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March 28th, 2018, 01:40 PM   #1
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Way to transcendence

Can both infinite, rational terms of series -- and finite, irrational coefficients of polynomials -- generate and eventually identify transcendental numbers?
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March 28th, 2018, 03:45 PM   #2
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Identifying Trancendental Numbers

ok the first link I would recommend is the work of the late Alan Baker:

https://en.wikipedia.org/wiki/Baker%27s_theorem

And a good paper by Michel Waldschmidt entitled "Transcendence of Periods:
The State of the Art" that has a lot of good theorems for deciding whether a number is transcendental or not that follow from Baker's theorem of linear independence.

This file size is too large for the limit on this forum, so if you cannot find a copy on the net, pm me and I will email you a copy.

Last edited by Adam Ledger; March 28th, 2018 at 03:48 PM. Reason: meh
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March 28th, 2018, 04:06 PM   #3
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This file size is too large for the limit on this forum ....
Fermat tried that dodge in 1637.
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March 28th, 2018, 05:54 PM   #4
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Well sometimes the old ways are the best.

But we can all rest easy now that a teenager with what appears to be a severe Adderall induced psychosis has shown us that simple elementary proof in a recent thread
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March 28th, 2018, 08:52 PM   #5
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i^i is actually a real transcendental number.

Here is how you can compute the value of i^i = 0.207879576...

1. Since e^(ix) = Cos x + i Sin x, then let x = Pi/2.
2. Then e^(iPi/2) = i = Cos Pi/2 + i Sin Pi/2; since Cos Pi/2 = Cos 90 deg. = 0. But Sin 90 = 1 and i Sin 90 deg. = (i)*(1) = i.
3. Therefore e^(iPi/2) = i.
4. Take the ith power of both sides, the right side being i^i and the left side = [e^(iPi/2)]^i = e^(-Pi/2).
5. Therefore i^i = e^(-Pi/2) = .207879576...

From Cliff Pickover, The 15 Most Famous Transcendental Numbers - Cliff Pickover
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March 28th, 2018, 08:57 PM   #6
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Baker's theorem deserves more contemplation.
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March 28th, 2018, 10:13 PM   #7
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Great page, thanks for the link. I love Chaitin's number. It's an easily understood example of a real number that can't possibly be computable.

I looked up Alan Baker, I'd never heard of him before. Won the Fields medal in 1970 for his work on transcendentals.

I was fascinated to discover that it's unproven whether $\pi^e$ is transcendental.
Thanks from Adam Ledger

Last edited by Maschke; March 28th, 2018 at 10:21 PM.
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March 29th, 2018, 10:50 AM   #8
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One great theorem that arises from Schwarz's Lemma in the work of Schneider on Abelian functions is this:

Let a and b be rational numbers which are not integers and such
that a + b is not an integer. Then B(a,b) is transcendental

https://en.wikipedia.org/wiki/Beta_function
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