Experimental mathematics: Computing power leads to insights

Experimental mathematics
Mathematicians often work with matrices, which are arrays of numbers. When written on a page, a matrix can look like a sea of numbers, so any patterns that might occur in the numbers can be difficult to discern. More and more, mathematicians are turning to graphical representations of matrices, like the two examples here. By using color and form to indicate the values of the numbers in the matrix, these graphical representations can instantly give a sense of the patterns in the matrix. The first picture is a representation of a matrix in which the numbers exhibit a clear pattern; the second picture, by contrast, is a matrix in which the numbers are random. Graphic by David Bailey and Jonathan Borwein. Request their permission before reproducing the graphic.

In his 1989 book "The Emperor's New Mind", Roger Penrose commented on the limitations on human knowledge with a striking example: He conjectured that we would most likely never know whether a string of 10 consecutive 7s appears in the digital expansion of the number pi. Just 8 years later, Yasumasa Kanada used a computer to find exactly that string, starting at the 22869046249th digit of pi. Penrose was certainly not alone in his inability to foresee the tremendous power that computers would soon possess. Many mathematical phenomena that not so long ago seemed shrouded and unknowable, can now be brought into the light, with tremendous precision.

In their article "Exploratory Experimentation and Computation", to appear in the November 2011 issue of the Notices of the American Mathematical Society, David H. Bailey and Jonathan M. Borwein describe how modern has vastly expanded our ability to discover new mathematical results. "By computing mathematical expressions to very high precision, the computer can discover completely unexpected relationships and formulas," says Bailey.

Mathematics, the Science of Patterns

A common is that mathematicians' work consists entirely of calculations. If that were true, computers would have replaced mathematicians long ago. What mathematicians actually do is to discover and investigate patterns---patterns that arise in numbers, in abstract shapes, in transformations between different mathematical objects, and so on. Studying such patterns requires subtle and sophisticated tools, and, until now, a computer was either too blunt an instrument, or insufficiently powerful, to be of much use in mathematics. But at the same time, the field of mathematics grew and deepened so much that today some questions appear to require additional capabilities beyond the .

"There is a growing consensus that human minds are fundamentally not very good at mathematics, and must be trained," says Bailey. "Given this fact, the computer can be seen as a perfect complement to humans---we can intuit but not reliably calculate or manipulate; computers are not yet very good at intuition, but are great at calculations and manipulations."

Although mathematics is said to be a "deductive science", mathematicians have always used exploration, whether through calculations or pictures, to test ideas and gain intuition, in much the same way that researchers in inductive sciences carry out experiments. Today, this inductive aspect of mathematics has grown through the use of computers, which have vastly increased the amount and type of exploration that can be done. Computers are of course used to ease the burden of lengthy calculations, but they are also used for visualizing mathematical objects, discovering new relationships between such objects, and testing (and especially falsifying) conjectures. A mathematician might also use a computer to explore a result to see whether it is worthwhile to attempt a proof. If it is, then sometimes the computer can give hints about how the proof might proceed. Bailey and Borwein use the term "experimental mathematics" to describe these kinds of uses of the computer in mathematics.

Exploring Prime Numbers via Computers

Their article gives several examples of experimental mathematics; the computations of the digits of pi mentioned above is one of them. Another example is provided by computer explorations of a mathematical problem known as Giuga's Conjecture. This conjecture proposes that, for any positive integer n, one can check definitively whether n is prime by calculating a certain sum in which n appears in the exponent of the summands. That sum would have a certain value, call it S, if and only if n is prime; stated differently, that sum would not have the value S if and only if n is composite. Although the conjecture dates to 1950, it has never been proved and seems out of reach by conventional mathematical methods.

However, Bailey and Borwein, along with their collaborators, were able to use computers to show that any number that is an exception to Giuga's Conjecture must have more than 3,678 distinct prime factors and be more than 17,168 decimal digits long. That is, any shorter composite number cannot result in the value S. This does not prove Giuga's Conjecture is true, but it is a compelling piece of evidence in favor of the conjecture's truth. This kind of empirical evidence is sometimes just what is needed to generate enough confidence for a mathematician to dedicate energy to seeking a full proof. Without such confidence, the inspiration to push through to a proof might not be there.

