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Mathematics is the study of numbers, shapes and patterns. The word is sometimes shortened to maths (in England, Australia, Ireland, and New Zealand) or math (in the United States and Canada). Mathematics is useful for solving problems that occur in the real world, so many people besides mathematicians study and use mathematics. Today, some mathematics is needed in many jobs. Business, science, engineering, and construction need some knowledge of mathematics.

Mathematics solves problems by using logic. Mathematicians often use 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 is just as important as the fact that it is true. Using deduction is what makes mathematics thinking different from other kinds of thinking. Mathematics includes the study of:

  • Numbers (example 3+6=9)
  • Structure: how things are organized.
  • Place: where things are and their arrangement.
  • Change: how things become different.

Mathematics uses logic, paper, and calculator. These things are used to create general rules, 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 solves many problems at the same time as these rules can be used on other problems.

A proof gives a reason why a rule in mathematics is correct. This is done by using certain other rules that everyone agrees are correct, which are called axioms. A rule that has a proof is sometimes called a theorem. Experts in mathematics perform research to create new theorems. Sometimes experts find an idea that they think is a theorem but cannot find a proof for it. That idea is called a conjecture, until a proof is found or it's proven to be incorrect.

Sometimes, mathematics finds and studies rules or ideas in the real world 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.

Number[change | change source]

Mathematics includes the study of number, or quantity.
Natural numbers Integers Rational numbers Real numbers Complex numbers
Ordinal numbers Cardinal numbers Arithmetic operations Arithmetic relations Functions

Structure[change | change source]

Some areas of mathematics study the structure that an object has.
Elliptic curve simple.svg Free module.png Eigenvectoren.pdf Lattice of the divisibility of 60.svg 6n-graf.svg
Number theory Abstract algebra Linear algebra Order theory Graph theory

Shape[change | change source]

Some areas of mathematics study the shapes of things.
Torus.jpg Pythagorean.svg Sin.svg Osculating circle.svg Koch curve.svg
Topology Geometry Trigonometry Differential geometry Fractal geometry

Change[change | change source]

Some areas of mathematics study the way things change.
Integral as region under curve.svg Vector field.svg LimSup.png
Calculus Vector calculus Analysis
Damping 1.svg Limitcycle.svg LorenzAttractor.png
Differential equations Dynamical systems Chaos theory

Applied mathematics[change | change source]

Applied mathematics uses mathematics to solve problems of other areas such as engineering, physics, and computing.
Numerical analysisOptimizationProbability theoryStatisticsMathematical financeGame theoryMathematical physicsFluid dynamics - computational algorithms

Famous theorems[change | change source]

These theorems have interested mathematicians and people who are not mathematicians.

Pythagorean theoremFermat's last theoremGoldbach's conjectureTwin Prime ConjectureGödel's incompleteness theoremsPoincaré conjectureCantor's diagonal argumentFour color theorem – Zorn's lemma – Euler's IdentityChurch-Turing thesis

These are theorems and conjectures that have greatly changed mathematics.

Riemann hypothesisContinuum hypothesisP Versus NPPythagorean theoremCentral limit theoremFundamental theorem of calculusFundamental theorem of algebraFundamental theorem of arithmeticFundamental theorem of projective geometryclassification theorems of surfacesGauss-Bonnet theoremFermat's last theorem

Foundations and methods[change | change source]

Progress in understanding the nature of mathematics also influences the way mathematicians study their subject.

Philosophy of MathematicsMathematical intuitionismMathematical constructivismFoundations of mathematics – Set theorySymbolic logicModel theoryCategory theoryLogicReverse MathematicsTable of mathematical symbols

History and the world of mathematicians[change | change source]

Mathematics in history, and the history of mathematics.

History of mathematicsTimeline of mathematicsMathematiciansFields medalAbel PrizeMillennium Prize Problems (Clay Math Prize)International Mathematical UnionMathematics competitionsLateral thinkingMaths and gender

Name[change | change source]

The word "mathematics" comes from the Greek word "μάθημα" (máthema). The Greek word "μάθημα" means "science, knowledge, or learning".

Often, the word "mathematics" is made shorter into maths (in British English) or math (in American English). The short words maths or math are often used for arithmetic, geometry or simple algebra by students and their schools.

Awards in mathematics[change | change source]

There is no Nobel prize in mathematics. Mathematicians can receive the Abel prize and the Fields Medal for important works.

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

Mathematical tools[change | change source]

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

Older tools
Newer tools

Other websites[change | change source]