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