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The Heretic

Transcendence of sqrt(2)^sqrt(2)^sqrt(2)

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I read a while ago that sqrt(2)^sqrt(2)^sqrt(2) hadn't been proved transcendental yet. I was wondering if it still hadn't but I haven't been able to find much on the subject. Does anyone know/have links? [edited by - The Heretic on April 20, 2003 11:25:15 AM]

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Well if you think about it, you can never get an exact value of an irrational number. So how can you make any claim about raising an irrational number to an irrational power?

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quote:
Original post by Malone1234
Well if you think about it, you can never get an exact value of an irrational number. So how can you make any claim about raising an irrational number to an irrational power?


There are ways, for example, assuming it is algebraic and forming a contradiction.

EDIT: I found a small error in my proof.

[edited by - The Heretic on April 20, 2003 1:17:14 PM]

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Can''t you rewrite transcendental or irrational powers in terms of polar imaginary numbers? By the way, is sqrt(2) transcendental?

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quote:
Original post by vanillacoke
is sqrt(2) transcendental?


Transcendental: a real or complex number that is not the root of any polynomial that has positive degree and rational coefficients.

x^2-2

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Simply put, transcendental functions are not algebraic functions. It means that you can't express it as a finite equation with elementary operations (here and here). For example, trigometric functions and the exponential function.

quote:
sqrt(2)^sqrt(2)^sqrt(2)

That evaluates to a constant, how can it be a transcendental function?

quote:
is sqrt(2) transcendental?

Once again, that makes no sense, it's a constant!

[edited by - Zipster on April 20, 2003 5:26:44 PM]

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