Repeating Digit Patterns in Square Roots 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.
 A: 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:
$$x=\frac{p^2}{q^2}$$
A: Just take your favorite rational number with a repeating decimal:
$$\frac{1}{11} = .09090909\ldots.$$
Square it:
$$\frac{1}{121} = .00826446280991735537190082644628099173553719\ldots,$$
and you've created an example.  This one has period 21 and its square root has period 2.
A: Every rational number has either eventually repeating or terminating decimals.
Therefore if your $x$ is a square of a rational number, $\sqrt x$  will either terminate or have eventually repeating decimals. For example $ 1/36 =.0277777777\ldots$ and its square root is $$ \sqrt {  .0277777777\ldots} = 0.166666666\ldots$$
or  
$$\sqrt {.049382716049382716049\ldots} = 0.22222222222222222222 \ldots$$
A: Any repeating digit pattern must be rational, so take a rational number $\frac{p}{q}$, and square it to get $\frac{p^2}{q^2}$, and there you have it!
It's worth noting that, in the examples you gave, $\sqrt{0.1111...} = \sqrt{\frac{1}{9}} = \sqrt{\frac{1}{3^2}} = \frac{1}{3}$, and $\sqrt{0.44444...} = \sqrt{\frac{4}{9}} = \sqrt{\frac{2^2}{3^2}} = \frac{2}{3}$.