Impact on Education

In addition to discussing state-of-the-art uses of computers in mathematics, the article also touches on the need to refashion mathematics education to give students the tools of experimental mathematics. "The students of today live, as we do, in an information-rich, judgment-poor world in which the explosion of information, and of tools, is not going to diminish," says Borwein. "So we have to teach judgment (not just concern with plagiarism) when it comes to using what is already possible digitally. Additionally, it seems to me critical that we mesh our software design---and our teaching style more generally---with our growing understanding of our cognitive strengths and limitations as a species."

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Oct 13, 2011
As somebody who has had problems with visualizing math applications, I have to say I like this very much. Staring at a page full of expressions and equations can be like trying to read a bowl of soup. It seems to be in the same mindset of Feynman and Penrose's work with inventing " translational " diagrams.

When I was in art school I used to try to visualize equations and expressions overlaying what I was drawing, to try and help myself understand maths through physical examples.

Apples, oranges, Bob and Alice just don't seem to cut it anymore.

Oct 13, 2011
Crap, I erased what I wrote.

Penrose was thinking about this from the wrong perspective.

A sequence of 10 7's appears in pretty much any irrational number, if you solve it to enough digits.

If you have an infinite number of digits in a non-repeating string, then there are an infinite number of sub-strings, of an infinite number of lengths, and all possible sub-strings will eventually appear at least one time.

If this were not true, then the number would not be irrational. If the main string EVER starts to repeat itself exactly, then you could find a ratio of real numbers to represent it, and it wouldn't be rational.



Is not irrational.

It takes several digits before it starts to repeat, but then it repeats entirely, exactly.

This is 3/13.

So you can say with absolute certainty that there exists a string of 10, 11, 50, and 100, etc, consecutive 7's in the decimals of pi.

Oct 13, 2011
I saw a proof similar to this involving the "apocalyptic powers" in a book by Cliffor Pickover, which was not exactly the same problem, but it dealt with the notion of "the first power of 2 which contains the 3 consecutive 6's '666'."

then he did "double apocalyptic powers" and "triple apocalyptic powers" and so on.

Basicly, he proved that any number of consecutive digits substring can be found in some power of 2 if it is large enough.

Although you can use the computer to find specific solutions to such searches, the computer is not necessary to prove the assertion correct.

You can prove it correct intuitively by simply realizing that every combination of digits will eventually appear for some power of 2...

Oct 13, 2011
That's not necessarily true, Nanobanano. In order for PI to have that property, it must be a normal number. While it is true that almost all real numbers are normal, to my knowledge nobody has proven that PI has this property (though it is likely).
Keep in mind that many, if not most, well defined numbers are not normal. PI has a very simple definition, and there exist algorithms to compute the nth digit, so it is well defined. So, on the one hand, while evidence is overwhelming that PI is normal based on empirical calculations, we also have overwhelming evidence that most well defined numbers aren't normal. Therefore, I would advise against saying things with certainty about whether certain sequences appear in PI, unless you have theoretical motivation for doing so.

Oct 13, 2011
The quantum effect in the "real" world... Absolutely NOTHING is exact. Wait - even THAT may not be exact...

Oct 13, 2011
We use numbers to make the real world fit into our minds. Have you ever noticed that a simple proof can be harder than it initially seemed at a glance. I have never seen a computer that can tell the difference other than ones or zeros; maybe this is where the difference is. I always thought that high end language could do that job though. I bet python would be quick to pick up. yes, just run some python scripts... ?Surely, you could write that kind of logic in code.. and then run the scripts on host computers participating in a folding project.

Oct 13, 2011
Local campuses could implement a folding project on all computers using WiFi. Professors would have full access to a dedicated interface for the computing system. Of course a basic set of tools should be provided to the end-user. A user interface with an automatic script generator would be nice along with examples on how to use the system; maybe a Microsoft-like walk through.

Oct 15, 2011
Nanobanano your proof of repeating 7s is not correct.
The easiest way for me to see that is to construct an irrational number that only has 0s and 1s in its decimal representation.
A quick way to do that is to look at any irrational number in its base 2 representation. As you point out, the 0s and 1s will not form a recurring pattern. Then just take this representation and use it as a decimal number instead. viola, an irrational number that has no 7s at all in its decimal representation.

Oct 17, 2011

I can't follow your argument at all.

Consider the decimal expansion:


(That's one zero, then two, then three and so on.)

It's irrational. It never repeats. Ten sevens never appear in it. Not even one seven appears in it.

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