Diophantine Approximation

The number $\pi$ has some very interesting properties when considered relative to rational numbers. To 15 decimal places, $\pi$ is given by

3.141592653589793...

For simple calculations, it is widely known that 22/7 = 3.142857... is a good approximation of $\pi$, valid to 2 decimal places. It is also true that 355/113=3.14159292... is accurate to 6 decimal places. For a relatively small denominator 113, we obtain accuracy up to a large number of decimal places. This kind of consideration is an example of the problem of Diophantine approximation: how close can irrational numbers be approximated by rational numbers. For instance, we may ask the question: Given an irrational number $\theta$ and a positive integer N, are there a constant C>0 and infinitely rational numbers p/q such that

$$\vert\theta - {p\over q}\vert \leq {C\over q^N} .$$

If N is very large, this inequality would say that p/q is very close to $\theta$ relative to the size of the denominator q. The subject of Diophantine approximation and continued fractions is concerned with finding these good rational approximations.