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Algebra

Algebra is a branch of mathematics concerning the study of structure, relation and quantity. The name is derived from the treatise written by the muslim Persian mathematician, astronomer, astrologer and geographer, Muhammad bin Mūsā al-Khwārizmī titled Kitab al-Jabr wa-l-Muqabala (meaning "The Compendious Book on Calculation by Completion and Balancing"), which provided symbolic operations for the systematic solution of linear and quadratic equations.

Together with geometry, analysis, combinatorics, and number theory, algebra is one of the main branches of mathematics. Elementary algebra is often part of the curriculum in secondary education and provides an introduction to the basic ideas of algebra, including effects of adding and multiplying numbers, the concept of variables, definition of polynomials, along with factorization and determining their roots.

Algebra is much broader than elementary algebra and can be generalized. In addition to working directly with numbers, algebra covers working with symbols, variables, and set elements. Addition and multiplication are viewed as general operations, and their precise definitions lead to structures such as groups, rings and fields.

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A loxodromic representation of the Riemann sphere.

The Riemann sphere is a way of extending the plane of complex numbers with one additional point at infinity, in a way that makes expressions such as

$\frac{1}{0} = \infty$

well-behaved and useful, at least in certain contexts. It is named after 19th century mathematician Bernhard Riemann. It is also called the complex projective line, denoted CP¹.

On a purely algebraic level, the complex numbers with an extra infinity element constitute a number system known as the extended complex numbers. Arithmetic with infinity does not obey all of the usual rules of algebra, and so the extended complex numbers do not form a field. However, the Riemann sphere is geometrically and analytically well-behaved, even near infinity; it is a one-dimensional complex manifold, also called a Riemann surface.

In complex analysis, the Riemann sphere facilitates an elegant theory of meromorphic functions. The Riemann sphere is ubiquitous in projective geometry and algebraic geometry as a fundamental example of a complex manifold, projective space, and algebraic variety. It also finds utility in other disciplines that depend on analysis and geometry, such as quantum mechanics and other branches of physics.

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Image credit: Karthik Narayanaswami

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A tetrahedron can be placed in 12 distinct positions by rotation alone. These are illustrated above in the cycle graph format, along with the 180° edge (blue arrows) and 120° vertex (reddish arrows) rotations that permute the tetrahedron through those positions.

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Did you know?

...that it is impossible to devise a single formula involving only polynomials and radicals for solving an arbitrary quintic equation?
...that it is possible for a three dimensional figure to have a finite volume but infinite surface area? An example of this is Gabriel's Horn.
...that the Gudermannian function relates the regular trigonometric functions and the hyperbolic trigonometric functions without the use of complex numbers?
...that the classification of finite simple groups was not completed until the mid 1980s?
...that a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers?

Topics in algebra

General	Elementary algebra	Key concepts	Linear algebra
Abstract algebra Algebraic number theory Algebraic geometry Topological algebra Algebraic combinatorics Commutative algebra Homological algebra Abstract algebra topics	Elementary algebra Function (mathematics) Equation Square root, cube root, Nth root Polynomial Linear equation Quadratic equation Cubic equation Simultaneous equations Partial fraction Associativity, Commutativity	Identity element Inverse element Commutativity Associativity Distributivity Isomorphism The Isomorphism theorems Homomorphism	Linear algebra Matrix, Matrix theory Determinant Vector, Vector space Inner product, Inner product space Hilbert space Linear algebra glossary Linear algebra topics
Algebraic structures	Groups	Rings and Fields	Other
Algebraic structure (list) Group, Group theory Magma, quasigroup, loop, semigroup, monoid Lattice Rng, Semiring Ring, Ring theory Field, Field theory Module, Comodule Algebra over a ring, Algebra over a field Lie algebra Jordan algebra Universal algebra	Group, Group theory Order Subgroup, Normal subgroup Abelian group Simple group Finite simple groups Classification of finite simple groups Lie group Symmetry group Group representation Computational group theory Topological group Glossary of group theory List of group theory topics	Ring, Ring theory Field, Field theory Factor ring Ideal Ring homomorphism Banach algebra Clifford algebra Glossary of ring theory Glossary of field theory	Order theory Well-ordered Domain theory Glossary of order theory List of order topics Category theory (Portal) Tensor algebra K-theory Module theory Representation theory Computer algebra system Normed algebra

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