# Algebras - Not Distant Origins?, The Arabic Innovations, European Developments To The Seventeenth Century, Developments With Equations From Descartes To Abel

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theory mathematics mode algebraic

The word *algebra* refers to a theory, usually mathematical, which is dominated by the use of words (often abbreviated), signs, and symbols to represent the objects under study (such as numbers), means of their combination (such as addition), and relationships between them (such as inequalities or equations). An algebra cannot be characterized solely as the determination of unknowns, for then most mathematics is algebra.

For a long time the only known algebra, which was and is widely taught at school, represented numbers and/or geometrical magnitudes, and was principally concerned with solving polynomial equations; this might be called "common algebra." But especially during the nineteenth century other algebras were developed.

The discussion below uses a distinction between three modes of algebraic mathematics that was made in 1837 by the great nineteenth-century Irish algebraist W. R. Hamilton (1805–1865): (1) The "practical" is an algebra of some kind, but it only provides a useful set of abbreviations or signs for quantities and operations; (2) In the "theological" mode the algebra furnishes the epistemological basis for the theory involved, which may belong to another branch of mathematics (for example, mechanics); (3) In the "philological" mode the algebra furnishes in some essential way the formal language of the theory.

Lack of space prevents much discussion of the motivations and applications of algebras. The most important were geometries, the differential and integral calculus, and algebraic number theory.

## Additional Topics

A similar judgment applies to ancient Chinese ways of solving systems of linear equations. While their brilliant collection of rules can be rendered in terms of the modern manipulation of matrices, they did not create matrix theory. …

Common algebra is a theory of manipulating symbols representing constant and unknown numbers and geometrical magnitudes, and especially of expressing polynomial equations and finding roots by an algorithm that produces a formula. Its founders were the Arabs (that is, mathematicians usually writing in Arabic) from the ninth century, the main culture of the world outside the Far East. Some of the in…

Common algebra came into an awakening Europe during the thirteenth century. Among the various sources involved, Latin translations of some Arab authors were important. A significant homegrown source was the Italian Leonardo Fibonacci, who rendered the theory into Latin, with res, census, and cubus denoting the unknown and its powers. He and some translators of Arabic texts also adopted the Indian …

René Descartes's (1596–1650) Géométrie (1637) was an important publication in the history algebra. While its title shows his main concern, in it he introduced analytic geometry, representing constants and also variable geometric magnitudes by letters. He even found an algebraic means of determining the normal to a curve. Both this method and the representation of varia…

Lagrange's algebraic ambitions inspired some new algebras from the late eighteenth century onward. The names used below are modern. Firstly, in differential operators, the process of differentiating a function in the calculus was symbolized by D, with the converse operation of integration taken as 1/ D, with 1 denoting the identity operation; similarly, finite differencing was symbolized by…

At the end of the nineteenth century some major review works appeared. The German David Hilbert (1862–1943) published in 1897 a long report on algebraic number theory. The next year the Englishman Alfred North Whitehead (1861–1947)
put out a detailed summary of several of them in his large book A Treatise on Universal Algebra, inspired by Grassmann but covering also Boole's l…

The proliferation of algebras has been nonstop: the classification of mathematics in the early twenty-first century devotes twelve of its sixty-three sections of mathematics to algebras, and they are also present in many other branches, including computer science and cryptography. The presence or absence in an algebra of properties such as commutativity, distributivity, and associativity is routin…

Corry, Leo. Modern Algebra and the Rise of Mathematical Structures. Basel, Switzerland, and Boston: Birkhäuser, 1996. Cournot, Antoine Augustin. De l'origine et des limites da la correspondance entre l'algèbre et la géométrie. Paris and Algiers: Hachette, 1847. Crowe, Michael J. A History of Vector Analysis: The Evolution of the Idea of a Vertical System. Notre Da…

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