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:The true root will lie between <math>x_n \!</math> and <math>\frac{S}{x_n}</math>. So when their absolute difference is less than the required value, then <math>x_n \!</math> (or better still <math>x_{n+1} \!</math>) is close enough. [[User:JRSpriggs|JRSpriggs]] 20:45, 30 November 2007 (UTC)
== Pell's equation ==
In this section, it's stated that:
*In either case, <math>\frac{p_n}{q_n}</math> is a rational approximation satisfying
:<math>\left|\frac{p_n}{q_n} - \sqrt{S}\right| < \frac{1}{q_n^2 \cdot \sqrt{S}}.</math>
In article [[Liouville number]] it's said that:
* In [[number theory]], a '''Liouville number''' is a [[real number]] ''x'' with the property that, for any positive [[integer]] ''n'', there exist integers ''p'' and ''q'' with ''q'' > 1 and such that
:<math>0< \vert x- \frac{p}{q} \vert < \frac{1}{q^{n}}. </math>
I think it should be stated that <math>\sqrt{S}\,</math> is '''not''' a Liouville number, because it satisfies that property for ''n = 2'' but not for all ''n'' (does it fail for ''n = 3''?). [[User:Albmont|Albmont]] ([[User talk:Albmont|talk]]) 18:29, 5 August 2008 (UTC)
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