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Re: approximation by rationals
Matt Herman wrote: > hi, > > there is a number theory theorem which states that if |z-(a/b)|< 1/(2 > b^2), then a/b is a convergent of z. If this is true, then why when I go > into mathematica and set z=E and a/b = 106/39, which is a convergent > of E, the inequality is false? Is this a problem in mathematica's > numerical evaluation, or a problem in the theorem? Is the theorem only > valid when z is a quadratic irrational? Because the theorem works for > z=Pi. Also the stronger form of the theorem is |zb-a|<1/2b^2. This is > false when z=E. Again the same question as above. The reason is that the theorem you quote is a sufficient requirement that a rational number be a convergent but it is not necessary. If a/b is a convergent to z then a somewhat weaker relation always holds: |z-(a/b)| < 1/b^2 Mark Sofroniou, Wolfram Research.