Transcendental numbers: interesting perspectives

[loom91]

Hi,

I'm presenting a proof that e is transcendental, but I want to spice up the presentation a bit by exploring some interesting perspectives, such as connections to topology, abstract algebra etc. I've already selected the Lindemann–Weierstrass theorem as a possible candidate. Can you suggest some other avenues? This is a first-year UG class, so the concepts used cannot be so technical as to be unexplainable in a timeframe of an hour. Thanks!

Molu

 

[Mark44]

Something that's stuck with me for a long time is this equation:
$e^{i\pi} + 1 = 0$
In it appear five of the most important symbols in mathematics. It might strike some of them as very odd that e and $\pi$ cannot be represented with a finite number of decimal digits, and that i is as unreal as a number can get, yet all three combine to make -1.

Maybe some of the less jaded among these students will appreciate this.

 

[loom91]

Ok, we could talk about that, but that's rather too basic for a 2nd sem undergrad class. Most of the students have already encountered that material in high-school, they would appreciate something slightly more challenging.

 

[loom91]

Anything else?

 

[Dick]

Liouville numbers (the first historic transcendentals), irrationality measures? Why does e have irrational measure 2 like algebraic numbers? Continued fraction expansions? Stuff like that?

 

[loom91]

Ah, that's what I was looking for. Though irrationality measure seems a very difficult thing to wrap your mind around, so I don't know if we can effectively explain that in a short time period, but let's see. Incidentally, is there any particularly intuitive reason for e having the irrationality measure 2? Thanks.

 

[Dick]

It must come from looking at the continued fraction expansion. I've never looked at it, but it might not be too hard. There's a reference in here http://planetmath.org/encyclopedia/IrrationalityMeasure.html

 

[loom91]

Any possible connections with complex analysis?