# Station 8G**Wild Exploration**

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

Station 8G

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

We’ve seen that $\dfrac {1}{2} = 0.5$, $\dfrac {1}{4} = 0.25$ 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 $0.322222$...$= 0.3\overline{2}$ a fraction?

Is $0.17\overline{023}$ a fraction?

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

You can either play with some of the optional stations below or go to the next island!