# How can a repeating decimal be a rational number?

Mar 24, 2018

A rational number is any number that can be written as a fraction with an integer over a denominator which is also an integer.

#### Explanation:

For example...

Irrational number is $\pi \approx 3.1415926535 \ldots ,$ and since nothing repeats, it cannot be written as one integer over a denominator as a fraction.

A rational number is $39.3939393939 \ldots \mathmr{and} 369.3693693693 \ldots$

Any $2$ numbers that repeat go over $99$ in a fraction.

Any $3$ numbers that repeat go over $999$ in a fraction.

So $39.3939393939 \ldots$as a fraction would be $39 \frac{39}{99}$,

and $369.3693693693 \ldots$ as a fraction would be $369 \frac{369}{999}$.

The simplest example is $\frac{1}{3}$ which gives $0.333333 \ldots$

All recurring decimals come from dividing a fraction, so they are rational.