Introduction

Nowadays we recognize written algebra by the presence of letters (called variables) standing for unspecified numbers, and especially by the presence of equations involving those letters. These two features—letters and equations—reveal the techniques of algebra, but algebra itself is not these techniques. Rather, algebra consists of problems in which the goal is to find a number knowing certain indirect information about it. If you were told to multiply 7 by 3, then add 26 to the product, you would be doing arithmetic, that is, you would be given not only the data, but also told which operations you must perform (multiplication followed by addition). But if you were asked for a number having the property that if it is multiplied by 3 and 26 is added to the product, the result is 57, you would be facing an algebra problem. In an algebra problem, the operations and some of the data are given to you, but these operations are not for you to perform. Rather, you assume someone else has performed them, and you need to find the number(s) on which they were performed. Using this definition, we can recognize algebra problems in very ancient texts that contain no equations at all.

But how do such problems arise?Why were people interested in solving them? Those are questions that any student who looks beyond the horizon of tomorrow's homework assignment is bound to ask. In the following paragraphs, we shall look at some examples and see if we can answer such questions.