approximation by rationals

*To*: mathgroup at smc.vnet.net*Subject*: [mg23487] approximation by rationals*From*: "Matt Herman" <Henayni at hotmail.com>*Date*: Fri, 12 May 2000 22:54:26 -0400 (EDT)*Sender*: owner-wri-mathgroup at wolfram.com

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. Matt

**Follow-Ups**:**Re: approximation by rationals***From:*Ken Levasseur <Kenneth_Levasseur@uml.edu>