3rd century BC Greek mathematician Euclid (holding calipers), as imagined by Raphael in this detail from The School of Athens (1509–1511)
A strange attractor arising from a differential equation. Differential equations are an important area of mathematical analysis with many applications in science and engineering.
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.
Archimedes used the method of exhaustion to compute the area inside a circle by finding the area of regular polygons with more and more sides. This was an early but informal example of a limit, one of the most basic concepts in mathematical analysis.
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".

Mathematics is an area of knowledge that includes such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and the spaces in which they are contained (geometry), and quantities and their changes (calculus and analysis).

- Mathematics

Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.

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

16 related topics

Alpha

Archimedes used the method of exhaustion to calculate the area under a parabola.

Calculus

Archimedes used the method of exhaustion to calculate the area under a parabola.
Alhazen, 11th-century Arab mathematician and physicist
Isaac Newton developed the use of calculus in his laws of motion and gravitation.
Gottfried Wilhelm Leibniz was the first to state clearly the rules of calculus.
Maria Gaetana Agnesi
The logarithmic spiral of the Nautilus shell is a classical image used to depict the growth and change related to calculus.

Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations.

In mathematics education, calculus denotes courses of elementary mathematical analysis, which are mainly devoted to the study of functions and limits.

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.

Typically, this occurs in mathematical analysis, where "a function from X to Y " often refers to a function that may have a proper subset of X as domain.

An illustration of Desargues' theorem, a result in Euclidean and projective geometry

Geometry

An illustration of Desargues' theorem, a result in Euclidean and projective geometry
A European and an Arab practicing geometry in the 15th century
Woman teaching geometry. Illustration at the beginning of a medieval translation of Euclid's Elements, (c. 1310).
An illustration of Euclid's parallel postulate
Acute (a), obtuse (b), and straight (c) angles. The acute and obtuse angles are also known as oblique angles.
A sphere is a surface that can be defined parametrically (by  or implicitly (by x2 + y2 + z2 − r2 = 0.)
Visual checking of the Pythagorean theorem for the (3, 4, 5) triangle as in the Zhoubi Suanjing 500–200 BC. The Pythagorean theorem is a consequence of the Euclidean metric.
The Koch snowflake, with fractal dimension=log4/log3 and topological dimension=1
A tiling of the hyperbolic plane
Differential geometry uses tools from calculus to study problems involving curvature.
A thickening of the trefoil knot
Quintic Calabi–Yau threefold
Discrete geometry includes the study of various sphere packings.
The Cayley graph of the free group on two generators a and b
Bou Inania Madrasa, Fes, Morocco, zellige mosaic tiles forming elaborate geometric tessellations
The Pythagoreans discovered that the sides of a triangle could have incommensurable lengths.

Geometry is, with arithmetic, one of the oldest branches of mathematics.

Two of the master geometers of the time were Bernhard Riemann (1826–1866), working primarily with tools from mathematical analysis, and introducing the Riemann surface, and Henri Poincaré, the founder of algebraic topology and the geometric theory of dynamical systems.

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 completeness property of the reals is the basis on which calculus, and, more generally mathematical analysis are built.

The failure of a function to be continuous at a point is quantified by its oscillation.

Continuous function

The failure of a function to be continuous at a point is quantified by its oscillation.
The graph of a cubic function has no jumps or holes. The function is continuous.
For a Lipschitz continuous function, there is a double cone (shown in white) whose vertex can be translated along the graph, so that the graph always remains entirely outside the cone.
A right-continuous function
A left-continuous function

In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function.

Continuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real and complex numbers.

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.

To solve equations from either family, one uses algorithmic or geometric techniques that originate from linear algebra or mathematical analysis.

Portrait by Jakob Emanuel Handmann (1753)

Leonhard Euler

Portrait by Jakob Emanuel Handmann (1753)
1957 Soviet Union stamp commemorating the 250th birthday of Euler. The text says: 250 years from the birth of the great mathematician, academician Leonhard Euler.
Stamp of the former German Democratic Republic honoring Euler on the 200th anniversary of his death. Across the centre it shows his polyhedral formula, in English written as "v − e + f = 2".
Euler's grave at the Alexander Nevsky Monastery
Map of Königsberg in Euler's time showing the actual layout of the seven bridges, highlighting the river Pregel and the bridges.
An Euler diagram
Euler portrait on the sixth series of the 10 Franc banknote
Euler portrait on the seventh series of the 10 Franc banknote
Illustration from Solutio problematis... a. 1743 propositi published in Acta Eruditorum, 1744
The title page of Euler's Methodus inveniendi lineas curvas.
Euler's 1760 world map.
Euler's 1753 map of Africa.

Leonhard Euler (, ; 15 April 1707 – 18 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in many other branches of mathematics such as analytic number theory, complex analysis, and infinitesimal calculus.

Euler is well known in analysis for his frequent use and development of power series, the expression of functions as sums of infinitely many terms, such as

A Venn diagram illustrating the intersection of two sets

Set theory

Branch of mathematical logic that studies sets, which can be informally described as collections of objects.

Branch of mathematical logic that studies sets, which can be informally described as collections of objects.

A Venn diagram illustrating the intersection of two sets
Georg Cantor
An initial segment of the von Neumann hierarchy

Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concerned with those that are relevant to mathematics as a whole.

Set theory as a foundation for mathematical analysis, topology, abstract algebra, and discrete mathematics is likewise uncontroversial; mathematicians accept (in principle) that theorems in these areas can be derived from the relevant definitions and the axioms of set theory.

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.

In geometric topology, most commonly studied are Morse functions, which yield handlebody decompositions, while in mathematical analysis, one often studies solution to partial differential equations, an important example of which is harmonic analysis, where one studies harmonic functions: the kernel of the Laplace operator.

Bass-guitar time signal of open-string A note (55 Hz)

Harmonic analysis

Bass-guitar time signal of open-string A note (55 Hz)
Fourier transform of bass-guitar time signal of open-string A note (55 Hz)

Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms (i.e. an extended form of Fourier analysis).

One of the most modern branches of harmonic analysis, having its roots in the mid-20th century, is analysis on topological groups.