Question

How can a repeating decimal be a rational number?

Answer 1

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 ~~ 3.1415926535...,` and since nothing repeats, it cannot be written as one integer over a denominator as a fraction.

A rational number is `39.3939393939...or 369.3693693693...`

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...`as a fraction would be `39 39/99`,

and `369.3693693693...` as a fraction would be `369 369/999`.

The simplest example is `1/3` which gives `0.333333...`

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