VI

Generalizations of Number

One great peculiarity of mathematics is the

set of allied ideas which have been invented in connection with the integral numbers from which we started. These ideas may be called extensions or generalizations of number. In the first place there is the idea of fractions. The earliest treatise on arithmetic which we possess was written by an Egyptian priest, named Ahmes, between 1700 and 1100 ,

and it is probably a copy of a much older work. It deals largely with the properties of fractions. It appears, therefore, that this concept was developed very early in the history of mathematics. Indeed the subject is a very obvious one. To divide a field into three equal parts, and to take two of the parts, must be a type of operation which had often occurred. Accordingly, we need not be surprised that the men of remote civilizations were familiar with the idea of two-thirds, and

with allied notions. Thus as the first generalization of number we place the concept of fractions. The Greeks thought of this subject rather in the form of ratio, so that a Greek would naturally say that a line of two feet in length bears to a line of three feet in length the ratio of $2$ to $3$. Under the influence of our algebraic notation we would more often say that one line was two-thirds of the other in length, and would think of two-thirds as a numerical multiplier.

In connection with the theory of ratio, or

fractions, the Greeks made a great discovery, which has been the occasion of a large amount of philosophical as well as mathematical thought. They found out the existence of "incommensurable" ratios. They proved, in fact, during the course of their geometrical investigations that, starting with a line of any length, other lines must exist whose lengths do not bear to the original length the ratio of any pair of integers–-or, in other words, that lengths exist which are not any exact fraction of the original length.

For example, the diagonal of a square cannot be expressed as any fraction of the side of the same square; in our modern notation the length of the diagonal is $\sqrt{2}$ times the length of the side. But there is no fraction which exactly represents $\sqrt{2}$. We can approximate

to $\sqrt{2}$ as closely as we like, but we never exactly reach its value. For example, ${\displaystyle \frac{49}{25}}$ is just less than $2$, and ${\displaystyle \frac{9}{4}}$ is greater than $2$, so that $\sqrt{2}$ lies between ${\displaystyle \frac{7}{5}}$ and ${\displaystyle \frac{3}{2}}$. But the best systematic way of approximating to $\sqrt{2}$ in obtaining a series of decimal fractions, each bigger than the last, is by the ordinary method of extracting the square root; thus the series is $1$, ${\displaystyle \frac{14}{10}}$, ${\displaystyle \frac{141}{100}}$, ${\displaystyle \frac{1414}{1000}}$, and so on.

Ratios of this sort are called by the Greeks incommensurable. They have excited from the time of the Greeks onwards a great deal of philosophic discussion, and the difficulties connected with them have only recently been cleared up.