Mathematics is the science of structure, order, and relation that evolved through counting, measuring, and describing shapes. It utilizes logical reasoning and quantitative calculation, with increasing idealization and abstraction of its subject matter. Since the 17th century, mathematics has been essential to the physical sciences and technology, and it has become increasingly important in the quantitative aspects of life sciences. All mathematical systems use combinations of axioms and theorems that can be logically deduced from them. The most basic notion in mathematics is the equation, which states that two mathematical expressions are equal. Branches of mathematics include algebra , arithmetic , calculus , geometry , and trigonometry .

What are numbers and how are they used in math?

Numbers are mathematical objects used to count, measure, and label. They are the foundation of mathematics, with various types including natural numbers , integers , rational numbers , real numbers , and complex numbers . In math, numbers are used in: * Algebra Numbers and symbols represent quantities and mathematical operations. * Calculus Numbers calculate rates of change and accumulate small parts to find totals. * Analysis Numbers form number systems where arithmetic operations are applied. * Geometry Numbers are paired with space to relate algebraic expressions to geometric objects.

What are addition, subtraction, multiplication, and division?

Arithmetic involves fundamental operations on numbers.

What is geometry in mathematics?

Geometry is a branch of mathematics that deals with the shape of objects, their spatial relationships, and the properties of the space around them. It is one of the oldest branches of mathematics, with its origins in practical problems such as surveying and construction. The word "geometry" comes from Greek words that mean "Earth measurement". Geometry is not limited to the study of flat surfaces (plane geometry) and rigid three-dimensional objects (solid geometry). Even abstract thoughts and images can be represented and developed in geometric terms. From its beginnings in ancient Egypt and Mesopotamia around 3100 BCE, geometry has evolved into a broad and abstract field that explores the nature of space and shape.

What is algebra and how are variables used in math?

Algebra is a branch of mathematics that uses symbols to represent numerical values or quantities in equations. It generalizes arithmetic operations by applying them to these abstract symbols rather than specific numbers. These operations include addition, subtraction, multiplication, and division. A key characteristic of algebra is the use of variables, which are symbols, usually letters, that stand in for unknown numerical values. Commonly used variables are ( x ) and ( y ), and they can represent real-number unknowns, time (( t )), radius (( r )), or arc length (( s )). In the equation ( ax^2 + bx + c = 0 ), ( x ) is the variable, while ( a ), ( b ), and ( c ) are coefficients.

How is mathematics used to solve real-life problems?

Mathematics offers tools for describing reality and analyzing problems in many fields. Algebra uses symbols to represent quantities and relationships, providing flexibility in problem-solving. Calculus helps to calculate rates of change and totals, with applications in physics, economics, and more. Statistics provides tools for understanding uncertainty and is applied across sciences, business, and policy. Geometry, which originally meant "Earth measurement," now has applications in astronomy, cartography, and other fields. Math is a tool in understanding phenomena. Mathematical models can analyze anything that can be expressed mathematically.

mathematics

Introduction
Ancient mathematical sources
Mathematics in ancient Mesopotamia
- The numeral system and arithmetic operations
- Geometric and algebraic problems
- Mathematical astronomy
Mathematics in ancient Egypt
- The numeral system and arithmetic operations
- Geometry
- Assessment of Egyptian mathematics
Greek mathematics
- The development of pure mathematics
  - The pre-Euclidean period
  - The Elements
  - The three classical problems
- Geometry in the 3rd century bce
  - Archimedes
  - Apollonius
  - Applied geometry
- Later trends in geometry and arithmetic
  - Greek trigonometry and mensuration
  - Number theory
  - Survival and influence of Greek mathematics
Mathematics in the Islamic world (8th–15th century)
- Origins
- Mathematics in the 9th century
- Mathematics in the 10th century
- Omar Khayyam
- Islamic mathematics to the 15th century
European mathematics during the Middle Ages and Renaissance
- The transmission of Greek and Arabic learning
- The universities
- The Renaissance
Mathematics in the 17th and 18th centuries
- The 17th century
  - Institutional background
  - Numerical calculation
  - Analytic geometry
  - The calculus
    - The precalculus period
    - Newton and Leibniz
- The 18th century
  - Institutional background
  - Analysis and mechanics
  - History of analysis
- Other developments
  - Theory of equations
  - Foundations of geometry
Mathematics in the 19th century
- Projective geometry
- Making the calculus rigorous
- Fourier series
- Elliptic functions
- The theory of numbers
- The theory of equations
- Gauss
- Non-Euclidean geometry
- Riemann
- Riemann’s influence
- Differential equations
- Linear algebra
- The foundations of geometry
- The foundations of mathematics
Mathematics in the 20th and 21st centuries
- Cantor
- Mathematical physics
- Algebraic topology
- Developments in pure mathematics
- Mathematical physics and the theory of groups
- Probabilistic mathematics

References & Edit History Quick Facts & Related Topics

Images

A chart showing ancient Egyptian hieroglyphic numerals, with symbols representing 1, 10, 100, 1,000, 10,000, 100,000, and 1,000,000, plus an example using these symbols to form the number 258,458.

