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,
|
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.
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 