Algebra had its beginnings in the evolution of the meaning of number. Slowly there emerged symbols called “numerals,” such as 1, 2, and 3, which are used to stand for certain numbers. Then relationships of numbers to one another were observed, and it was discovered that, by giving enough information about the relation of some unknown number to certain known numbers, the unknown could be determined. This “bringing together of known and unknown parts” was called al-jebr in Arabic and, later on, algebra in Medieval Latin.
It took many centuries for mathematicians, chiefly in Europe, to invent algebraic notation similar to that in current usage. Then problems of “reunion of parts” could be more elegantly stated in the form of algebraic equations. Initially, the numbers in these equations were restricted to the positive integers or “natural numbers,” but such equations as
could not be solved without introducing fractions. It was not until the seventeenth century that such numbers as -2 were admitted into the “fraternity of numbers,” so that such equations as
have solutions in numbers. It was largely through the vigorous pursuit of solutions to equations throughout the eighteenth and the early nineteenth centuries that the systems of numbers that we shall study in this book were developed.
Early in the nineteenth century a new kind of algebra emerged. The essential distinction between the old “classical algebra” and the new “modern algebra” lies in the study of the structure of a mathematical system, rather than in its particular members. The structure of a system can be defined loosely to be the basic properties of the operations that can be performed on its members.
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