## Why are Non-terminating, Repeating Decimals Rational

Last night, I received a Facebook message from a Grade 8 student asking why non-terminating repeating decimals are rational. I am posting the answer here for reference.

Rational numbers is closed under addition. That is, if we add two rational numbers, we are guaranteed that the sum is also a rational number. The proof of this is quite easy, so I leave it as an exercise for advanced high school students.

Before discussing non-terminating decimals, let me also note that terminating decimals are rational. I think this is quite obvious because terminating decimals can be converted to fractions (and fractions are rational). For example, $0.842$ can be expressed as $\displaystyle\frac{842}{1000}$.

Further, terminating decimals can be expressed as sum of fractions. For example, $0.842$ can be expressed as $\frac{8}{10} + \frac{4}{100} + \frac{2}{1000}$.

Since rational numbers is closed under addition, the sum of any number of fractions is also a fraction. This shows that all terminating decimals are fractions.  » Read more 