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Re: approximation by rationals

• To: mathgroup at smc.vnet.net
• Subject: [mg23512] Re: approximation by rationals
• From: Mark Sofroniou <marks at wolfram.com>
• Date: Tue, 16 May 2000 02:44:58 -0400 (EDT)
• Organization: Wolfram Research Inc
• References: <8fih6t$50k@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

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.