Day 11, Mat194: Mathematical Sciences Seminar

Today:

• Announcements:
  • This week we have our last "spring break" days, and I hope that you will be able to enjoy some relaxation and recreation as we gear up for one more month of classes.

  • The Kentucky MAA (Mathematical Association of America) section were this past weekend. Some of you attended, and some of you did not.

    In any event, your short papers summarizing KYMAA talks are due tonight. Recall that I asked you for "thoughtful, one-to-two page summary of what you learned during each talk. Include a summary of what the presenter discussed, and your reactions to their presentation as well as the material."

  • We need to start preparing for our autobiographical mathematical histories, so you might start thinking about that. Review the details in the syllabus, and ask any questions.

• This week:
  • The Kentucky MAA (Mathematical Association of America) section meetings are a wonderful time for me. They provide an opportunity to get away from the routine for a weekend; mingle with mathematician, statistician, and otherician friends; and see some interesting talks and meet new people.

  • This year, although it was different in many ways, I still found much inspiration. One of the talks, by Ken Ono, concerned the life of Ramanujan (1887-1920; a great Indian mathematician, who died almost 100 years ago, too young).

    In Ken's talk I learned something very interesting: that \[ 1 + 2 + 3 + 4 + \ldots = -\frac{1}{12} \] This result was discovered by Ramanujan, but had been postulated earlier by other great mathematicians (including Riemann). So that's how they did math in the old days...:)

    Of course they may have been doing things in somewhat different fashion than we imagine; this is related to Grandi's series (born in 1671): \[ A = 1 - 1 + 1 - 1 + 1 - 1 + \ldots \] Of course in our calculus classes we say that "the series is divergent." But evidently that didn't stop Ramanujan....

    Your classmate Caeser Bao, in one of our discussions, used the Grandi series to prove that $0 = 1$:

    			0=1
    			0=0+0+0+0+0+0+...
    			0=(1-1)+(1-1)+(1-1)+(1-1)+(1-1)+(1-1)+...
    			0=1+(-1+1)+(-1+1)+(-1+1)+(-1+1)+(-1+1)+...
    			0=1+0+0+0+0+0+0+0+0+0+0+0+0+0+0+0...
    			0=1
    			

    I was quite surprised, of course, because I always thought that $0=2$:

    			0=2
    			0=0+0+0+0+0+0+...
    			0=(1-1)+(1-1)+(1-1)+(1-1)+(1-1)+(1-1)+...
    			0=1+(-1+1)+(-1+1)+(-1+1)+(-1+1)+(-1+1)+...
    			0=1+(1-1)+(1-1)+(1-1)+(1-1)+(1-1)+...
    			0=1+1+(-1+1)+(-1+1)+(-1+1)+(-1+1)+(-1+1)+...
    			0=1+1+0+0+0+0+0+0+0+0+0+0+0+0+0+0...
    			0=2
    			

    Clearly things are not what they seem, when it comes to the infinite.

    I had always considered these sorts of things "mathematical jokes", but it turns out that they can be deadly serious; even the stuff of modern day String Theory.

  • This week:
    1. Read:

       • the article The Ramanujan Summation: $1 + 2 + 3 + \ldots = -1/12$ (here's a local pdf); and

       • this article in the New York Times: In the End, It All Adds Up to $-1/12$ (local copy).

    2. Then watch one or both of these Numberphile videos, which try to fill in some of the mathematical detail (the second is a little more advanced than the first -- they're both good):
       • Why -1/12 is a gold nugget

       • Riemann Hypothesis, about the "zeta function": \[ \zeta(s)=\frac{1}{1^s}+\frac{1}{2^s}+\frac{1}{3^s}+\frac{1}{4^s}+\ldots \]

    3. Then participate in the on-line discussion on Canvas.

• Links:
  • Nothing to report.