

Title: Trivial Approximation Post by Sir Col on Jun 17^{th}, 2008, 12:02pm A rational approximation, m/n, of an irrational number, a, is defined as trivial if 1/n^{2} < a  m/n < 1/n. If a  m/n < 1/n^{2} then it is defined as a reasonable approximation. (Technically a  m/n < 1/n^{3} is defined as a good approximation, but we shall not concern ourselves with these for this problem.) For example, sqrt(2)  17/12 = 0.002453... < 1/12^{2} = 0.0069444..., and so 17/12 is considered to be a nontrivial approximation. However, if we round sqrt(2) to two decimal places, sqrt(2) = 1.41 (2 d.p.) and sqrt(2)  141/100 = 0.00421356... < 1/100 = 0.01, which is only a trivial approximation. If sqrt(x) is rounded to k > 1 decimal places, does this always leads to a trivial approximation? (Edited to correct m,n mixup pointed out by Hippo.) 

Title: Re: Trivial Approximation Post by Hippo on Jun 17^{th}, 2008, 12:48pm There should be 1/n on place of 1/m and so on in the definition ... otherewise the examples do not correspond to the definition. 

Title: Re: Trivial Approximation Post by Eigenray on Jun 18^{th}, 2008, 12:48am on 06/17/08 at 12:02:03, Sir Col wrote:
If x is a positive integer below 245, then [hide]yes[/hide]. For some larger values, [hide]no[/hide]. I'm not sure about the rest though. E.g., given x and r, can we always find a bound on k for which [hide]x*10^{2k}  r is a square[/hide]? 

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