# Mathematics

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

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 maths (in British Commonwealth countries) or math (in North America).[1]

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.[2][3]

## Problem-solving in mathematics

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 mathematics thinking different from other kinds of scientific thinking, which might rely on experiments or on interviews.[4]

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.[5] These rules can be called theorems (if they have been proved) or conjectures (if it is not known if they are true yet).[6] Most mathematicians use non-logical and creative reasoning in order to find a logical proof.[7]

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.

## Areas of study in mathematics

### 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.
 $\displaystyle{ 0, 1, 2, 3, \ldots }$ $\displaystyle{ \ldots, -1, 0, 1, \ldots }$ $\displaystyle{ \frac{1}{2}, \frac{2}{3}, 0.125,\ldots }$ $\displaystyle{ \pi, e, \sqrt{2},\ldots }$ $\displaystyle{ 1+i, 2e^{i\pi/3},\ldots }$ Natural numbers Integers Rational numbers Real numbers Complex numbers $\displaystyle{ 0, 1, \ldots, \omega, \omega + 1, \ldots, 2\omega, \ldots }$ $\displaystyle{ \aleph_0, \aleph_1, \ldots }$ $\displaystyle{ +,-,\times,\div }$ $\displaystyle{ \gt ,\ge, =, \le, \lt }$ $\displaystyle{ f(x) = \sqrt x }$ Ordinal numbers Cardinal numbers Arithmetic operations Arithmetic relations Functions, see also special functions

### Structure

Structural mathematics studies objects' and constructions' shape and integrity. These are areas of algebra and calculus.
 Number theory Abstract algebra Linear algebra Order theory Graph theory

### Shape

Some areas of mathematics study the shapes of things. Most of these areas are part of the study of geometry.
 Topology Geometry Trigonometry Differential geometry Fractal geometry

### Change

Some areas of mathematics study the way things change. Most of these areas are part of the study of analysis.
 Calculus Vector calculus Analysis Differential equations Dynamical systems Chaos theory

### Applied mathematics

Applied math uses symbolic logic to solve problems in areas like engineering and physics.
Numerical analysisOptimizationProbability theoryStatisticsMathematical financeGame theoryMathematical physicsFluid dynamics - computational algorithms

## Famous theorems

These theorems and conjectures have interested mathematicians and amateurs alike:

Pythagorean theoremFLTGoldbach's conjectureTwin Prime ConjectureGödel's incompleteness theoremsPoincaré conjectureCantor's diagonal argumentFour color theoremZorn's lemmaEuler's IdentityChurch-Turing thesis

These theorems and hypotheses have greatly changed mathematics:

Central limit theorem classification theorems of surfacesContinuum hypothesisFourier TheoremFundamental theorem of calculusFundamental theorem of algebraFundamental theorem of arithmeticFundamental theorem of projective geometryGauss-Bonnet theorem - Kantorovich theoremP Versus NPPythagorean theoremRiemann hypothesis

These are a few conjectures that have been called "revolutionary":

Beal Conjecture (a generalization of FLT) – Birch and Swinnerton-Dyer ConjectureCollatz ConjectureGoldbach's ConjectureHodge ConjecturePoincaré Conjecture

## Foundations and methods

Set theorySymbolic logicModel theoryCategory theoryLogicTable of mathematical symbols

### History and the world of mathematicians

History of mathematics – Timeline of mathematics – MathematiciansFields medalAbel PrizeMillennium Prize Problems (Clay Math Prize) – International Mathematical Union – Mathematics competitions – Lateral thinking – Mathematics and gender

## Awards in mathematics

There is no Nobel Prize in mathematics. Mathematicians can receive the Abel prize and the Fields Medal for important works.[8][9]

The Clay Mathematics Institute has said it will give one million dollars to anyone who solves one of the Millennium Prize Problems.

## Mathematical tools

There are many tools used to do math or find answers to math problems.

Older tools

## References

1. Waldman, Katy (2014-12-08). "Why Do Brits Say Maths and Americans Say Math?" (in en-US). Slate. . Retrieved 2018-07-30.
2. "Thinking of a Career in Applied Mathematics? | SIAM". www.siam.org. Retrieved 2018-07-30.
3. Wigner, Eugene (February 1960). "The Unreasonable Effectiveness of Mathematics in the Natural Sciences". Communications in Pure and Applied Mathematics. 13 (1): 1–14. Bibcode:1960CPAM...13....1W. doi:10.1002/cpa.3160130102. S2CID 6112252. Archived from the original on 2018-08-10. Retrieved 2018-08-07.
4. "The science checklist applied: Mathematics". undsci.berkeley.edu. Retrieved 2018-08-05.
5. "The Role of Generalization in the Advanced Mathematical Thinking" (in en-US). AMS Grad Blog. 2016-08-21. Retrieved 2018-08-07.
6. Houston, Kevin (2009). How to Think Like a Mathematician. Cambridge University Press. p. 99. ISBN 978-0-521-71978-0.
7. Thurston, William (April 1994). "On proof and progress in mathematics". Bulletin of the American Mathematical Society. 30 (2): 161–177. arXiv:math/9404236. Bibcode:1994math......4236T. doi:10.1090/S0273-0979-1994-00502-6.
8. Ronan, Mark (2014-08-13). "The Fields Medal is the greatest prize in maths" (in en-GB). The Daily Telegraph. . Retrieved 2018-08-07.
9. Bellos, Alex (2018-03-20). "Abel Prize 2018: Robert Langlands wins for 'unified theory of maths'". the Guardian. Retrieved 2018-08-07.