How can a repeating decimal be a rational number?

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Answer 1 · 2018-03-24T19:44:49

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

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.