# Thread: How many correct decimals required ...

1. ## How many correct decimals required ...

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

Originally Posted by Problem
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).
Originally Posted by Attempted proof
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.

2. Made a little mistake, correcting ...