Station 8G
Wild Exploration

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

Let's go!

Question 1

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?

Question 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.