Rhetorical Algebra
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This type of algebra goes back to the Babylonians. Rhetorical algebra was all oral. There were no symbols to use and writing materials may or may not have existed in different cultures. This does not mean that "easy" problems were the only ones described. Keep in mind that writing materials and symbols do not add intelligence or problem solving ability to people. Though arithmetic began for practical reasons, an art developed to cultivate the science. Naturally, consideration of the abstract began to develop. People began studing mathematics for the sake of mathematics, not out of necessity. This is how algebra ultimately evolved from arithmetic. But, the people of this time period were limited in their ability to write and the materials to write on. Some of the easiest obtainable material was clay tablets, which had to be just the right consistency at that moment to allow an impression to be made with some sort of stylus, then baked in the sun to hold that shape. And they simply wrote out exactly what they said. They did not have symbols or abbreviations. They didn't know what +, -, *, or = meant!

276 B.C. - 197 B.C. -- Eratosthenes of Cyrene

He is noted for a device known as the sieve, used for finding all prime numbers less than a given number n.

Arabian math was mostly rhetorical, even though their years correspond to much after this above date. The Arabic algebras (the best known is Al-Khowarizmi's) expain rules for computing, give process of casting out nines, and give rules of false position and double false position. Other explanations given in these were square and cube roots, fractions, and the rule of three. In Al-Khowarizmi's algebra, he gives four elementary operations, solves linear and quadratic equations, contains some geometric mensuration, and gives problems on inheritance. Omar Khayyam was a great Moslem mathematician who gave a geometric solution of cubic equations.

The Arabs were the first to recognize irrational roots of quadratic equation. They also saw the existence of two solutions of a quadratic equation (Euclid did not).

They introduced the rule of false position to European scholars.

790 - 850 -- Al-Khowarizmi

1048 - 1122 -- Omar Khayyam

He was a great poet of his day, author of the exquisite Rubaiyat. He found a geometric solution of cubic equations.

Evolution of Algebra

Babylonian and Egyptian Mathematics

Fractions of Babylon and Egypt

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