A chart displays Egyptian hieratic numerals for units, tens, hundreds, thousands, tens of thousands, and hundreds of thousands, with each value represented by a unique red symbol in handwritten script.

Diagram explaining the Egyptian seked, showing a pyramid with labeled dimensions and a triangle illustrating the run and rise. A formula box defines seked as the ratio of run to rise, with an example calculation.

Map of the Greco-Roman world highlighting the birthplaces of notable ancient mathematicians, marked with the pi symbol and labeled with their names. Cities such as Athens, Alexandria, Syracuse, and Miletus are shown around the Mediterranean, with modern country boundaries outlined. An inset provides detail for the region of Greece and western Anatolia.

A mathematical diagram showing the intersection of a parabola and a hyperbola, with labeled equations and axes, illustrating Menaechmus's method for doubling the volume of a cube using conic sections.

A red sphere fits perfectly inside a transparent blue cylinder, illustrating Archimedes' relationship between the surface area and volume of a sphere and its circumscribing cylinder, with labeled formulas and dimensions.

Diagram showing a double cone intersected by planes at different angles, producing the conic sections: a horizontal plane creates a circle, a tilted plane creates an ellipse, a plane parallel to the cone's side forms a parabola, and a steep plane produces a hyperbola. Each section is labeled, with the axis and apex of the cone indicated.

A geometric diagram showing a conchoid curve: several lines radiate from point P, each intersecting a horizontal line LN. Equal distances marked from LN on each line form a smooth, shell-like red curve above LN.

A geometric diagram showing angle trisection using a conchoid curve, with labeled lines, points, and a shaded triangle illustrating how the angle is divided into three equal parts by intersecting the conchoid.

At a Glance

mathematics summary

mathematics

Also known as: math

Written by

Craig G. FraserAll

Fact-checked by

Britannica Editors

Last updated

Nov. 11, 2026 •History

News •

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AI challenges math by solving the unsolvable • Oct. 20, 2026, 3:10 PM ET (Phys.org)

mathematics, the science of structure, order, and relation that has evolved from elemental practices of counting, measuring, and describing the shapes of objects. It deals with logical reasoning and quantitative calculation, and its development has involved an increasing degree of idealization and abstraction of its subject matter. Since the 17th century, mathematics has been an indispensable adjunct to the physical sciences and technology, and in more recent times it has assumed a similar role in the quantitative aspects of the life sciences.

In many cultures—under the stimulus of the needs of practical pursuits, such as commerce and agriculture—mathematics has developed far beyond basic counting. This growth has been greatest in societies complex enough to sustain these activities and to provide leisure for contemplation and the opportunity to build on the achievements of earlier mathematicians.

All mathematical systems (for example, Euclidean geometry) are combinations of sets of axioms and of theorems that can be logically deduced from the axioms. Inquiries into the logical and philosophical basis of mathematics reduce to questions of whether the axioms of a given system ensure its completeness and its consistency. For full treatment of this aspect, see mathematics, foundations of.

This article offers a history of mathematics from ancient times to the present. As a consequence of the exponential growth of science, most mathematics has developed since the 15th century ce, and it is a historical fact that, from the 15th century to the late 20th century, new developments in mathematics were largely concentrated in Europe and North America. For these reasons, the bulk of this article is devoted to European developments since 1500.

This does not mean, however, that developments elsewhere have been unimportant. Indeed, to understand the history of mathematics in Europe, it is necessary to know its history at least in ancient Mesopotamia and Egypt, in ancient Greece, and in Islamic civilization from the 9th to the 15th century. The way in which these civilizations influenced one another and the important direct contributions Greece and Islam made to later developments are discussed in the first parts of this article.

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Numbers and Mathematics

India’s contributions to the development of contemporary mathematics were made through the considerable influence of Indian achievements on Islamic mathematics during its formative years. A separate article, South Asian mathematics, focuses on the early history of mathematics in the Indian subcontinent and the development there of the modern decimal place-value numeral system. The article East Asian mathematics covers the mostly independent development of mathematics in China, Japan, Korea, and Vietnam.

The substantive branches of mathematics are treated in several articles. See algebra; analysis; arithmetic; combinatorics; game theory; geometry; number theory; numerical analysis; optimization; probability theory; set theory; statistics; trigonometry.