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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
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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

1 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 $\sqrt 2$.

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I assume you mean, "There is no known way...". –  TonyK Feb 5 '13 at 18:12
    
@TonyK, Yes, it has not been proven to be an impossible task. See for example this page from Mathworld. –  Peter Phipps Feb 5 '13 at 19:33

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