A number is called irrational when it can't be expressed as a ratio of integers (equivalently, the decimal expansion is not periodic). The ancient Greeks proved, for instance, that the square root of 2 is irrational. The Greek geometers also were familiar with the number

but they never found a proof that π is irrational. Of course, you might think it is obvious π should be irrational. But how do you really prove it? Come to the first meeting of the UConn math club this year and find out!

To prove π is irrational, it turns out that the main tool is not geometry, but calculus. The method we use, which involves some definite integrals, will look very mysterious. To put the idea of that proof in perspective, we will also use definite integrals to prove the irrationality of e and its fractional powers (such as e^1/2 and e^7/3, but excluding the single case of e⁰=1). If time permits, we will see how the irrationality proof for powers of e is related to the irrationality proof for π through the use of complex numbers.