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# Mathematics

**Mathematics**, sometimes shortened to **maths** (in England, Australia, New Zealand and France) or **math** (in the United States, Canada and Germany) is the study of numbers, shapes and patterns. Mathematicians are people who learn about and discover such things in mathematics. Mathematics is useful for solving problems that occur in the real world, so many people besides mathematicians study and use mathematics. Today, mathematics is needed in many jobs. Business, science, engineering, and construction need some knowledge of mathematics.

Mathematicians solve 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 mathematical thinking different from other kinds of thinking.

## Contents

## About

*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 over time.

Mathematics often uses logic, paper, and a 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.

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 can not find a proof for it. That idea is called a *conjecture* until they find a proof.

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 beautiful. 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.

### Branches

Mathematics has the following main branches.

### Number

- Mathematics includes the study of number, or
*quantity*.

[math]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 numbers Integers Rational numbers Real numbers Complex numbers [math]0, 1, \ldots, \omega, \omega + 1, \ldots, 2\omega, \ldots[/math] [math]\aleph_0, \aleph_1, \ldots[/math] [math]+,-,\times,\div[/math] [math] \gt ,\ge, =, \le, \lt [/math] [math]f(x) = \sqrt x[/math] Ordinal numbers Cardinal numbers Arithmetic operations Arithmetic relations Functions

### Structure

- Some areas of mathematics study the structure that an object has.

### Shape

- Some areas of mathematics study the shapes of things.

### Change

- Some areas of mathematics study the way things change.

### Applied mathematics

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

- Numerical analysis – 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 conjectures that have greatly changed mathematics.

- Riemann hypothesis – Continuum hypothesis – 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 – 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

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 – Maths and gender

## 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.

## Awards in mathematics

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

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

Old:

- Abacus
- Order of Operations Calculator
- 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

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