Algebra may divided into "classical algebra" (equation solving or "find the unknown number" problems) and "abstract algebra", also called "modern algebra" (the study of groups, rings, and fields). Classical algebra has been developed over a period of 4000 years. Abstract algebra has only appeared in the last 200 years. The development of algebra is outlined in these notes under the following headings: Egyptian algebra, Babylonian algebra, Greek geometric algebra, Diophantine algebra, Hindu algebra, Arabic algebra, European algebra since 1500, and modern algebra. Since algebra grows out of arithmetic, recognition of new numbers - irrationals, zero, negative numbers, and complex numbers - is an important part of its history. The development of algebraic notation progressed through three stages: the rhetorical (or verbal) stage, the syncopated stage (in which abbreviated words were used), and the symbolic stage with which we are all familiar. The materials presented here are adapted from many sources including Burton, Kline's Mathematical Development From Ancient to Modern Times, Boyer's A History of Mathematics , and the essay on "The History of Algebra" by Baumgart in Historical Topics for the Mathematics Classroom - the 31st yearbook of the N.C.T.M.
Much of our knowledge of ancient Egyptian mathematics, including algebra, is based on the Rhind papyrus. This was written about 1650 B.C. and is thought to represent the state of Egyptian mathematics of about 1850 B.C. They could solve problems equivalent to a linear equation in one unknown. Their method was what is now called the "method of false position." Their algebra was rhetorical, that is, it used no symbols. Problems were stated and solved verbally. The Cairo Papyrus of about 300 B.C. indicates that by this time the Egyptians could solve some problems equivalent to a system of two second degree equations in two unknowns. Egyptian algebra was undoubtedly retarded by their cumbersome method of handling fractions.
The mathematics of the Old Babylonian Period (1800 - 1600 B.C.) was more advanced that that of Egypt. Their "excellent sexagesimal [numeration system]. . . led to a highly developed algebra" [Kline]. They had a general procedure equivalent to solving quadratic equations, although they recognized only one root and that had to be positive. In effect, they had the quadratic formula. They also dealt with the equivalent of systems of two equations in two unknowns. They considered some problems involving more than two unknowns and a few equivalent to solving equations of higher degree. There was some use of symbols, but not much. Like the Egyptians, their algebra was essentially rhetorical. The procedures used to solve problems were taught through examples and no reasons or explanations were given. Also like the Egyptians they recognized only positive rational numbers, although they did find approximate solutions to problems which had no exact rational solution.
Greek Geometrical Algebra
The Greeks of the classical period, who did not recognize the existence of irrational numbers, avoided the problem thus created by representing quantities as geometrical magnitudes. Various algebraic identities and constructions equivalent to the solution of quadratic equations were expressed and proven in geometric form. In content there was little beyond what the Babylonians had done, and because of its form geometrical algebra was of little practical value. This approach retarded progress in algebra for several centuries. The significant achievement was in applying deductive reasoning and describing general procedures.
The later Greek mathematician, Diophantus (fl. 250 A.D.), represents the end result of a movement among Greeks (Archimedes, Apollonius, Ptolemy, Heron, Nichomachus) away from geometrical algebra to a treatment which did not depend upon geometry either for motivation or to...