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.