# How many correct decimals required ...

#### Bacterius

Hello,
somehow I'm stuck on a seemingly simple problem.

Problem said:
Given that $$\displaystyle n$$ is an integer, and $$\displaystyle q$$ is some rational $$\displaystyle 0 < q < 1$$ such that $$\displaystyle \sqrt{nq}$$ yields an integer, can you set up a relationship linking the number of correct decimals of the nonrepeating part of $$\displaystyle q$$, and the error bound of $$\displaystyle \sqrt{nq}$$ (hint : use orders of magnitude).
Attempted proof said:
Apart from the particular (and trivial) case $$\displaystyle k = 0$$, knowing $$\displaystyle k$$ decimals of $$\displaystyle q$$ gives $$\displaystyle q - 10^{-k} < a < q + 10^{-k}$$, so $$\displaystyle \sqrt{n\left (a - 10^{-k} \right )} < \sqrt{nq} < \sqrt{n\left (a + 10^{-k} \right )}$$ which gives the error bound for $$\displaystyle a \approx q$$.
Am I right ? Empirical tests seem to confirm this but I like to have more people look over my work because one always makes mistakes on his own.

Last edited:

#### Bacterius

Made a little mistake, correcting ...