We’ve seen that \(\dfrac{1}{2}=0.5\) and \(\dfrac{1}{4}=0.25\) and \(\dfrac{1}{8}=0.125\) each have finite decimal expansions. (We’re ignoring infinite repeating zeros now.)

Of course, all finite decimal expansions give fractions with finite decimal expansions! For example, \(0.37\) is the fraction \(\dfrac{37}{100}\), showing that \(\dfrac{37}{100}\) has a finite decimal expansion.

What must be true about the integers \(a\) and \(b\) (or true just about \(a\) or just about \(b\)) for the fraction \(\dfrac{a}{b}\) to have a finite decimal expansion?

 We have seen that \(\dfrac{1}{3}=0.\overline{3}\) and \(\dfrac{4}{7}=0.\overline{571428}\) , for example, and that every fraction gives a decimal expansion that (eventually) repeats, perhaps with repeating zeros.

Is the converse true? Does every infinitely repeating decimal fraction correspond to a number that is a fraction?

Is \(0.\overline{17}\) a fraction? If so, which fraction is it?

Is \(0.\overline{450}\) a fraction? If so, which fraction is it?

Is \(0.32222\cdots = 0.3\overline{2}\) a fraction? Is \(0.17\overline{023}\) a fraction?

Indeed, does every repeating decimal have a value that is a fraction?