## ::Algebra

### ::concepts

Algebra::''a''    Theory::algebra    Numbers::which    Group::''b''    Equation::title    First::element

{{#invoke:redirect hatnote|redirect}} {{#invoke:Hatnote|hatnote}} {{#invoke:Pp-move-indef|main}} {{#invoke:Protection banner|main}} The quadratic formula expresses the solution of the degree two equation [itex]ax^2 + bx +c=0[/itex] in terms of its coefficients [itex]a, b, c[/itex], where [itex]a[/itex] is not zero.

Algebra (from Arabic and Farsi "al-jabr" meaning "reunion of broken parts"<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>) is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols;<ref>I. N. Herstein, Topics in Algebra, "An algebraic system can be described as a set of objects together with some operations for combining them." p. 1, Ginn and Company, 1964</ref> it is a unifying thread of almost all of mathematics.<ref>I. N. Herstein, Topics in Algebra, "...it also serves as the unifying thread which interlaces almost all of mathematics." p. 1, Ginn and Company, 1964</ref> As such, it includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra, the more abstract parts are called abstract algebra or modern algebra. Elementary algebra is generally considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine and economics. Abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians. Much early work in algebra, as the Arabic origin of its name suggests, was done in the Near East, by Persian mathematicians such as al-Khwārizmī (780 – 850) and Omar Khayyam (1048–1131).<ref>Omar Khayyám</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

Elementary algebra differs from arithmetic in the use of abstractions, such as using letters to stand for numbers that are either unknown or allowed to take on many values.<ref name=citeboyer /> For example, in [itex]x + 2 = 5[/itex] the letter [itex]x[/itex] is unknown, but the law of inverses can be used to discover its value: [itex]x=3[/itex]. In [itex]E=mc^2[/itex], the letters [itex]E[/itex] and [itex]m[/itex] are variables, and the letter [itex]c[/itex] is a constant. Algebra gives methods for solving equations and expressing formulas that are much easier (for those who know how to use them) than the older method of writing everything out in words.

The word algebra is also used in certain specialized ways. A special kind of mathematical object in abstract algebra is called an "algebra", and the word is used, for example, in the phrases linear algebra and algebraic topology.

A mathematician who does research in algebra is called an algebraist.

Algebra sections
Intro   Etymology    Different meanings of \"algebra\"    Algebra as a branch of mathematics    History    Areas of mathematics with the word algebra in their name    Elementary algebra    Abstract algebra    See also    Notes    References    External links

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