Mathematics (math or maths for short) is the study of quantity (how many things there are), structure (how things are organized), space (where things are) and change.

Because mathematics looks at such general topics, it can use fewer details. The simplest example of this is numbers. In the real world, two apples plus two apples make four apples, two bricks plus two bricks make four bricks; so in mathematics, this becomes the general statement "two plus two equals four". This is called arithmetic.

Another very simple example comes from logic. If all blackbirds are black, and one bird is not black, it is not a blackbird. If all snow is white, and another thing is not white, it is not snow. In math, we make this idea abstract by saying: if A is a subset of B then "not B" is a subset of "not A". For example, if you have something that is not black, it cannot be a blackbird.

By finding a general way to say something, mathematics solves many problems at the same time. The examples of snow and blackbirds are easy to understand without math, but harder situations can be much easier to understand with math. Sometimes, mathematics studies rules or ideas that have not yet been found in the real world. If the rules are chosen because they are simple, later on they may be found in the real world, so studying the rules might help us understand the world better.

The word "mathematics" comes from the Greek μάθημα (máthema), and means "science, knowledge, or learning". Often, it is abbreviated to maths (math in American English), especially when describing arithmetic, geometry or basic algebra, as taught to teenagers (as students).

Mathematical prediction is the basis of hard science, which almost always uses equations to predict events in physics. The philosophy of science explores this. However, creating mathematics is clearly a human science. The philosophy of mathematics explores these issues, and others regarding how mathematics fits in with philosophy, ethics and real life.

Important themes in mathematics

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See also: List of mathematics topics

Here is a possible grouping of mathematical areas and topics.

Quantity

This is about counting and measuring, and the ways to find such measurements.

{| style="border:1px solid #999; text-align:center;" cellspacing="20" $1,\; 2,\; \backslash ldots$ $0,\; 1,\; -1,\; \backslash ldots$ $\backslash frac\{1\}\{2\},\; \backslash frac\{2\}\{3\},\; 0.125,\backslash ldots$ $\backslash pi,\; e,\; \backslash sqrt\{2\},\backslash ldots$ $i,\; 3i+2,\; e^\{i\backslash pi/3\},\backslash ldots$ Natural numbers Integers Rational numbers Real numbers Complex numbers

Number – Natural number – Integers – Rational numbers – Real numbers – Complex numbers – Hypercomplex numbers – Quaternions – Octonions – Sedenions – Hyperreal numbers – Surreal numbers – Ordinal numbers – Cardinal numbers – p-adic numbers – Integer sequences – Mathematical constants – Number names – Infinity – Base

Change

Ways to express and handle change in mathematical functions, and changes between numbers.

{| style="border:1px solid #999; text-align:center;" cellspacing="20" || Vectorfield_jaredwf.png || $\backslash int\; 1\_S\backslash ,d\backslash mu=\backslash mu(S)$ Arithmetic Calculus Vector calculus Analysis || LorenzAttractor.png Differential equations Dynamical systems Chaos theory

Arithmetic – Calculus – Vector calculus – Analysis – Differential equations – Dynamical systems – Chaos theory – List of functions

Structure

Express ideas of size, symmetry, and mathematical structure.

Abstract algebra – Number theory – Algebraic geometry – Group theory – Monoids – Analysis – Topology – Linear algebra – Graph theory – Universal algebra – Category theory – Order theory – Measure theory

Spatial relations

A more visual variant of mathematics.

{| style="border:1px solid #999; text-align:center;" cellspacing="15" || Pythagorean.svg || Sin 1to1.png || OsculatingCircle.png || Koch_curve.gif Topology Geometry Trigonometry Differential geometry Fractal geometry

Topology – Geometry – Trigonometry – Algebraic geometry – Differential geometry – Differential topology – Algebraic topology – Linear algebra – Fractal geometry

Discrete mathematics

Discrete mathematics is about objects, that can only have specific states.

{| style="border:1px solid #999; text-align:center;" cellspacing="15" || Fsm_moore_model_door_control.jpg || Caesar3.png || 6n-graf.png Naive set theory Theory of computation Cryptography Graph theory

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.

Mechanics – Numerical analysis – Optimization – Probability – Statistics – Financial mathematics – Game theory – Mathematical biology – Cryptography – Information theory – Fluid dynamics

Famous theorems and conjectures

These theorems have interested mathematicians and non-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

Important theorems and conjectures

See list of theorems, list of conjectures for more

These are theorems and conjectures that have changed the face of mathematics throughout history.

Riemann hypothesis – Continuum hypothesis – P=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

Foundations and methods

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

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

History and the world of mathematicians

See also list of mathematics history topics

Mathematics in history, and the history of mathematics.

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

Mathematics and other fields

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

Mathematical tools

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Tools that are used to do mathematics or to calculate.

Old:

• Abacus
• Napier's bones, Slide Rule
• Ruler and Compass
• Mental calculation

New:

• Calculators and computers
• Programming languages
• Computer algebra systems (listing)
• Internet shorthand notation
• statistical analysis software
  • SPSS
  • SAS programming language
  • R programming language

Mathematics

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