Mathematics

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Mathematics is the study of numbers, patterns, and shapes. People who are experts at mathematics are called mathematicians.

Mathematics is used to find practical solutions to real world problems. A common saying is "Do the math," meaning check the answer and see that it is right. For example, if we want to know whether or not a particular medicine will help to cure a disease, mathematics is used to answer that question.

The method mathematicians use to solve problems is called deduction. Deduction is the use of reason to discover new ideas from old. For example, from the simple idea of a triangle mathematicians discovered the Pythagorean theorem, that in a right triangle the sum of the squares of the two shorter sides is equal to the square of the longest side. Thus, if we know that the shorter sides of a right triangle are 3 inches and 4 inches, we can deduce that the longest side is 5 inches, because the square of 3 is 9, the square of 4 is 16, and the square of 5 is 25; and 9 + 16 = 25.

Mathematics is necessary in business, science, engineering, construction work, and in many other jobs.

1 About
2 Famous theorems
3 Foundations and methods
- 3.1 History and the world of mathematicians
4 Name
5 Awards in mathematics
6 Mathematical tools
7 Notes

[change] About

Mathematics includes the study of:

Number: how many things there are.
Structure: how things are organized.
Place: where things are and their arrangement.
Change: how things become different over time.

Mathematics uses logic to study these things and to create general rules, which are an important part of mathematics. These rules leave out information that isn't important so that a single rule can cover many situations. By finding general rules, mathematics solves many problems at the same time.

A proof gives a reason why a rule in mathematics is correct. The reason can include 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, mathematics finds and studies rules or ideas that have not yet been found in the real world. Often in mathematics, ideas and rules are chosen because they are considered simple or beautiful. 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. This means that studying the rules and ideas of mathematics can help us know the world better.

[change] Number

Mathematics includes the study of number, or quantity.

$0,1, 2, \ldots$	$0, 1, -1, \ldots$	$\frac{1}{2}, \frac{2}{3}, 0.125,\ldots$	$\pi, e, \sqrt{2},\ldots$	$i, 1+i, 2e^{i\pi/3},\ldots$
Natural numbers	Integers	Rational numbers	Real numbers	complex numbers
$+,-,\times,\div$	$π$	$\omega, \omega + 1, \ldots, 2\omega, \ldots$	$\aleph_0$
Arithmetic	Mathematical constant	Ordinal number	Cardinal number

[change] Structure

Some areas of mathematics study the structure that an object possesses.

					$(()(()()))$
Number theory	Abstract algebra	Linear algebra	Order theory	Graph theory	Operand theory

[change] Place

Some areas of mathematics study the arrangement of things.


Topology	Geometry	Trigonometry	Differential geometry	Fractal geometry

[change] Change

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

		$\int 1_S\,d\mu=\mu(S)$
Calculus	Vector calculus	Analysis
$\frac{d^2}{dx^2} y = \frac{d}{dx} y + c$
Differential equations	Dynamical systems	Chaos theory

[change] Discrete mathematics

Discrete mathematics is about objects that can only be certain arrangements, called states, but cannot have any other arrangements:

$\begin{matrix} (1,2,3) & (1,3,2) \\ (2,1,3) & (2,3,1) \\ (3,1,2) & (3,2,1) \end{matrix}$
Combinatorics	Naïve set theory	Theory of computation	Cryptography	Graph theory

[change] Applied mathematics

Applied mathematics uses mathematics to solve problems of other areas such as engineering and physics.

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

[change] Famous theorems

See list of theorems, list of conjectures for more

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 conjectures that have greatly changed mathematics.

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 – Fermat's last theorem

[change] 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

[change] 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

[change] Name

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 math or maths are often used for arithmetic, geometry or simple algebra by young students and their schools.

[change] Awards in mathematics

There is no Nobel prize in mathematics. The most important awards a mathematician can receive are the Abel prize and the Fields medal.

The Clay mathematics institute has said they will give one million dollars to any mathematician who solves one of the Millennium Prize Problems

[change] Mathematical tools

Tools that are used to do mathematics or find answers to mathematics problems.

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

[change] Notes

Category: Mathematics

Hidden category: Complex pages