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approximation by rationals

  • To: mathgroup at
  • Subject: [mg23487] approximation by rationals
  • From: "Matt Herman" <Henayni at>
  • Date: Fri, 12 May 2000 22:54:26 -0400 (EDT)
  • Sender: owner-wri-mathgroup at


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.


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