Hardy and Wright in chapter on continued fractions prove the following theorem

Let $x$ be an irrational number whose simple continued fraction representation is given by $[a_0,a_1,\ldots,a_n,\ldots] \big( \text{ie.} x=a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + {_\ddots}}}} \big)$. Then $ \bigl|x-\frac{p_n}{q_n} \bigl| < \frac{1}{q_n^2} $, where $\frac{p_n}{q_n}$ is the $n^{\text{th}}$ convergent of $x$.

In the section on approximation of irrationals by rationals they say if we have an irrational say $x$ and an integer $q$ we want to know an upper bound for $\epsilon$ ( a positive number ) such that $\bigl|x-\frac{p}{q}\bigl| \le \epsilon $ ( $p$ is an integer ) ie. a relation of sorts mentioned in the above theorem.

My question is why study this sort of relationship between $q$ and $\epsilon$ at all ? Is it not good enough if I approximate an irrational, say up to say $10$ decimal places ?

  • 1
    $\begingroup$ No, it is not good enough. Have a look here. $\endgroup$ – Dietrich Burde Dec 25 '16 at 20:02

Surprisingly, studying these really "cheap"/"efficient" approximations is important when trying to solve some number theory problems, like Pell's equation.

  • $\begingroup$ I am self studying Hardy and Wright. I have not reached the chapter on Pell's equation yet. But thanks I will again look back once I reach chapter on Pell's Equations. $\endgroup$ – sashas Dec 25 '16 at 20:09

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.