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Transcendental Numbers 2

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By Michael Doré on Wednesday, July 04, 2001 - 08:14 pm:

The topic of transcendental numbers has cropped up quite a lot recently. See for example here. During the next day or so, I'm going to state three results which are useful for showing various numbers are transcendental, and will prove the first two theorems, since the proofs are quite interesting, and not impossible to understand. In particular the second theorem will prove that e and p are transcendental (and hence irrational), which is a frequently occuring theme. I'd be grateful for any feedback if there are any unclear points, or for any other reason.

Firstly, the definition.

A number x (real or complex) is algebraic if and only if there exists a polynomial P with integer coefficients such that P(x) = 0. A number which is not algebraic is said to be transcendental. If x is algebraic then the minimal polynomial of x is the polynomial P with minimal degree such that P(x) = 0.

Here are the three theorems.

1) Liouville's Theorem

If x is algebraic then there exists c > 0 such that for all integer p and natural q we have:

|x - p/q| > c/qⁿ

where n is the degree of x. (That is, the degree of the minimal polynomial P such that P(x) = 0.)

Equivalently, if the quantity |x - p/q|qⁿ can be made arbitrarily small then x is transcendental.

This theorem effectively shows if a number is very "close" to being rational (by which I mean it has very good rational approxiations) then it is transcendental.

For example, using this theorem, it is a straightforward exercise to show that the series:

1/2^1! + 1/2^2! + ...

is transcendental. The reason is because the terms of the series are decreasing in absolute value so rapidly that the series has excellent rational approximations hence Liouville's theorem shows it's transcendental.

In fact this number was the first number ever to be proved to be transcendental. I'm not sure, but I think that before they'd proved it was transcendental, most people believed all numbers were algebraic. Later Cantor stepped on the scene and showed that "almost all" numbers are transcendental.

2) Lindemann's theorem

This one shows that if a₁,...,a_n are distinct algebraic numbers and b₁,...,b_n are non-zero and algebraic then b₁exp(a₁) + ... + b_nexp(a_n) is non-zero.

Three useful facts that follow immediately from this theorem are:

a) e,p are transcendental.

b) e^x, cos x, sin x, tan x are transcendental if x is algebraic and not 0.

c) e^pix,cos(px),sin(px),tan(px) are transcendental if x is algebraic and irrational.

3) Gelfond-Schneider

If a,b are algebraic with a not equal to 0,1 and b not rational then a^b is transcendental.

I will not be proving this one as the proof is incredibly complicated, intricate and uses complex analysis. To give some idea, the proof was first obtained in the 1930s. It was one of the 20 problems Hilbert proposed in 1900.

It follows pretty quickly from this theorem that:

a) sqrt(2)^sqrt(2) is transcendental (this has been asked at least 3 times before!)

b) e^p is transcendental (to prove this, use the fact that e^pi = -1).

It is not however known whether p^e is transcendental or not (or even whether it's irrational). Nor is it known whether p + e or indeed any individual linear combination of p and e is transcendental.

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