# Algebra

Algebra 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; 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 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.

## Quotes

* Algebra is the offer made by the devil to the mathematician. The devil says: `I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvellous machine.'
** Michael Atiyah (2004). Collected works. Vol. 6. Oxford Science Publications. The Clarendon Press Oxford University Press. ISBN 978-0-19-853099-2.
* The historical associations of the word algebra almost substantiate the sordid character of the subject. The word comes from the title of a book written by... Al Khawarizmi. In this title, al-jebr w' almuqabala, the word al-jebr meant transposing a quantity from one side of an equation to another and muqabala meant simplification of the resulting expressions. Figuratively, al-jebr meant restoring the balance of an equation... When the Moors reached Spain... algebrista... came to mean a bonesetter... and signs reading Algebrista y Sangrador (bonesetter and bloodletter) were found over Spanish barber shops. Thus it might be said that there is a good historical basis for the fact that the word algebra stirs up disagreeable thoughts.
** Morris Kline, Mathematics and the Physical World (1959), p. 69
* [T]he sciences that are expressed by numbers or by other small signs, are easily learned; and... this facility rather than its demonstrability is what has made the fortune of algebra.
** Julien Offray de La Mettrie, Man a Machine (1747)
* The motives which give rise to the use of alphabetic letters as symbols of number in preference to any other system of symbols, arbitrarily selected for the same purpose, are principally the following. First, As they have no numerical signification in themselves, they are subject to no ambiguity, having in reference to numbers no other signification than they are defined to have in the outset of each problem… Secondly, Being familiar to the eye, the tongue, the hand, and the mind, that is, having a well-known form and name, they are easily read, written, spoken, remembered, and discriminated from one another, which could not be the case were they mere arbitrary marks, formed according to the caprice of each individual who used them. Thirdly, The order in which the letters are arranged in the alphabet, facilitates the classification of them into groups much more easy to survey and comprehend in the expressions which arise…, and thereby renders the investigator much less likely to omit any of them by an imperfect enumeration.
** Charles Hutton and Olinthus Gregory, A Course of Mathematics (1836)

## See also

* Abstract algebra
* Algebraic geometry
* Algebraic topology
* Commutative algebra
* History of algebra
* Homological algebra

## External links

* Archive.org books
** Elements of Algebra (1822) by Leonhard Euler. Translated from the French, with the notes of Bernoulli and the additions of De La Grange. 3d ed., carefully revised and corrected by John Hewlett. To which is prefixed a memoir of the life and character of Euler.
** Elements of Algebra (1837) by Augustus De Morgan Preliminary to the Differential Calculus and fit for the Higher Classes of Schools in which the Principles of Arithmetic are Taught. 2nd edition.