# Open ended investigation: Exploring fractions in different bases

This project is related to the repeating decimals project I’ve outlined before (see this post) but with a twist.

What do fractions look like in bases other than 10 (for example: base 7)?

Instead of decimals, what would septimals look like? When we write 10.356, what we really mean is $10 + \frac{3}{10} + \frac{5}{100} + \frac{6}{1000}$. In septimal notation, $7 + \frac{4}{7}$  would be written 10.4. Can you extend this example?

Here’s a trickier question: how is 0.5 in decimal notation represented in septimal notation?

Is there a relationship between the number of digits it takes a fraction to repeat when it is written in expanded (in base ten, we call these decimals) form and the base in which it is being represented? Is it possible to write numbers in any rational base, for example 1.5? What about base $\pi$?

## 2 thoughts on “Open ended investigation: Exploring fractions in different bases”

1. Garrett G |

Doesn’t septimal (base 7) notation use only the digits 0 – 6? I’m a little confused by your 7.4 example. In base 7, wouln’t 7 4/7 be represented as 1*7^1 + 0*7^0 + 4*7^(-1) = 10.4?

• David Wees |

You are absolutely right! Hah, I have no idea how I made that mistake. I’ll update the post. Thank you for pointing my mistake out.