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-   -   Way to transcendence (http://mymathforum.com/number-theory/343767-way-transcendence.html)

Loren March 28th, 2018 01:40 PM

Way to transcendence
 
Can both infinite, rational terms of series -- and finite, irrational coefficients of polynomials -- generate and eventually identify transcendental numbers?

Adam Ledger March 28th, 2018 03:45 PM

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.

Maschke March 28th, 2018 04:06 PM

Quote:

Originally Posted by Adam Ledger (Post 590916)
This file size is too large for the limit on this forum ....

Fermat tried that dodge in 1637.

Adam Ledger March 28th, 2018 05:54 PM

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

Loren March 28th, 2018 08:52 PM

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

Loren March 28th, 2018 08:57 PM

Baker's theorem deserves more contemplation.

Maschke March 28th, 2018 10:13 PM

Quote:

Originally Posted by Loren (Post 590939)

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.

Adam Ledger March 29th, 2018 10:50 AM

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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