# π Is Transcendental

Teaching History of Math this past semester gave me an excuse to read carefully two Dover Publications books that I have owned since high school, but only skimmed then. Imagine my delight to discover that if you are given a theorem that is hard to prove beforehand, you can prove that $\ \pi$ is transcendental in just a couple of lines. The hard theorem gives many other corollaries too, corollaries that I’ve known in my gut but never had a handle on how to prove.

Here are the details, from p. 76 of Felix Klein’s book Famous Problems of Elementary Geometry. You can read it on-line at Google Books.

Theorem (Lindemann): Over the complex field, in the equation $\ c_0 + c_1 e^{k_1} + c_2 e^{k_2} + ... + c_n e^{k_n} ~=~ 0,$ not all of the $\ c_i$
and $\ k_i$ can be algebraic, assuming that at least one $\ k_i \neq 0.$

Corollary 1:
$\ \pi$ is transcendental.

Proof:
$\ 1 + e^{\pi i} ~=~ 0,$ and 1 is trivially algebraic, so $\ \pi i$ is transcendental, so then also is $\ \pi .$

Corollary 2: In the equation $\ y = \sin {x},$ if x is algebraic and $\ \neq 0,$ then y is transcendental. If x is a non-zero rational multiple of $\ \pi ,$ then y is algebraic.

Proof: Note first that by a power series argument, for example, $\sin x = \frac {e^{i x} - e^{- i x} } {2 i}.$ Therefore, $\ (*)~~ 2 i \sin {x} - e^{i x} + e^{-i x} ~=~ 0.$ If x is algebraic then, Lindemann’s theorem applied to (*) shows that
$\ y ~=~ \sin x$ is transcendental. If x is a rational multiple of $\ \pi ,$ then x is transcendental in a simple proof by contradiction from Corollary 1, hence y cannot be transcendental, again by Lindemann’s theorem.

Similarly, we can show that in $\ y = \ln x$ only one of x or y can be algebraic (excluding the y = 0 case, as Lindemann’s theorem does, because $\ x ~=~ e^y$). Similarly for all of the rest of the trig functions and inverse trig function, which can also be expressed as rational functions of exponentials.

And finally, what is the heart of the proof of Lindemann’s theorem in the first place? It rests on the fact that $\ e^x$ as a power series has factorials in denominators, making any attempt to make it satisfy an algebraic equation a failure.

Tags: ,