I read somewhere that
$$e^{\pi\sqrt{163}}$$
is almost an integer and strangely enough this isn't just a random coincidence but rather there exists some general theory
http://en.wikipedia.org/wiki/Heegner_number
behind the occurences of these almost integers (and their relation to other areas of number theory)
Surely there are many other strange identities such as:
$$\sqrt{2} \approx \frac{3}{5} + \frac{\pi}{7 -\pi}$$
I'm guessing that this "coincidence" is probably similar to the earlier example a special case of some general theory that relates rational expressions of pi to algebraic integers.
Can someone point me in the right direction if not explain it here itself?