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Irrationality of e+pi and e*pi

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Date: 09/24/2001 at 00:38:32
From: Andy Weck
Subject: Irrationality of e+pi and e*pi

I have read that it is unknown if either E + Pi or E * Pi is an
irrational number. However, it is provable that at most one of the
two numbers is rational. How do you prove this? I thought that the
proof would involve summing the two numbers in some manner and then
proving that the result is irrational. (This way you know at least one
of the addends must be irrational.) But I can't figure out how to
prove that any combination of the two numbers is irrational.
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Date: 09/24/2001 at 10:59:22
From: Doctor Luis
Subject: Re: Irrationality of e+pi and e*pi

Hi Andy,

Thanks for the very interesting question. Like you, I was unable to
find a combination e+pi and e*pi that proved immediately the
irrationality of either number. There's hardly anything that can
easily be proved by using that approach. However, given further
knowledge about both both pi and e, your claim can be proved rather
elegantly.

Now, e and pi are rather peculiar numbers. It turns out that, in
addition to being irrational numbers, they are also transcendental
numbers. Basically, a number is transcendental if there are no
polynomials with rational coefficients that have that number as a
root.

Clearly, p(x) = (x-e)*(x-pi) is a polynomial whose roots are e and pi,
so its coefficients cannot all be rational, by the definition of
transcendental numbers. Expanding that expression, we get

(x-e)*(x-pi) = x^2 - (e+pi)*x + e*pi

This means that 1, -(e+pi), e*pi cannot all be rational. If all the
coefficients were rational, we would have found a polynomial with
rational coefficients that had e and pi as roots, and that has been
proven impossible already. Hermite proved that e is transcendental in
1873, and Lindemann proved that pi is transcendental in 1882. In fact,
Lindemann's proof was similar to Hermite's proof and was based on the
fact that e is also transcendental.

In other words, at most one of e+pi and e*pi is rational. (We know
that they cannot both be rational, so that's the most we can say).

Great question. I hope this explanation helped!

- Doctor Luis, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
College Number Theory
High School Number Theory

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