From Arithmetic to Algebra

Abstract

In our relationship to numbers in our daily lives we are interested in specific numbers and not just numbers in a general or abstract way. Thus, each of us wants to know such things as our blood pressure, bank balance, and Social Security number, but as long as we limit ourselves to specific numbers, the usefulness of arithmetic itself is limited. If we write the symbol “5” we think of it as representing five and only five things, but since the laws of arithmetic apply to any number of things, we extend our arithmetic by operating not with specific numbers but with entities that may represent any number or numbers. This extension of our arithmetic to include symbols that may stand for or represent any numbers is called algebra. As we shall see, the rules of algebra are identical to those of arithmetic, but they apply to numbers combined with symbols rather than to numbers alone. Since the symbols in algebra stand for numbers, even though the numbers are not specific, algebra is the arithmetic of symbols.

Algebra begins with the unknown and ends with the unknowable.

—ANONYMOUS