# Decimal Expansion

We know all we need to know about geometric series. As a nifty bonus, we can use geometric series to better understand infinite repeating decimals. This is something you can rub in your former math teachers' faces.

### Sample Problem

Does 0.99999... equal 1?

Answer.

First we have to figure out what 0.99999... means. If we break it up by individual decimal places,

0.99999... = 0.9 + 0.09 + 0.009 + 0.0009 + 0 .00009 + ....

Convert each term to a fraction. Now

This is an infinite geometric series with and . The sum of the series is

We've just proved that

.99999... = 1.

Haha! Take that 8th-grade math teacher that took off 5 points from your test when you wrote '1' instead of '0.9999...'.

The same trick can be used to turn other infinite repeating decimals back into rational numbers.