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Algebra

 Algebra is one of the broad areas of mathematics, together with number theory, geometry, and analysis. In the most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics. It includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic part 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.

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 many values. For example, in x + 2 = 5 the letter x is an unknown, but by applying additive inverses can reveal its value: x = 3. Algebra gives methods for writing formulas and solving equations that are much clearer and easier 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 as a word comes from the Arabic al-jabr, meaning "reunion of broken parts," bonesetting from the title of the early 9th century book Il al-jabr wa I-muqālaba (The Science of Restoring and Balancing) by the Persian mathematician and astronomer al-Khwarizmi. In his work, the term al-jabr referred to the operation of moving a term from one side of an equation to the other, while al-muqābala "balancing" referred to adding equal terms on both sides. Shortened to just algeber or algebra in Latin, the word eventually entered the English language during the 15th century, from either Spanish, Italian, or Medieval Latin. It originally referred to the surgical procedure of setting broken or dislocated bones. The mathematical meaning was first recorded (in English) in the 16th century.

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