## Why are Non-terminating, Repeating Decimals Rational

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
$\frac{8}{10} + \frac{4}{100} + \frac{2}{1000}$.