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Early Greek Algebra

The algebra of the early Greeks (of the Pythagoreans and Euclid, Archimedes, and Apollonius, 500-200 B.C.) was geometric because of their logical difficulties with irrational and even fractional numbers and their practical difficulties with Greek numerals, which were somewhat similar to Roman numerals and just as clumsy. It was natural for the Greek mathe­maticians of this period to use a geometric style for which they had both taste and skill.

The Greeks of Euclid's day thought of the product ab (as we write it nowadays) as a rectangle of base b and height a and they referred to it as "a rectangle contained by CD and DE". Some centuries later, another Greek, Diophantus, made a start toward modern symbolism in his work Diophantine Equations by introducing abbreviated words and avoiding the rather cumbersome style of geometric algebra, Diophantus introduced the syncopated style of writing equations.

Hindu and Arabic Algebra

Little is known about Hindu maths before the fourth or fifth century A.D. because few records of the ancient period have been found. India was subjected to numerous invasions, which facilitated the exchange of ideas. Babylonian and Greek accomplishments, in particular, were apparently known to Hindu mathematicians. The Hindus solved quadratic equations by "completing the square" and they accepted negative and irrational roots; they also realized that a quadratic equation (with real roots) has two roots. Hindu work on indeterminate equations was superior to that of Diophantus; the Hindus attempted to find all possible integral solutions and were perhaps the first to give general methods of solution. One of their most outstanding achievements was the system of Hindu (often called Arabic) numerals.

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