Mathematics

3rd century BC Greek mathematician Euclid (holding calipers), as imagined by Raphael in this detail from The School of Athens (1509–1511)
The distribution of prime numbers is a central point of study in number theory. This Ulam spiral serves to illustrate it, hinting, in particular, at the conditional independence between being prime and being a value of certain quadratic polynomials.
The quadratic formula expresses concisely the solutions of all quadratic equations
Rubik's cube: the study of its possible moves is a concrete application of group theory
The Babylonian mathematical tablet Plimpton 322, dated to 1800 BC.
Archimedes used the method of exhaustion, depicted here, to approximate the value of pi.
The numerals used in the Bakhshali manuscript, dated between the 2nd century BC and the 2nd century AD.
A page from al-Khwārizmī's Algebra
Leonardo Fibonacci, the Italian mathematician who introduced the Hindu–Arabic numeral system invented between the 1st and 4th centuries by Indian mathematicians, to the Western World.
Leonhard Euler created and popularized much of the mathematical notation used today.
Carl Friedrich Gauss, known as the prince of mathematicians
The front side of the Fields Medal
Euler's identity, which American physicist Richard Feynman once called "the most remarkable formula in mathematics".

Area of knowledge that includes such topics as numbers , formulas and related structures (algebra), shapes and the spaces in which they are contained (geometry), and quantities and their changes (calculus and analysis).

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

171 related topics

Alpha

The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface.

Manifold

The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface.
The surface of the Earth requires (at least) two charts to include every point. Here the globe is decomposed into charts around the North and South Poles.
Figure 1: The four charts each map part of the circle to an open interval, and together cover the whole circle.
Four manifolds from algebraic curves: circles, parabola,  hyperbola,  cubic.
The chart maps the part of the sphere with positive z coordinate to a disc.
A finite cylinder is a manifold with boundary.
Möbius strip
The Klein bottle immersed in three-dimensional space
A Morin surface, an immersion used in sphere eversion

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point.

Argument terminology used in logic

Logic

Study of correct reasoning or good arguments.

Study of correct reasoning or good arguments.

Argument terminology used in logic
Aristotle, 384–322 BCE.
A depiction from the 15th century of the square of opposition, which expresses the fundamental dualities of syllogistic.

Logic is studied in and applied to various fields, such as philosophy, mathematics, computer science, and linguistics.

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

Topological space

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

In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance.

Addition of functions: The sum of the sine and the exponential function is

Vector space

[[File: Vector add scale.svg|200px|thumb|right|Vector addition and scalar multiplication: a vector

[[File: Vector add scale.svg|200px|thumb|right|Vector addition and scalar multiplication: a vector

Addition of functions: The sum of the sine and the exponential function is
A typical matrix
Commutative diagram depicting the universal property of the tensor product.
The succeeding snapshots show summation of 1 to 5 terms in approximating a periodic function (blue) by finite sum of sine functions (red).
An affine plane (light blue) in R3. It is a two-dimensional subspace shifted by a vector x (red).

In mathematics, physics, and engineering, a vector space (also called a linear space) is a set whose elements, often called vectors, may be added together and multiplied ("scaled") by numbers called scalars.

Schematic depiction of a function described metaphorically as a "machine" or "black box" that for each input yields a corresponding output

Function (mathematics)

Schematic depiction of a function described metaphorically as a "machine" or "black box" that for each input yields a corresponding output
The red curve is the graph of a function, because any vertical line has exactly one crossing point with the curve.
A function that associates any of the four colored shapes to its color.
The function mapping each year to its US motor vehicle death count, shown as a line chart
The same function, shown as a bar chart
Graph of a linear function
Graph of a polynomial function, here a quadratic function.
Graph of two trigonometric functions: sine and cosine.
Together, the two square roots of all nonnegative real numbers form a single smooth curve.
A composite function g(f(x)) can be visualized as the combination of two "machines".
A simple example of a function composition
Another composition. In this example, {{math|1=(g ∘ f )(c) = #}}.

In mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the set Y is called the codomain of the function.

The permutations of the Rubik's Cube form a group, a fundamental concept within abstract algebra.

Abstract algebra

The permutations of the Rubik's Cube form a group, a fundamental concept within abstract algebra.
Algebraic structures between magmas and groups. For example, monoids are semigroups with identity.

In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures.

A symbol for the set of real numbers

Real number

A symbol for the set of real numbers
Real numbers

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line (or alternatively, a quantity that can be represented as an infinite decimal expansion).

The first use of an equals sign, equivalent to 14x + 15 = 71 in modern notation. From The Whetstone of Witte by Robert Recorde of Wales (1557).

Equation

The first use of an equals sign, equivalent to 14x + 15 = 71 in modern notation. From The Whetstone of Witte by Robert Recorde of Wales (1557).
Illustration of a simple equation; x, y, z are real numbers, analogous to weights.
The solutions –1 and 2 of the polynomial equation x2 – x + 2 = 0 are the points where the graph of the quadratic function y = x2 – x + 2 cuts the x-axis.
The Nine Chapters on the Mathematical Art is an anonymous 2nd-century Chinese book proposing a method of resolution for linear equations.
The blue and red line is the set of all points (x,y) such that x+y=5 and -x+2y=4, respectively. Their intersection point, (2,3), satisfies both equations.
A conic section is the intersection of a plane and a cone of revolution.
Cartesian coordinate system with a circle of radius 2 centered at the origin marked in red. The equation of a circle is (x − a)2 + (y − b)2 = r2 where a and b are the coordinates of the center (a, b) and r is the radius.
A strange attractor, which arises when solving a certain differential equation

In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign.

Möbius strips, which have only one surface and one edge, are a kind of object studied in topology.

Topology

Möbius strips, which have only one surface and one edge, are a kind of object studied in topology.
A three-dimensional depiction of a thickened trefoil knot, the simplest non-trivial knot
The Seven Bridges of Königsberg was a problem solved by Euler.
A continuous transformation can turn a coffee mug into a donut. Ceramic model by Keenan Crane and Henry Segerman

In mathematics, topology (from the Greek words, and λόγος) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself.

Arithmetic tables for children, Lausanne, 1835

Arithmetic

Arithmetic tables for children, Lausanne, 1835
Leibniz's Stepped Reckoner was the first calculator that could perform all four arithmetic operations.
A scale calibrated in imperial units with an associated cost display.

Arithmetic is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers—addition, subtraction, multiplication, division, exponentiation, and extraction of roots.