Both are irrational and in fact transcendental numbers. In this little appendix we will prove that both numbers are irrational. We will not show that they are also transcendental numbers. The proofs will actually use tools from much later sections, but they might be interesting nevertheless, and the results fit well in this section. In fact, it is a transcendental number, but that requires a lot of work to prove.
R n is also an integer.
But we know that so that if n is large enough the left side is less than 1. R n must be a positive integer less than 1, which is a contradiction. Next we prove that Pi is irrational, which is a lot harder to do. But this implies that is an integer for any k . Moreover, since we also have Therefore is an integer for any k .