We will put the incommensurable ratios

with the fractions, and consider the whole set of integral numbers, fractional numbers, and incommensurable numbers as forming one class of numbers which we will call "real numbers." We always think of the real numbers as arranged in order of magnitude, starting from zero and going upwards, and becoming indefinitely larger and larger as we proceed. The real numbers are conveniently

represented by points on a line. Let $O X$ be pg76 any line bounded at $O$ and stretching away indefinitely in the direction $O X$ . Take any convenient point, $A$ , on it, so that $O A$ represents the unit length; and divide off lengths $A B$ , $B C$ , $C D$ , and so on, each equal to $O A$ . Then the point $O$ represents the number $0$ , $A$ the number $1$ , $B$ the number $2$ , and so on. In fact the number represented by any point is the measure of its distance from $O$ , in terms of the unit length $O A$ . The points between $O$ and $A$ represent the proper fractions and the incommensurable numbers less than $1$ ; the middle point of $O A$ represents $\frac{1}{2}$ , that of $A B$ represents $\frac{3}{2}$ , that of $B C$ represents $\frac{5}{2}$ , and so on. In this way every point on $O X$ represents some one real number, and every real number is represented by some one point on $O X$ .

The series (or row) of points along $O X$ ,

starting from $O$ and moving regularly in the direction from $O$ to $X$ , represents the real numbers as arranged in an ascending order

of size, starting from zero and continually increasing as we go on.

All this seems simple enough, but even at

this stage there are some interesting ideas to be got at by dwelling on these obvious facts. Consider the series of points which represent the integral numbers only, namely, the points, $O$ , $A$ , $B$ , $C$ , $D$ , etc. Here there is a first point $O$ , a definite next point, $A$ , and each point, such as $A$ or $B$ , has one definite immediate predecessor and one definite immediate successor, with the exception of $O$ , which has no predecessor; also the series goes on indefinitely without end. This sort of order is called the type of order of the integers; its essence is the possession of next-door neighbours on either side with the exception of No. 1 in the row. Again consider the integers and fractions together, omitting the points which correspond to the incommensurable ratios. The sort of serial order which we now obtain is quite different. There is a first term $O$ ; but no term has any immediate predecessor or immediate successor. This is easily seen to be the case, for between any two fractions we can always find another fraction intermediate in value. One very simple way of doing this is to add the fractions together and to halve the result. For example,

between $\frac{2}{3}$ and $\frac{3}{4}$ , the fraction $\frac{1}{2} (\frac{2}{3} + \frac{3}{4})$ , that is $\frac{17}{24}$ , lies; and between $\frac{2}{3}$ and $\frac{17}{24}$ the

fraction 1 2 ( 2 3 + 17 24 ) , that is 33 48 , lies; and so on indefinitely. Because of this property the series is said to be "compact." There is no end point to the series, which increases indefinitely without limit as we go along the line O X . It would seem at first sight as though the type of series got in this way from the fractions, always including the integers, would be the same as that got from all the real numbers, integers, fractions, and incommensurables taken together, that is, from all the points on the line O X . All that we have hitherto said about the series of fractions applies equally well to the series of all real numbers. But there are important