# Approximate irrational number using convergents of continued fractions

I would really appreciate it if anyone could help me with this problem:

Among the convergents of $\sqrt{15}$, find a rational number that approximates $\sqrt{15}$ with accuracy to four decimal places.

I know how to find the continued fraction and the convergents for $\sqrt{15}$. But I am just not sure how to determine how far I need to go in order to approximate that number with the requested accuracy.

Thanks!

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Are you familiar with a certain bound on the absolute value of the difference between a number $\alpha$ and one of its convergents? – Qiaochu Yuan May 22 '11 at 21:44
$|\alpha - p_{n}/q_{n}| \le |\alpha -a/b|<1/2b^2$ ? – user2467 May 22 '11 at 22:24
Of course, the lowbrow way to do this is to calculate $\sqrt{15}$ to 4 decimals, then calculate convergents to 4 decimals until you find one that works. – Gerry Myerson May 23 '11 at 0:02
Yes, that's what I did at first, but something tells that the author had something else in mind. – user2467 May 23 '11 at 0:56
@Daniel: yes, that one. Can you see how that answers your question? – Qiaochu Yuan May 23 '11 at 7:24