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

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