Algebraic Numbers
The term algebra comes from the Arabic al-jabr, meaning to combine. Al-Jabr refers to the mathematician's process of combining like terms to solve an equation.
Historical references to algebra date back to Greek history, spanning the years 540 B.C.-250 B.C. Euclid, Pythagoras, and their followers used algebraic problems with geometric proofs. This method of mixing algebra and geometry led to complex constructions. The Greek system of mathematics used layers of numbers, letters, and punctuation marks written above each other to express these complexities.
The separation of Greek algebra from geometry did not occur until around A.D. 250, when Diophantus of Alexandria demonstrated algebra. Diophantus was the first to use symbolic notation for algebraic expressions. This was similar to Babylonian algebra, except Babylonian mathematicians were usually limited to approximate solutions. Diophantus worked out equations with exact solutions. He developed a system that offered symbols for frequently used numbers, operations, and variables.
The Greeks and Babylonians influenced the practice of mathematics in India, where Brahmagupta, a Hindu scholar, practiced circa A.D. 630. Brahmagupta understood negative numbers, zero, second-degree equations, and equations with variables.
In A.D. 825, Al-Khwarizmi (c. 780-c. 850) wrote Hisab al-jabr w-al-musqabalah (The Science of the Transposition and Cancellation) in Baghdad. This was the title that gave algebra its name. Al-Khwarizmi's work helped to bring about the acceptance of algebraic equations in Europe.
During the thirteenth and fourteenth centuries, Arabic, Persian, and Hindu advances in algebra were received in Europe as the Renaissance in art, the Reformation in religion, and the discovery of the New World helped to bring about a revolution in mathematics in the fifteenth and sixteenth centuries. Algebra had to be developed before advances could be made in calculus and analytical geometry in the seventeenth century. Italy was the center of this rebirth, with over 200 mathematical texts published between 1472 and 1500.
In 1545 the Italian mathematician Girolamo Cardano (1501-1576) wrote Ars magna sive de regulis algebraicis, which sparked interest in algebra by offering what, by today's standards, would be a tedious collection of similar problems intended to illustrate the same principle. Ars magna also demonstrated a solution to a quartic equation by manipulating it into a form of a cubic equation.
François Viète, the author of an algebra text published in 1591, is recognized as "the father of modern algebraic notation." Viète's text is similar to modern texts of elementary algebra. Viète worked out a standard procedure for solving quadratic equations, cubic equations, and quartic equations.
The system of analytic geometry of Descartes (1596-1650) used letters at the end of the alphabet to indicate unknown quantities (x, y, z) and letters at the beginning of the alphabet (a, b, c) to indicate known quantities. Descartes' system of analyzing geometric problems algebraically helped to advance both algebra and geometry.
In 1797, German mathematician Carl Frederich Gauss (1777-1855) published his proof of the fundamental theorem of algebra. A cascade of algebraic discoveries would continue through the nineteenth and early twentieth centuries. In his A Treatise on Algebra, a textbook published in 1830, the British mathematician George Peacock claimed that algebra had been thought of as arithmetic using symbols, with letters replacing numbers. Peacock proposed new ways of thinking about algebra and different kinds of algebra altogether. During the nineteenth century, algebra became very abstract.
In another significant advance, mathematicians Ernest E. Krummer (1810-1893), J. W. Richard Dedekind (1831-1916), and Leopold Kronecker (1823-1891) created a theory of algebraic numbers, which laid the groundwork for modern abstract algebra.
The work of Nikolay Lobachevsky (1793-1856) and Janòs Bolyai (1802-1860) in the early 1830s led to advances in non-Euclidean geometry and abstract algebra. These developments fundamentally affected the approach to deductive reasoning. Elementary algebra was formalized during the 1830s and there was a ripple effect in scientific discovery.
The English mathematician and logician George Boole (1815-1864) invented Boolean algebra in 1847, basing it on the concept that ideas have only two possible states when stable: on/off, closed/open, yes/no, true/false. Boolean algebra and its close relative, binary theory, opened the door for the development of computer science.
B. Pierce (1809-1880) said in 1870 that "all relations are either qualitative or quantitative" in algebra and these two relations may be considered independently of one another or in some cases combined. This was a step toward abstract mathematical theory. The trend was away from devising elaborate theories, with mathematicians looking more and more at the interrelationships among theories.
Toward the end of the nineteenth century, linear algebra seemed to move in promising new directions but without significant advances. A new era of algebraic discovery began around 1910 with a breakthrough in the development of general methods of linear algebra. The 1920s and 1930s marked the modern development of abstract algebra. During this period, there was a competitive relationship between scholars in algebra and topology (the specialized study of geometric configurations) for dominance in mathematics.
After World War II, much of the research in mathematics became so abstract that it was impossible for nonmathematicians to understand. Abraham Adrian Albert studied nonassociative algebras, and there were advances in probability theory and factoring. Discoveries in four-dimensional spaces and computer science were taking mathematics in entirely new directions. In 1955, Henri-Paul Cartan (1904-) and Samuel Eilenberg brought a certain symmetry to the study of mathematics by devising the system of homological algebra, a combination of abstract algebra and algebraic topology that reunited the two competitive fields.
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