Portal:Algebra

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Algebra

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Algebra is a branch of mathematics concerning the study of structure, relation and quantity. The name is derived from the treatise written by the Persian mathematician, astronomer, astrologer and geographer, Muḥammad ibn Mūsā al-Khwārizmī titled Kitab al-Jabr al-Muqabala (meaning "The Compendious Book on Calculation by Completion and Balancing"), which provided 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.

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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Stereographic projection in 3D.png
3D illustration of a stereographic projection from the north pole onto a plane below the sphere.

The stereographic projection is a particular mapping (function) that projects a sphere onto a plane. The projection is defined on the entire sphere, except at one point — the projection point. Where it is defined, the mapping is smooth and bijective. It is also conformal, meaning that it preserves angles. On the other hand, it does not preserve area, especially near the projection point.

Intuitively, then, the stereographic projection is a way of picturing the sphere as the plane, with some inevitable compromises. Because the sphere and the plane appear in many areas of mathematics and its applications, so does the stereographic projection; it finds use in diverse fields including differential geometry, complex analysis, cartography, geology, and crystallography.

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Tetrahedral group ca

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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Topics in algebra

General Elementary algebra Key concepts Linear algebra
Algebraic structures Groups Rings and Fields Other

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