Mathematics

Mathematics is the study of numbers, shapes, and patterns. The word comes from the Greek μάθημα (máthema), meaning "science, knowledge, or learning", and is sometimes shortened to math or maths.

Ancient Greek mathematician Euclid (holding calipers), 3rd century BC, as imagined by Raphael in this detail from the School of Athens (1509–1511)

It is the study of:

Numbers: including how things can be counted.
Structure: including how things are organized, but also how they can be or could have been. This subfield is usually called algebra.
Place: where things are, and spatial arrangement, including arrangements of spaces themselves. This subfield is usually called geometry.
Change: how things become different. This subfield is usually called analysis.

Applied math is useful for solving problems in the real world. People working in business, science, engineering, and construction use mathematics.^[1]^[2]

Mathematics solves problems by using logic. One of the main tools of logic used by mathematicians is 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 (called a proof) is just as important as the fact that it is true, and this reason is often found using deduction. Using deduction is what makes mathematical thinking different from other kinds of scientific thinking, which might rely on experiments or on interviews.^[3]

Logic and reasoning are used by mathematicians 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 as these rules can be used on other problems.^[4] These rules can be called theorems (if they have been proven) or conjectures (if it is not known if they are true yet).^[5] Most mathematicians use non-logical and creative reasoning in order to find a logical proof.^[6]

Sometimes, mathematics finds and studies rules or ideas that we don't understand yet. Often in mathematics, ideas and rules are chosen because they are considered simple or neat. 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. Some examples of math problems are addition, subtraction, multiplication, division, calculus, fractions and decimals. Algebra problems are solved by evaluating certain variables. A calculator answers every math problem in the four basic arithmetic operations.

Number

Mathematics includes the study of numbers and quantities. It is a branch of science that deals with the logic of shape, quantity, and arrangement. Most of the areas listed below are studied in many different fields of mathematics, including set theory and mathematical logic. The study of number theory usually focuses more on the structure and behavior of the integers rather than on the actual foundations of numbers themselves, and so is not listed in this given subsection.

$1,2,3,\ldots$	$\ldots ,-1,0,1,\ldots$	${\frac {1}{2}},{\frac {2}{3}},0.125,\ldots$	$\tau ,\pi ,e,{\sqrt {2}},\ldots$	$1+i,2e^{i\pi /3},\ldots$
Natural numbers	Integers	Rational numbers	Real numbers	Complex numbers
$0,1,\ldots ,\omega ,\omega +1,\ldots ,2\omega ,\ldots$	$\aleph _{0},\aleph _{1},\ldots$	$+,-,\times ,/$	$<,\leq ,=,\geq ,>$	$f(x)={\sqrt {x}}$
Ordinal numbers	Cardinal numbers	Arithmetic operations	Arithmetic relations	Functions, see also special functions