Credits and References

The Inverse Symbolic Calculator was a project at the Centre for Experimental & Constructive Mathematics . It is based on independent earlier work by Peter and Jonathan Borwein and by Simon Plouffe. The project was conducted by Simon Plouffe with the help of Adam Van Tuyl (programming interface) and Paul Irvine as an Analyst/Programmer (and responsible for a great deal of the interface), Loki Jörgenson, research manager at CECM, David H. Bailey at NASA Ames Research Center, Peter Borwein, Associate Director of the CECM, Jonathan Borwein, Director of the CECM, and Yasumasa Kanada, University of Tokyo.

The first table of mathematical constants table is very old but we could say that the closest that first existed and resembles to what we do here is the Potter and Robinson table done in 1971. It was the first collection of math constants sorted in numerical order. It began (for me) in April of 1986. I was collecting articles from Mathematics of Computation, and was patiently entering the numbers into a computer database of real numbers. I entered all the tables I could find at that time. My goal was to make it as complete as possible in order to conduct some experiments with these numbers. I persevered for about 2 years, and then I communicated with Walter Gautschi and Daniel Shanks, editors of the `Unpublished Mathematical Tables' or UMT. I realized then that most of these tables could in fact be generated with a micro-computer and that it would have been difficult to automate the task of entering them in the database. Thus, I began to generate my own tables of real numbers and to collect as many articles I could find on that subject. Then in 1989, I obtained a copy of Jonathan and Peter Borwein's book A Dictionary of Real Numbers. I was truly encouraged by their efforts and was elated to learn that I was not alone in being interested in the question of identifying a real number. At that time I had about the same amount of entries as they did and was naively trying to make a system.

I continued to collect numbers until the Encyclopedia of Integer Sequences project intervened. It occupied most of my time, but it was finally released this year (Academic Press, April 1995) after 4 years of work with Neil J.A. Sloane. Needless to say, it was a fantastic experience. We (with the great help of François Bergeron of Université du Québec à Montréal, Bruno Salvy and Paul Zimmermann, both at INRIA in France), created the program GFUN, a Maple package for the manipulation of power series. Their contributions were crucial - they wrote most of the final version. That program could do things that were somewhat unusual, among them, it could guess a generating function from the first few terms of a series by using an algorithm of Padé/Cabay/Choi/Geddes for rational fractions (see 'convert/ratpoly' in MapleV and Journal of Experimental Mathematics, vol. 1, #3 (1992)). This very program (and a lot of others by Neil Sloane and his colleagues) is used presently in the sequence server at the Bell Labs (see Electronic Journal of Combinatorics, Vol 1, #1). Here is the link... Sequence server

All this created a need to jump to the other side, that is: The Real Numbers. The question is simple: if we can from a few terms of a series, using a good algorithm and some heuristic, guess what the generating function is, then, could that method perhaps be used for real numbers, that is, from a 'string' of digits, can we guess where the number comes from? This idea was milling around in the minds of Peter and Jon Borwein for a long time, and when we met (by mail first, mostly by accident (see Additions Theorems and Binary Expansions ), and then in person in January of this year) we realized that it would be very useful to have a system that could deal with this INVERSE problem. We quickly came up with a simple idea: a giant calculator with a very large display and ONE button. The button would do only one thing, answering the question "What is the number made of?" David H. Bailey helped with his programs of multiple precision MPFUN and PSLQ, a very fast implementation of the lattice algorithm.

Simon Plouffe

Research Associate at the CECM