# Portal:Algebra

## 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 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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The graph of a real-valued quadratic function of a real variable x, is a parabola. |

A **quadratic equation** is a polynomial equation of degree two. The general form is

where *a* ≠ 0 (if *a* = 0, then the equation becomes a linear equation). The letters *a*, *b*, and *c* are called coefficients: the quadratic coefficient *a* is the coefficient of *x*^{2}, the linear coefficient *b* is the coefficient of *x*, and *c* is the constant coefficient, also called the *free term*.

Quadratic equations are called *quadratic* because *quadratus* is Latin for "square"; in the leading term the variable is squared.

A quadratic equation has two (not necessarily distinct) solutions, which may be real or complex, given by the **quadratic formula**:

These solutions are *roots* of the corresponding quadratic function

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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?
- ...that the classification of finite simple groups is of more than 10000 pages.

*Topics in algebra*

General | Elementary algebra | Key concepts | Linear algebra |
---|---|---|---|

Algebraic structures | Groups | Rings and Fields | Other |

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Algebra |
Analysis |
Categorytheory |
Computerscience |
Cryptography |
Discretemathematics |
Geometry |

Logic |
Mathematics |
Numbertheory |
Physics |
Science |
Set theory |
Statistics |
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