Exploding Dots

8.8 Wild Explorations

Here’s are some “big question” investigations you might want to explore, or just think about. Have fun!

 EXPLORATION 1: WHICH FRACTIONS GIVE FINITE DECIMAL EXPANSIONS? 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?

 EXPLORATION 2: BACKWARDS? ARE REPEATING DECIMALS FRACTIONS?  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?

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