Why are Non-terminating, Repeating Decimals Rational

August 11, 2012 GB High School Mathematics, Question and Answer

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.

Now, for non-terminating decimals, a certain strategy is needed to show that it is a rational number. The strategy is to multiply the decimal to powers of $10$ and subtract them so that the repeating decimals are eliminated. For example, to show that $0.7345345 \cdots$ (with $345$ repeating indefinitely) is rational, we let $x = 0.7345345 \cdots$ .