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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.

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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: $$x=\frac{p^2}{q^2}$$

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  • $\begingroup$ In your example $0.11111\ldots=\frac{1^2}{3^2}$ and $0.444444\ldots=\frac{2^2}{3^2}$. $\endgroup$
    – Delta-u
    Apr 26, 2018 at 18:02
  • $\begingroup$ So if a rational is the quotient of two perfect squares, the square root is also a rational and has a repeating digit pattern. The next question to pop up, of course, is if a real can be approximated as closely as desired by a rational whose numerator and denominator are perfect squares, fourth powers, eighth powers, etc. But that is another job for another day. $\endgroup$
    – richard1941
    Apr 26, 2018 at 18:09
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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.

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  • $\begingroup$ That was the first thing I tried. Alas, my WP-34 calculator (simulated on an iPhone) only gives square roots to 20 digits. No wonder I didn't find it $\endgroup$
    – richard1941
    Apr 26, 2018 at 18:19
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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

$$\sqrt {.049382716049382716049\ldots} = 0.22222222222222222222 \ldots$$

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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 $\sqrt{0.44444...} = \sqrt{\frac{4}{9}} = \sqrt{\frac{2^2}{3^2}} = \frac{2}{3}$.

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  • $\begingroup$ I found that the subset of rational numbers whose numerator and denominator are squares contains members arbitrarily close to any real number. Thanks for the answers! $\endgroup$
    – richard1941
    May 4, 2018 at 13:57

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