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Sir Col
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 Trivial Approximation   « on: Jun 17th, 2008, 12:02pm » Quote Modify

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.)
 « Last Edit: Jun 17th, 2008, 1:04pm by Sir Col » IP Logged

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Hippo
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 Re: Trivial Approximation   « Reply #1 on: Jun 17th, 2008, 12:48pm » Quote Modify

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.
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Eigenray
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 Re: Trivial Approximation   « Reply #2 on: Jun 18th, 2008, 12:48am » Quote Modify

on Jun 17th, 2008, 12:02pm, 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 yes.  For some larger values, no.  I'm not sure about the rest though.  E.g., given x and r, can we always find a bound on k for which x*102k - r is a square?
 « Last Edit: Jun 18th, 2008, 1:13am by Eigenray » IP Logged
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