## 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?
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? |

## Resources

## Books

Take your understanding to the next level with easy to understand books by James Tanton.

BROWSE BOOKS

## Guides & Solutions

Dive deeper into key topics through detailed, easy to follow guides and solution sets.

BROWSE GUIDES

## Donations

Consider supporting G'Day Math! with a donation, of any amount.

Your support is so much appreciated and enables the continued creation of great course content. Thanks!

## Ready to Help?

Donations can be made via PayPal and major credit cards. A PayPal account is not required. Many thanks!

DONATE