 05 217 385   # Station  8GWild Exploration Here are some “big question” investigations you might want to explore, or just think about. Have fun!

## WHICH FRACTIONS GIVE FINITE DECIMAL EXPANSIONS?

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?

## 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 $0.322222$...$= 0.3\overline{2}$ 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!

### Let's Go Wild! 