Diary for 01:090:101:21, Experimental Math

Second meeting, September 15

I'll describe one traditional way, very old, to write a formula for 1+2+3+...+n when n is a positive integer. And then, in front of the astonished (well, maybe) class, I will attempt to show how such a formula could be discovered and proved using a tool like Maple. The discovery part is interesting and fun, but the word proved is one which takes some understanding and belief, and, to me, proving such a result with a machine is not at all obvious. I do not mean computing the sum of the first 276 integers (that's 38,226) or the first 10,017 integers (that's 50,175,153) but an effort to get an algebraic method which shortcuts the whole summation and allows the sum to be computed briefly and efficiently. And this algebraic method will be valid for any positive integer n. So we will need to use Maple carefully. I should then hand out a discussion of the some of the commands used, maybe without the errors I'll probably commit in class.

Student challenge; student work
I'd like students to "discover" in a similar way a formula for the sum of the first n squares. That is, find some simpler (to write, and to compute!) expression for 12+22+32+...+n2. I'll try to walk around the room and "facilitate" your explorations. I will strongly urge you to work together and to discuss what you are doing. In fact, it may be useful for students to work in pairs.

Depending on the pace of the class, there may be time for a student (or a pair of students) to show the whole class a discovered solution. I think that would be neat.

Further historical background
Formulas for the sums of integers, and squares of integers, etc. have been known for a long time. Besides their obvious interest as sheer curiosities, these formulas can be used to compute certain quantities with the ideas of calculus. Historically, the first systematic listing of these formulas seems to be due to Johann Faulhaber about three and a half centuries ago. Faulhaber found formulas for the sums of powers of integers when the power ranged from 1 to 25. Remember, this is all a long time ago, and he must have done it by hand. A biography of him (Johann Faulhaber, 1580-1635) declares

He gives the formulae in the form of a secret code, which was common practice at the time. ... Faulhaber had the correct formulae up to k=23, but his formulae for k=24 and k=25 appear to be wrong.
The person who verified this statement, and who had to "decode" Faulhaber's concealed formulas, was Donald Knuth. It is nice to refer to a living person, and Donald Knuth, 1938-- is probably the world's most eminent computer scientist. His original training was in mathematics, but, perhaps in spite of that, his writing is very informative and can be quite entertaining. Much of Knuth's 1993 paper, Johann Faulhaber and sums of powers, could probably be understood by most of the students in this class. It has the following paragraph:
Faulhaber's cryptomath. Mathematicians of Faulhaber's day tended to conceal their methods and hide results in secret code. Faulhaber ends his book [2] with a curious exercise of this kind, evidently intended to prove to posterity that he had in fact computed the formulas for sums of powers as far as ∑n25 although he published the results only up to ∑n17.
In much scientific and technical work, the ∑ symbol is used as an abbreviation for summation, that is, for adding up things. Knuth discusses both the formulas and the encryption scheme and later concludes (rather sadly, I think), "Therefore we have no evidence that Faulhaber's calculations beyond ∑n23 were reliable." The computations Faulhaber would have done, by hand, are difficult and involve rather large integers. Any participant in today's class can use the techniques introduced today together with Maple to verify a sum formula for 25th powers and this formula would have integers like 3,606,964,705 and 1,181,820,455 (really!).

More history, including names of formulas
Probably the type of formulas discussed here should use Faulhaber's name, but they don't. They frequently are called Bernoulli formulas after some member(s) of a 17th century Swiss family of mathematicians, probably Jacob Bernoulli, 1654-1705. The formulas can be written in terms of what are called the Bernoulli numbers. Much more than almost anyone would want to know about these numbers is here, with a briefer Wikipedia article here.

The Wikipedia article contains an assertion which I've also seen elsewhere and is a bit remarkable. The case can be made that the first computer program was written to compute Bernoulli numbers, and therefore to get the summation formulas discussed in this class. Augusta Ada King, Countess of Lovelace (!), 1815-1852, was the daughter of Lord Byron, an English poet. See here for another biography. She became acquainted with Charles Babbage, 1791-1871. Babbage invented and supervised the construction of the Difference Engine, an early digital computational device, and planned an Analytical Engine which had many logical features in common with the stored program digital computers first built 75 years later. The latter's design was partially based on complicated Jacquard looming machines! Countess Lovelace wrote essays about these machines and described how to compute Bernoulli numbers with such a machine, a rather remarkable achievement for that time. She died of cancer at age 37. I've always thought she was an inspiration for the marvelously talented young English woman mathematician in Tom Stoppard's wonderful play, Arcadia.

