Rationality problems

Since number theory was first concerned with the properties of integers and rational numbers, it became natural to ask what other kinds of numbers existed. Remember that a rational number is the quotient of two integers. Whether or not there exist other numbers (which we will call ``irrational'') is a very nontrivial question to answer. In fact, when the Greeks, whose mathematical viewpoint was primarily geometrical, first proved to themselves that the diagonal of a square was an irrational multiple of the side of the square, they were horrified that nature could allow what they considered such an imperfection. In modern language, what they showed was that $\sqrt{2}$ is not a rational number. Later on, we shall reproduce their proof. It is now a problem of profound interest and difficulty to determine whether certain numbers that occur naturally in all parts of mathematics are rational or not. It was only at the very end of the nineteenth century that it was shown that $\pi$ and e are irrational. In the 1930's, a pair of Russian mathematicians Gelfond and Schneider proved that $2^{\sqrt{2}}$ and $e^\pi$ are irrational. Actually, they proved somewhat more, which we shall describe below. It is still unknown whether $\pi^e$ or even $\pi+e$ are rational or not.