is needed in many jobs. People working in [[business]], [[science]], [[engineering]], and [[construction]] need some knowledge of mathematics.<ref>{{Cite web|url=https://www.siam.org/Students-Education/Programs-Initiatives/Thinking-of-a-Career-in-Applied-Mathematics|title=Thinking of a Career in Applied Mathematics? {{!}} SIAM|website=www.siam.org|language=en-US|access-date=2018-07-30}}</ref><ref>{{Cite journal|last=Wigner|first=Eugene|date=February 1960|title=The Unreasonable Effectiveness of Mathematics in the Natural Sciences|url=https://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html|journal=Communications in Pure and Applied Mathematics|volume=13|pages=1-14|via=}}</ref>
-== Problem-solving in mathematics ==
-Mathematics solves problems by using [[logic]]. One of the main tools of logic used by mathematicians is [[Deductive reasoning|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 (mathematics)|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 mathematics thinking different from other kinds of scientific thinking, which might rely on [[Experiment|experiments]] or on [[Interview|interviews]].<ref>{{Cite web|url=https://undsci.berkeley.edu/article/mathematics|title=The science checklist applied: Mathematics|website=undsci.berkeley.edu|access-date=2018-08-05}}</ref>
-Logic and reasoning are used by mathematicians to create general [[rule]]s, which are an important part of mathematics. These rules leave out [[information]] that is not important so that a single rule can cover many situations. By finding general rules, mathematics [[solution|solves]] many [[problem]]s at the same time as these rules can be used on other problems.<ref>{{Cite news|url=https://blogs.ams.org/mathgradblog/2016/08/21/role-generalization-advanced-mathematical-thinking/|title=The Role of Generalization in the Advanced Mathematical Thinking|date=2016-08-21|work=AMS Grad Blog|access-date=2018-08-07|language=en-US}}</ref> These rules can be called [[Theorem|theorems]] (if they have been proved) or [[Conjecture|conjectures]] (if it is not known if they are true yet).<ref>{{Cite book|title=How to Think Like a Mathematician|last=Houston|first=Kevin|publisher=Cambridge University Press|year=2009|page=99|isbn=978-0-521-71978-0}}</ref> Most mathematicians use non-logical and creative reasoning in order to find a logical proof.<ref>{{Cite journal|last=Thurston|first=William|date=April 1994|title=On proof and progress in mathematics|url=https://arxiv.org/pdf/math/9404236.pdf|journal=Bulletin of the American Mathematical Society|volume=30|pages=161-177|arxiv=math/9404236|via=arXiv}}</ref>
-Sometimes, mathematics finds and studies rules or ideas that we don't understand yet. Often in mathematics, ideas and rules are [[choice|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 [[knowledge|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 [[variable]]s. A [[calculator]] answers every math problem in the four basic [[arithmetic]] operations.
-== Areas of study in mathematics ==
-=== Number ===
-:Mathematics includes the study of numbers and quantities.It is a branch of science the deals with 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 subsection.
-:{| style="border:1px solid #999; text-align:center;" cellspacing="20"
-| <math>0, 1, 2, 3, \ldots</math> || <math> \ldots, -1, 0, 1,  \ldots</math> || <math>\frac{1}{2}, \frac{2}{3}, 0.125,\ldots</math> || <math>\pi, e, \sqrt{2},\ldots</math> || <math> 1+i, 2e^{i\pi/3},\ldots </math>
-|-
-| [[Natural number]]s || [[Integer]]s || [[Rational number]]s || [[Real number]]s || [[Complex number]]s
-|-
-|| <math>0, 1, \ldots, \omega, \omega + 1, \ldots, 2\omega, \ldots</math> || <math>\aleph_0, \aleph_1, \ldots</math> || <math>+,-,\times,\div</math> || <math> >,\ge, =, \le, < </math> || <math>f(x) = \sqrt x</math>
-|-
-|| [[Ordinal number]]s || [[Cardinal number]]s || [[Arithmetic|Arithmetic operations]] || [[Arithmetic|Arithmetic relations]] || [[Function (mathematics)|Functions]]
-|}
-=== Structure ===
-:Many areas of mathematics study the structure that an object has. Most of these areas are part of the study of [[algebra]].
-:{| style="border:1px solid #999; text-align:center;" cellspacing="30"
-| [[File:Elliptic curve simple.svg|96px]] || [[File:Free module.png|96px]] || [[File:Eigenvectoren.pdf|96px]] ||[[File:Lattice of the divisibility of 60.svg|96px]] || [[File:6n-graf.svg|128px]]
-|-
-| [[Number theory]] || [[Abstract algebra]] || [[Linear algebra]] || [[Order theory]] || [[Graph theory]]
-|}
-=== Shape ===
-:Some areas of mathematics study the shapes of things. Most of these areas are part of the study of [[geometry]].
