Mathematics

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 how these three things change.

Math tries to leave details out so it can make a rule about lots of things at once. Numbers are a good example. 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 idea "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 what will happen in physics. The philosophy of science explores this. However, creating mathematics is a human science. The philosophy of mathematics talks about these ideas, and other ideas about how mathematics fits in with philosophy, ethics and real life.

This Test2 page for Mathematics is somehow often vandalized; with the last revision even being blank, likely due to the school subject class of mathematics being too controversial among children and teenagers.

Important themes in mathematics

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.

$1,2,\ldots$	$0,1,-1,\ldots$	${\frac {1}{2}},{\frac {2}{3}},0.125,\ldots$	$\pi ,e,{\sqrt {2}},\ldots$
Natural numbers	Integers	Rational numbers	Real numbers

Number – Natural number – Integers – Rational numbers – Real numbers – Complex numbers – Ordinal numbers – Cardinal numbers – Integer sequences – Mathematical constants – Number names – Infinity – Base

Change

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

$36\div 9=4$			$\int 1_{S}\,d\mu =\mu (S)$
Arithmetic	Calculus	Vector calculus	Analysis
${\frac {d^{2}}{dx^{2}}}y={\frac {d}{dx}}y+c$
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.

		File:Sin 1to1.png	File:OsculatingCircle.png
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 be certain ways, called states:

File:Venn A intersect B.png	File:Fsm moore model door control.jpg	File:Caesar3.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 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

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