Experimental mathematics

Experimental mathematics is a special way of studying math where computers help us explore numbers, shapes, and patterns. Instead of just writing proofs on paper, mathematicians use computers to test ideas and discover new facts. This approach helps them make guesses, called conjectures, about how math works.

Many important math ideas started with experiments. For example, mathematicians might use a computer to check thousands of examples to see if a pattern always holds true. This can lead to new theories and deeper understanding.

Famous mathematician Paul Halmos said that math isn’t just about proving things — it also involves trying things out, making guesses, and learning from what happens. Just like scientists in a lab, math explorers use experiments to find out what is true. This fun and creative part of math helps everyone, even young students, see that math can be full of discovery and surprise.

History

Mathematicians have always used experiments to explore numbers and patterns. Early records, like those from Babylonian mathematics, show lists of numbers that helped them understand math better. But as math grew more complex, mathematicians began sharing only their final, formal results, often forgetting the early examples that inspired them.

Experimental math became more important in the twentieth century with the invention of computers. These powerful tools let mathematicians perform calculations faster and more precisely than ever before. One famous example is the discovery of the Bailey–Borwein–Plouffe formula in 1995, which helps find the binary digits of π. This discovery was made by running computer searches, and only later was a proper proof created.

Objectives and uses

The goals of experimental mathematics are to help people understand and enjoy math more, to test ideas, and to find new patterns. This kind of math uses computers to explore and check guesses, making complex topics easier to grasp for everyone, from experts to beginners.

Experimental math can help discover new connections, show hidden principles through graphs, and decide if a result is worth proving with traditional methods. It also makes solving problems faster by using computers instead of long hand calculations, and it can confirm results that were found through careful analysis.

Tools and techniques

Experimental mathematics uses numerical methods to find approximate values for integrals and infinite series. Arbitrary precision arithmetic helps calculate these values very accurately, often with more than 100 digits. This precision helps scientists tell the difference between a random pattern and a real mathematical connection.

When looking for exceptions to a rule or testing many cases, distributed computing can split the work across several computers. Scientists also use mathematical software with built-in checks to make sure the results are correct and not affected by computer errors.

Applications and examples

Experimental mathematics helps mathematicians explore ideas using computers. For example, it can be used to look for special numbers or patterns that might solve big math questions. Projects like the Great Internet Mersenne Prime Search aim to find new Mersenne primes, which are rare and interesting numbers.

Scientists also use computers to test theories and find surprising patterns. For instance, Edward Lorenz discovered a special pattern called the Lorenz attractor while studying weather models. These explorations often lead to new discoveries and deeper math proofs.

Plausible but false examples

Main article: mathematical coincidence

Sometimes, math looks like it follows a pattern, but it doesn’t quite work out perfectly. One famous example is a complex math problem that seems to equal π⁄8, but when you check really far out, the numbers start to differ.

Another example involves special kinds of math expressions called cyclotomic polynomials. For numbers up to 10,000, these expressions seemed to follow a certain rule. But when mathematicians checked even larger numbers, they found one—14,235—where the rule didn’t hold true anymore.

Practitioners

Many mathematicians and computer scientists have helped shape the field of experimental mathematics. Some of the key figures include Fabrice Bellard, David H. Bailey, Jonathan Borwein, David Epstein, Helaman Ferguson, Ronald Graham, Thomas Callister Hales, Donald Knuth, Clement Lam, Oren Patashnik, Simon Plouffe, Eric Weisstein, Stephen Wolfram, Doron Zeilberger, and A.J. Han Vinck. Their work has shown how using computers can help explore and understand complex mathematical ideas.