The Transcendence of

In defining algebraic and transcendental numbers we pointed out that it could be shown that transcendental numbers exist. One way of achieving this would be the demonstration that some specific number is transcendental.

In 1851 Liouville gave a criterion that a complex number be algebraic; using this, he was able to write down a large collection of transcendental numbers. For instance, it follows from his work that the number is transcendental; here the number of zeros between successive ones goes as .

This certainly settled the question of existence. However, the question whether some given, familiar numbers were transcendental still persisted. The first success in this direction was by Hermite, who in 1873 gave a proof that is transcendental. His proof was greatly simplified by Hilbert. The proof that we shall give here is a variation, due to Hurwitz, of Hilbert's proof.

The number offered greater difficulties. These were finally overcome by Lindemann, who in 1882 produced a proof that is transcendental. One immediate consequence of this is the fact that it is impossible, by straightedge and compass, to square the circle, for such a construction would lead to an algebraic number such that . But if is algebraic then so is , in virtue of which would be algebraic, in contradiction to Lindemann's result.

In 1934, working independently, Gelfond and Schneider proved that if and are algebraic numbers and if is irrational, then is transcendental. This answered in the affirmative the question raised by Hilbert whether was transcendental.

For those interested in pursuing the subject of transcendental numbers further, we would strongly recommend the charming books by C. L. Siegel, entitled Transcendental Numbers, and by I. Niven, Irrational Numbers.

To prove that is irrational is easy; to prove that is irrational is much more difficult. Johann Lambert proved that is irrational in 1761. For a very clever and neat proof of the latter, see the paper by Niven entitled ``A simple proof that is irrational,'' Bulletin of the American Mathematical Society, Vol. 53 (1947), page 509.

Now to the transcendence of . Aside from its intrinsic interest, its proof offers us a change of pace. Up to this point all our arguments have been of an algebraic nature; now, for a short while, we return to the more familiar grounds of the calculus. The proof itself will use only elementary calculus; the deepest result needed, therefrom, will be the mean value theorem.

THEOREM 5.2.1: The number is transcendental.