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MathGroup Archive 2000

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Re: approximation by rationals

  • To: mathgroup at smc.vnet.net
  • Subject: [mg23495] Re: [mg23487] approximation by rationals
  • From: Ken Levasseur <Kenneth_Levasseur at uml.edu>
  • Date: Sun, 14 May 2000 16:59:58 -0400 (EDT)
  • Organization: UMass Lowell
  • References: <200005130254.WAA04912@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Matt:

I'm aware of two theorems that are close to what you cite:

Hurwitz' Theorem says that among any three consecutive convergents of
irrational x, one of them, a/b, satisfies the inequality
   |x-a/b|<1/(Sqrt[5]b^2)

This implies that an infinite number, but not all, satisfy the weaker
inequality you want.

A second theorem is the converse of what you state:  If |x-a/b|<1/(2b^2)
then a/b is convergent.

Both of these theorems are in Schumer's Intro. to Number Theory, PWS,
1996 and I'm sure they are in many other basic texts.  Hope this helps.

Ken Levasseur
Math. Sci.
UMass Lowell

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.
> 
> Matt


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