Mathematics

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Mathematics is something people do to work with abstract symbols. It is the study of sets and 'connections between sets'. The 'connections between sets' are called functions or operators.

[edit] What is Mathematics?

Mathematics is the study of:

number and quantity

Number and quantity is a measure of how many things there are.

structure

Structure shows how things are organized.

place

Place shows the position of things. Place tells where things are.

change

In mathematics, people also study how quantity, structure and place change or evolve.

[edit] Abstraction

General rules are part of math. The most basic rules in mathematics are called axioms. General rules are useful for many things at once, not just one. Math leaves out information so it can make a rule about lots of things at once. This is called abstraction.

[edit] Number Example

Numbers are a good example of an abstraction. In the real world, two apples plus two apples make four apples. Two bricks plus two bricks make four bricks. A general rule for both the apples and bricks is "two plus two equals four". Ideas that everyone cannot see, such as four, is called abstraction.

This type of rule is a part of arithmetic.^[1]

[edit] Logic Example

Another example of abstraction comes from logic.

If all fish have gills, and an animal does not have gills, it is not a fish. If food is always edible, and another thing is not edible, then it is not food. Math can make a general rule for both fish and food.

The mathematical way of saying this for the fish and food observations, is:

if A is a subset of B then "not B" is a subset of "not A".

[edit] General Rules in Mathematics

By finding general rules, mathematics solves many problems at the same time. The examples of fish and food are easy to understand without math. Math helps people understand and answer harder problems.

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 simple or beautiful. After, these ideas and rules might be found in the real world. This has happened many times in the past. Therefore, studying the rules and ideas of mathematics can help us know the world better.

[edit] 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 shortened to maths (math in American English). The short words math or maths are often used for arithmetic, geometry or basic algebra by young students and their schools.

[edit] Mathematics and Science

Mathematics is used in science to predict what will happen.

[edit] Example of Mathematics in Science

For example, Tom drops a brick. The brick falls to the ground. Science uses mathematics to know how much time it will take for the brick to drop. Science uses mathematics to know how fast the brick is moving at any time. Science uses mathematics to know where the brick is at any time.

The type of science used to know the position of the brick is geometry. Physics is used to know what the brick will do when it is dropped (gravity). This is called prediction.

[edit] Parts of Mathematics

[edit] Quantity

Quantity is about counting and measurements.

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

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

[edit] 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

[edit] Spatial relations

A more visual variant of mathematics.


Topology	Geometry	Trigonometry	Differential geometry	Fractal geometry

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

[edit] 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

[edit] Discrete mathematics

Discrete mathematics is about objects that can only be certain ways, called states, but not in between states:


Naïve set theory	Theory of computation	Cryptography	Graph theory

Combinatorics – Naive set theory – Theory of computation– Cryptography – Graph theory

[edit] 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

[edit] 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

[edit] 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 – Fermat's last theorem

[edit] 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

[edit] 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

[edit] Mathematics and other fields

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

[edit] 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

[edit] Notes

↑ Arithmetic is a set of rules to help with counting.

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