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## Answers from our experts to important scientific questions

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Q: Where did algebra originate?

A: Algebra originated in the ancient Middle East. By 2000 B.C., temple scribes, both in Egypt and in Mesopotamia, the ancient land between the Tigris and Euphrates Rivers, discovered rules to find unknown quantities in such problems as the following example from Egypt: A quantity, 2/3 of it, and 1/2 of it are added together to get 33. What is the quantity?

However, it was in the great Mesopotamian cities, such as Nineveh and Babylon, where temple scribes, as early as 2000 B.C., discovered how to solve more complicated equations, of the second degree, such as the one we would write as x2 = 3x + 10. They had no algebraic symbolism such as our "x"s and "plus" signs, and they phrased their problems in a geometrical language. So what we call x2 they thought of as a geometrical square of unknown side, and our 3x would be a rectangle of unknown length and width 3. Despite their geometric interpretation of the equation, however, they carried out exactly the same steps in solving such equations as we do today.

The Babylonians also discovered the rules of algebra for expanding expressions such as (a + b)2 or (a  b)2, and they pioneered the study of finding solutions of so-called indeterminate equations. (These are equations with infinitely many solutions.) For example, in the second millennium B.C, the Babylonians not only knew the solution x = 3, y = 4, and z = 5 for the famous equation x2 + y2 = z2, but they also had a list of 15 different solutions for it in which x, y, and z are integers.

To these ancient accomplishments, the Greeks after the time of Euclid (around 300 B.C.) added more methods for finding solutions of indeterminate equations. And one Greek, Diophantus of Alexandria (250 A.D.), contributed so much to the study of these equations that they are now called, after him, diophantine equations. He also introduced some limited symbolism into algebra.

In the early ninth century A.D., scientists of medieval Islam, such as Muhammad al-Khwarizmi, organized ancient algebra into a deductive science at the courts of the caliphs in Baghdad. The science was called, in Arabic, al-jabr walmuqabala, often translated as "Restoration and Cancellation," and from the Latin version of this we got our word algebra. (And, by the way, the Latin version of al-Khwarizimi's name, algorismi, gave us our word 'algorithm.') By the end of the tenth century these scientists had worked out the full content of what would, today, be a very good high school advanced algebra course. They even discovered the binomial theorem for expanding expressions such as (a + b)3, (a+b)4, etc. Thus, when Europeans learned of Greek and Arabic algebra through translations of the ancient books in the late middle ages they inherited a rich legacy of algebraic knowledge. This legacy prepared them well for the further development of the subject first in Italy and then in lands further north.

J. L. Berggren
Department of Mathematics
Simon Fraser University

References:

Asger Aaboe, "Episodes from the Early History of Mathematics," The Mathematical Association of America, 1975.

Howard Eves, "An Introduction to the History of Mathematics, 6E," Harcourt Brace College Publishers, 1990.

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