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

Question

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.

Answer 1

There is no way to determine the frequency of digits in the decimal expansions of irrational numbers other than to compute the number and count up the digits.

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 $\sqrt 2$.