A report on 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)

174 related topics with Alpha

Overall

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

Geometry

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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.

The word algebra comes from the title of a book by Muhammad ibn Musa al-Khwarizmi.

Algebra

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Quadratic formula.svg expresses the solution of the equation

Quadratic formula.svg expresses the solution of the equation

The word algebra comes from the title of a book by Muhammad ibn Musa al-Khwarizmi.
A page from Al-Khwārizmī's al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala
Italian mathematician Girolamo Cardano published the solutions to the cubic and quartic equations in his 1545 book Ars magna.
Linear algebra lecture at the Aalto University
Algebraic expression notation:
 1 – power (exponent)
 2 – coefficient
 3 – term
 4 – operator
 5 – constant term
 x y c – variables/constants
The graph of a polynomial function of degree 3

Algebra is one of the broad areas of mathematics.

A strange attractor arising from a differential equation. Differential equations are an important area of mathematical analysis with many applications in science and engineering.

Mathematical analysis

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A strange attractor arising from a differential equation. Differential equations are an important area of mathematical analysis with many applications in science and engineering.
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.

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.

Axiom

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Statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments.

Statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments.

As used in mathematics, the term axiom is used in two related but distinguishable senses: "logical axioms" and "non-logical axioms".

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

Calculus

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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.

Mathematical logic

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Mathematical logic is the study of formal logic within mathematics.

A Venn diagram illustrating the intersection of two sets

Set theory

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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.

Foundations of mathematics

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Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathematics.

This Togliatti surface is an algebraic surface of degree five. The picture represents a portion of its real locus.

Algebraic geometry

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This Togliatti surface is an algebraic surface of degree five. The picture represents a portion of its real locus.
Sphere and slanted circle

Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials.

Arithmetic tables for children, Lausanne, 1835

Arithmetic

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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.