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


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