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Homework Help: Transcendental numbers: interesting perspectives

  1. Jan 21, 2009 #1

    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!

  2. jcsd
  3. Jan 21, 2009 #2


    Staff: Mentor

    Something that's stuck with me for a long time is this equation:
    [itex]e^{i\pi} + 1 = 0[/itex]
    In it appear five of the most important symbols in mathematics. It might strike some of them as very odd that e and [itex]\pi[/itex] 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.
  4. Jan 24, 2009 #3
    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.
  5. Jan 26, 2009 #4
    Anything else?
  6. Jan 26, 2009 #5


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    Liouville numbers (the first historic transcendentals), irrationality measures? Why does e have irrational measure 2 like algebraic numbers? Continued fraction expansions? Stuff like that?
  7. Jan 28, 2009 #6
    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.
  8. Jan 28, 2009 #7


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    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 [Broken]
    Last edited by a moderator: May 3, 2017
  9. Feb 14, 2009 #8
    Any possible connections with complex analysis?
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