In the preceding article, "Arithmetic," we were concerned with particular numbers, which are expressed by symbols. "Sixty-seven" is a particular number. To do arithmetical problems in which sixty-seven plays a part, we use the symbols 6 and 7, combined as 67. We are now going to consider a branch of mathematics in which a symbol, such as the letter a, b, or c, stands not for a particular number, but for a whole class of numbers. This kind of mathematics is called algebra. Polynomial division is available online.
We can illustrate the difference between arithmetic and algebra by a very simple example:
In arriving at the final result, 10, we used the method of arithmetic, involving particular numbers, throughout.
Suppose now that we think of any number. Let us indicate "any number" by the symbol x, and let us go through the same operation as before:
Here we have been using the methods of algebra, because x can be replaced by any number. We could substitute for it 2, or 3, or 15, and the final result would always be 10. Polynomial division is available online.
When a generalized number, represented by a letter (such as a), is multiplied by a particular number (such as 5) or by another generalized number (such as b), we do not use multiplication signs, but indicate multiplication by putting these symbols close to one another. Thus a × b = ab; 5 × a = 5a; 5 × a × b = 5ab. We could not indicate the multiplication of two particular numbers in this way; 7 × 5 could not be given as 75, because 75 really stands for 70 + 5.
Let us consider another example. In the equation (2 + 3)2 = 25, we are dealing with the particular numbers 2 and 3, and the result is always 25. But suppose that instead of two particular numbers, we used the letters a and b to stand for any two numbers. We would then have the algebraic equation (a + b)2 = a2 + 2ab + b2. (The derivation of this equation will be explained later.)
What is significant about (a + b)2 = a2 + 2ab + b2 is that it indicates a general relationship that holds true for a great many particular numbers. If we substituted 3 for a and 2 for b, we could have (3 + 2)2 = 32 + (2 × 3 × 2) + 22 = 9 + 12 + 4 = 25. Or we could substitute 5 for a and 6 for b, giving (5 + 6)2 = 52 + (2 × 5 × 6) + 62 = 25 + 60 + 36 = 121. Polynomial division is available online.
Algebra, the mathematics of "any numbers," or variables, goes to the heart of the relationship between numbers. Generally speaking, it is concerned with particular numbers only insofar as they are applications of general principles. It is also used in the solution of certain specific problems in which we start out with one or more unknown quantities whose values are indicated by algebraic symbols.
An Ancient Discipline
The study of algebra goes back to antiquity. Recent discoveries have shown that the Babylonians solved problems in algebra, although they had no symbols for variables. They used only words to indicate such numbers, and for that reason their algebra has been referred to as rhetorical algebra. The Ahmes Papyrus, an Egyptian scroll going back to 1600 B.C., has a number of problems in algebra, in which the unknown is referred to as a hau, meaning "a heap." Polynomial division is available online.
Little further progress was made in algebra until we come to Diophantus, a 3rd-century A.D. Greek mathematician. He reduced problems to equations, representing the unknown quantity by a symbol suggesting the Greek letter Σ (sigma). He also introduced an interesting system of abbreviations, in which he used only the initial letters of words, after omitting all unnecessary words. Polynomial division is available online.