Rational numbers fit into this scheme by dividing up the intervals between the integers. For example, goes midway between 0 and 1, and goes midway between 1 and 2. The numbers and and divide the interval from 0 to 1 into three parts of equal length, and the numbers and divide the interval between 2 and 3 similarly. If you locate fractions with different denominators on the line, they may appear to be arranged somewhat irregularly.
However, if you fix a denominator, and label all points by numbers with that fixed denominator, then you get an evenly spaced set, with each unit interval divided up into the same number of subintervals. Thus all rational numbers, whatever their denominators, have well-defined places on the number line. In particular, decimals with one digit to the right of the decimal point partition each unit interval on the number line into subintervals of length and decimals with two digits to the right of the decimal point refine this to intervals of length with 10 of these fitting into each interval of length See Box 3–6.
Box 3–6 The Number System of Finite Decimals
Although they are not usually singled out explicitly, the finite decimals, such as 3, –104, 21.6, 0.333, 0.0125, and 3.14159, form a number system in the sense that you can add them and multiply them and get finite decimals. You can also subtract finite decimals, but you cannot always divide them. For example, cannot be exactly represented as a finite decimal, although it can be approximated by 0.333. The finite decimal system is intermediate between the integers and the rational numbers.
The advantage of working with finite decimals rather than all the rational numbers is that the usual arithmetic for integers extends almost without change. The only complication is that one must keep track of the decimal point. (This seemingly small complication is actually a large conceptual leap.) For example,