Fifty years ago, we were taught to calculate square roots one digit at a time with pencil and paper (and in my case, an eraser was also required). That was before calculators with a square root button were available. (And a square root button is an incredibly powerful doorway to the calculation of other functions.)
While trying to refresh that old skill, I found that the square root of 0.1111111…. was also a repeating decimal, 0.3333333…. Also, the square root of 0.4444444….. is 0.66666666…..
My question: are there any other rational numbers whose square roots have repeating digit patterns? (with the repeating digit other than 9 or 0, of course)? The pattern might be longer than the single digit in the above examples.
Best Answer
A real as repeating digit patterns iff it is a rational.
Your question is then under which conditions the square root of a rational $r$ is rational. The answer (see here) is that $r$ must be the square of a rational.
So $\sqrt{x}$ as a repeating pattern iff there exist integers $p$ and $q$ such that: $$x=\frac{p^2}{q^2}$$