# Relative frequency of $0$ in decimal repersentaion of $\sqrt{2}$ expansion

Demonstrate that the relative frequency of $0$ in the $\sqrt{2}$ is approximately $\frac{1}{10}$. The first $0$ is at the $13$ digit after the coma.

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There doesn't seem to be any question here. Is this a homework problem that you're expecting people to do for you without doing any work of your own? – Henning Makholm Feb 5 '13 at 13:38
How is the frequency of a digit in a number in decimal format defined? – Berci Feb 5 '13 at 13:54
Let me rephrase : i have to demonstrate that the frequency of 0 in the decimals of the square root of 2 is 1/10 – user61079 Feb 5 '13 at 13:58
A rather simple way to phrase this question is to prove that $\lim_{n \to \infty} \frac{f(n)}{n} = \frac{1}{10}$ where $f(n)$ counts the number of $0$'s in $\sqrt{2}$'s decimal representation in the first $n$ digits. However, this problem may be open I think. – dinoboy Feb 5 '13 at 15:46
Once you have solved this problem, you may claim your Fields Medal. – Gerry Myerson Feb 6 '13 at 3:35

You may find an expansion of $\sqrt 2$ in a book of tables or on the internet suitable for your purposes, or you could compute it yourself using Newton-Raphson, for example. But your conclusion can only be that the digit $0$ turns up/does not turn up one tenth of the time in up to so many decimal places of $\sqrt 2$.