-:{| style="border:1px solid #999; text-align:center;" cellspacing="20"
-| [[File:Torus.jpg|128px]] || [[File:Pythagorean.svg|128px]] || [[File:Sin.svg|160px]] || [[File:Osculating circle.svg|128px]] || [[File:Koch curve.svg|128px]]
-|-
-| [[Topology]] || [[Geometry]] || [[Trigonometry]] || [[Differential geometry]] || [[Fractal geometry]]
-|}
-=== Change ===
-:Some areas of mathematics study the way things change. Most of these areas are part of the study of [[Mathematical analysis|analysis]].
-:{| style="border:1px solid #999; text-align:center;" cellspacing="40"
-| [[File:Integral as region under curve.svg|128px]] || [[File:Vector field.svg|128px]] || [[File:LimSup.png|128px]]
-|-
-| [[Calculus]] || [[Vector calculus]] || [[Mathematical analysis|Analysis]]
-|-
-|| [[File:Damping 1.svg|128px]] || [[File:Limitcycle.svg|128px]] || [[File:LorenzAttractor.png|128px]]
-|-
-|| [[Differential equation]]s || [[Dynamical systems theory|Dynamical systems]] || [[Chaos theory]]
-|}
-=== Applied mathematics ===
-:[[Applied mathematics|Applied]] mathematics uses mathematics to solve problems of other areas such as [[engineering]], [[physics]], and [[computing]].
-:[[Numerical analysis]] – [[Optimization (mathematics)|Optimization]] – [[Probability theory]] – [[Statistics]] – [[Mathematical finance]] – [[Game theory]] – [[Mathematical physics]] – [[Fluid dynamics]] - [[computational algorithms]]
-== Famous theorems ==
-These theorems have interested mathematicians and people who are not mathematicians.
-:[[Pythagorean theorem]] – [[Fermat's last theorem]] – [[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 are theorems and [[conjecture]]s that have greatly changed mathematics.
-:[[Riemann hypothesis]] – [[Continuum hypothesis]] – [[P versus NP|P Versus NP]] – [[Pythagorean theorem]] – [[Central limit theorem]] – [[Fundamental theorem of calculus]] – [[Fundamental theorem of algebra]] – [[Fundamental theorem of arithmetic]] – [[Fundamental theorem of projective geometry]] – [[classification theorems of surfaces]] – [[Gauss-Bonnet theorem]] – [[Fermat's last theorem]]
-== Foundations and methods ==
-Progress in understanding the nature of mathematics also influences the way mathematicians study their subject.
-:[[Philosophy of mathematics|Philosophy of Mathematics]] – [[Mathematical intuitionism]] – [[Mathematical constructivism]] – [[Foundations of mathematics|Foundations of]] mathematics – [[Set theory]] – [[Symbolic logic]] – [[Model theory]] – [[Category theory]] – [[Logic]] – [[Reverse Mathematics]] – [[Table of mathematical symbols]]
-=== History and the world of mathematicians ===
-Mathematics in history, and the history of mathematics.
-:[[History of mathematics]] – [[Timeline of mathematics]] – [[Mathematician]]s – [[Fields Medal|Fields medal]] – [[Abel Prize]] – [[Millennium Prize Problems|Millennium Prize Problems (Clay Math Prize)]] – [[International Mathematical Union]] – [[Mathematics Competitions|Mathematics competitions]] – [[Lateral thinking]] – [[Mathematics and gender]]
-== Awards in mathematics ==
-There is no [[Nobel prize]] in mathematics. Mathematicians can receive the [[Abel Prize|Abel prize]] and the [[Fields Medal]] for important works.<ref>{{Cite news|url=https://www.telegraph.co.uk/news/science/11029161/The-Fields-Medal-is-the-greatest-prize-in-maths.html|title=The Fields Medal is the greatest prize in maths|last=Ronan|first=Mark|date=2014-08-13|access-date=2018-08-07|language=en-GB|issn=0307-1235}}</ref><ref>{{Cite web|url=https://www.theguardian.com/science/alexs-adventures-in-numberland/2018/mar/20/abel-prize-2018-robert-langlands-wins-for-unified-theory-of-maths|title=Abel Prize 2018: Robert Langlands wins for 'unified theory of maths'|last=Bellos|first=Alex|date=2018-03-20|website=the Guardian|language=en|access-date=2018-08-07}}</ref>
-The [[Clay Mathematics Institute]] has said it will give one million dollars to anyone who solves one of the [[Millennium Prize Problems]].
-== Mathematical tools ==
-There are many tools that are used to do mathematics or to find answers to mathematics problems.
-;Older tools
-* [[Abacus]]
-* [[Napier's bones]], [[slide rule]]
-* [[Ruler]] and [[Compass (drafting)|Compass]]
-* [[Mental calculation]]
-;Newer tools
-* [[Calculator]]s and [[computer]]s
-* [[Programming language]]s
-* [[Computer algebra system]]s ([[List of abstract algebra topics#computer algebra|listing]])
-* [[Internet shorthand notation]]
-* [[statistical analysis]] [[software]] (for example [[SPSS]])
-* [[SAS programming language]]
-* [[R programming language]]
 == References ==
 {{Reflist}}