The next homework assignment
This handout has some advice about how to discover and verify some summation formulas more briefly than what we've already done. There are no great improvements, but rather a desire to use some of Maple's facilities more so that there is less need for human intervention (and so less likelihood for human error!).
Read about what I'd like you to do. There are seven homework questions and seven solution teams. Please find the other people on your team and solve your question together and get the answer to me next time. And maybe send e-mail telling me if you want to present your solution.

First meeting, September 8

I will introduce myself, and then discuss the idea of mathematical creation (perhaps contrasting it with, say, poetic or divine inspiration) and the reality of trying lots and lots of examples.
Students in the course could have a voluntary sort of "term paper" assignment, which if completed will make them immortal (sort of). I'll try to explain this later. And students will be able to help ME with my research program. This is not likely to be an income-producing opportunity.
The major goal of this meeting is to begin students' familiarization with Maple. This program will be our primary tool for experimentation. Seven pages were distributed.

Links to other material mentioned in this lecture

  • Bernhard Riemann lived in the nineteenth century and created a large amount of really significant math. An unwary student can suppose that this was all done with magnificent intuition or inspiration or ... A long-standing problem associated with Riemann, a problem which is certainly one of the outstanding targets of current math research, concerns the Riemann zeta function. Riemann conjectured (means: "guessed", but "conjectured" is a neater word used in academic communities) some results about where this function equals 0. Consequences of this conjecture, if true, would be facts about prime numbers (primality testing, cryptography, etc.) and even facts about some models in theoretical physics. I always thought the conjecture was done with pure thought. Only a few years ago, deep in the mass of material left after Riemann's death, a huge number of numerical computations were discovered, and these computations apparently were important in his discovery of the conjecture. Some references:
    Biography of Bernhard Riemann, 1826-1866 He was born in Germany and died in Italy at age 40 of tuberculosis.
    How to make a million bucks!. The link is to a collection of math questions. Any person solving one of them would become very famous and rapidly rich. Here is very specific (complicated) information about the Riemann hypothesis, which is one of these problems.

  • Srinivasa Ramanujan lived a bit later than Riemann. He is the ideal example of a (seemingly) naive person from way outside the mainstream of mathematics whose genius permitted him to make fantastic contributions. Again, his life looks like fiction (there's actually been a recent novel about him). Many of his notebooks were found after his death (the last one, the lost notebook was discovered as recently as 1976). These notebooks contain a collection of intricate formulas and statements which were verified after great effort by others. Ramanujan also made incredible experiments in computation. In his society, paper was expensive, so very little evidence survives. But there is ample testimony to his sitting and making many computations on a slate, hours and hours of work every day! He is probably the patron saint (pardon me if this phrasing is offensive!) of experimental math. Some references:
    Biography of Srinivasa Ramanujan, 1887-1920 He was born and died in India at age 33, probably due to the combined effects of tuberculosis, dysentery, and hepatic amoebiasis. Ken Ono's article about a visit to India includes some information about Ramanujan's life and his daily hours of experimentation.

  • A general experimental math website. This page has many links to courses and activities at other universities and research centers.

  • Rutgers is a major research university. It has an active experimental mathematics community. I'll try to get a few of these people to come and talk to us about their work. One aspect of this community is an experimental math seminar. In general, the talks in this seminar are expositions of advanced scholarly work, and I would not expect many of our students to understand them. But you could always attend, sit in back, and then do your own work (I do that frequently!). On occasion, you may enjoy and understand a significant amount of a presentation. The web page of the seminar has brief descriptions of the talk. And you can discuss with me how understandable a talk might be!

    I handed out a request for student information. During class students worked on the pages below. At the end of class, I handed out the first homework assignment.

    Maple handouts
    Page 0:
    Goals of showing Maple to students;
    GUI compared to command line interface
    Page I: playing with arithmetic Page II: playing with algebra Page III: playing with calculus Page IV: playing with graphs

    Maintained by greenfie@math.rutgers.edu and last modified 9/3/2008.