The finite decimal system does allow division by 10 (and by its divisors, 2 and 5), and it may be characterized as the smallest number system containing the integers and allowing division by 10. Indeed, another way of representing finite decimals is as rational numbers with denominators that are powers of 10. For example, 21.6=216/10 and 0.0125=125/10,000.

It may not seem a huge gain to be able to divide by 10. What is the point of enlarging the system of integers to the system of finite decimals? It is that arithmetic can remain procedurally similar to the arithmetic of whole numbers, and yet finite decimals can be arbitrarily small and, as a consequence, can approximate any number as closely as you wish. This process is best illustrated by using the number line.

The integers occupy a discrete set of points on the number line, each separated from its neighbors on either side by one unit distance:

The finite decimals with at most one digit to the right of the decimal point label the positions between the integers at the division points:

If you allow two digits to the right of the decimal point, these tenths are further subdivided into hundredths.

As you can see, space between these numbers is already rather small. It would be very difficult to draw a picture of the next division, defined by decimals with three digits to the right of the decimal point. Nonetheless, you can imagine this subdivision process continuing on and on, giving finer and finer partitions of the line.



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