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        riddles >> medium >> Trivial Approximation
        
(Message started by: Sir Col on Jun 17^th, 2008, 12:02pm)

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/n² < |a - m/n| < 1/n.
If |a - m/n| < 1/n² then it is defined as a reasonable approximation.
(Technically |a - m/n| < 1/n³ 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... < 1/12² = 0.0069444..., and so 17/12 is considered to be a non-trivial 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 mix-up 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 sqrt(x) is rounded to k > 1 decimal places, does this always leads to a trivial approximation?

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]?