When you are first introduced to the real numbers you are almost forcefully told that they can be approximated by rationals numbers. The voice in your head goes something like ‘we can get as close as we like to an irrational by using rationals’. This is normally expressed in the statement which reads is dense in where ‘dense’ is a very precise mathematical statement.

It’s quite easy to think we are done here, but what invokes a good approximation? Dirichlet’s approximation is not only good in the sense you can control how close you want your rational approximation to be but it gives a splitting, in terms of behavior, with respect to rational and irrational numbers.

**Theorem: **For we can find infinitely where such that . Moreover, if only finitely many approximations exist.

**Proof of theorem**: We are fist going to introduce some notation. let the integral part of and the fractional part of will be denoted and defined as and respectively.

Let be irrational and be a positive integer. The numbers gives us points between and that are distributed among the intervals for . Applying the pigeon hole principle means there has to be at least one interval containing at least two numbers and such that with and .

Computing the difference of and gives us which lies in the interval meaning that as the LHS is an integer. Setting and gives the property and so computing the difference between and gives . The last inequality follows from .

To see the split in behaviour we first assume is irrational and that there are finitely many approximations . Since is irrational by the fact $latex \overline{\mathbb{Q}} = \mathbb{R}$ we can fine (a positive integer) such that for all we have which contradicts the established facts above.

In the second case when is a rational number, say with and if then by direct computation so we can only have finitely many such approximations as we require .