Algebra is one of the broad parts of mathematics, together with number theory, geometry and analysis. For historical reasons, the word "algebra" has several related meanings in mathematics, as a single word or with qualifiers. •
As a single word without article, "algebra" names a broad part of mathematics (see below). •
As a single word with article or in plural, "algebra" denotes a specific mathematical structure. See algebra (ring theory) and algebra over a field. •
With a qualifier, there is the same distinction:
Without article, it means a part of algebra, like linear algebra, elementary algebra (the symbol-manipulation rules taught in elementary courses of mathematics as part of primary and secondary education), or abstract algebra (the study of the algebraic structures for themselves). •
With an article, it means an instance of some abstract structure, like a Lie algebra or an associative algebra. •
Frequently both meanings exist for the same qualifier, like in the sentence: Commutative algebra is the study of commutative rings, that all arecommutative algebras over the integers. •
Sometimes "algebra" is also used to denote the operations and methods related to algebra in the study of a structure that does not belong to algebra. For example algebra of infinite series may denote the methods for computing with series without using the notions of infinite summation, limits andconvergence.
The Hellenistic mathematician Diophantus has traditionally been known as "the father of algebra" but debate now exists as to whether or not Al-Khwarizmi deserves this title instead.Those who support Diophantus point to the fact that the algebra found in Al-Jabr is more elementary than the algebra found in Arithmetica and that Arithmetica is syncopated while Al-Jabr is fully rhetorical. Those who support Al-Khwarizmi point to the fact that he gave an exhaustive explanation for the algebraic solution of quadratic equations with positive roots, and was the first to teach algebra in an elementary form and for its own sake, whereas Diophantus was primarily concerned with the theory of numbers. Al-Khwarizmi also introduced the fundamental concept of "reduction" and "balancing" (which he originally used the term al-jabr to refer to), referring to the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation. Other supporters of Al-Khwarizmi point to his algebra no longer being concerned "with a series of problems to be resolved, but an exposition which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study." They also point to his treatment of an equation for its own sake and "in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems." Al-Khwarizmi's work established algebra as a mathematical discipline that is independent of geometry and arithmetic.
18th century Math
Most of the late 17th Century and a good part of the early 18th were taken up by the work of disciples of Newtonand Leibniz, who applied their ideas on calculus to solving a variety of problems in physics, astronomy and engineering. The period was dominated, though, by one family, the Bernoulli’s of Basel in Switzerland, which boasted two or three generations of exceptional mathematicians, particularly the brothers, Jacob and Johann. They were largely responsible for further developingLeibniz’s infinitesimal calculus - paricularly through the generalization and extension of calculus known as the "calculus of variations" - as well asPascal and Fermat’s probability and number theory. Basel was also the home town of the greatest of the 18th Century mathematicians, Leonhard Euler, although, partly due to the difficulties in getting on in a...
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