# Number-Crunching: Taming Unruly Computational Problems from Mathematical Physics to Science Fiction

## Synopsis

How do technicians repair broken communications cables at the bottom of the ocean without actually seeing them? What's the likelihood of plucking a needle out of a haystack the size of the Earth? And is it possible to use computers to create a universal library of everything ever written or every photo ever taken? These are just some of the intriguing questions that best-selling popular math writer Paul Nahin tackles in Number-Crunching. Through brilliant math ideas and entertaining stories, Nahin demonstrates how odd and unusual math problems can be solved by bringing together basic physics ideas and today's powerful computers. Some of the outcomes discussed are so counterintuitive they will leave readers astonished.

Nahin looks at how the art of number-crunching has changed since the advent of computers, and how high-speed technology helps to solve fascinating conundrums such as the three-body, Monte Carlo, leapfrog, and gambler's ruin problems. Along the way, Nahin traverses topics that include algebra, trigonometry, geometry, calculus, number theory, differential equations, Fourier series, electronics, and computers in science fiction. He gives historical background for the problems presented, offers many examples and numerous challenges, supplies MATLAB codes for all the theories discussed, and includes detailed and complete solutions.

Exploring the intimate relationship between mathematics, physics, and the tremendous power of modern computers, Number-Crunching will appeal to anyone interested in understanding how these three important fields join forces to solve today's thorniest puzzles.

## Excerpt

I didn’t become a mathematician because mathematics was so full of
beautiful and difficult problems that one might waste one’s power in
pursuing them without finding the central problem.
—Albert Einstein

Mathematics is trivial, but I can’t do my work without it.
—Richard Feynman

These two views, from two of the most famous physicists of the last century, on the relationship between physics and mathematics, are quite different. Both won the Nobel Prize (Einstein in 1921 and Feynman in 1965), and their words, however different, deserve some thought. Einstein’s are ones without sting, carefully crafted to perhaps even bring a surge of pride to the practitioners of the rejected mathematics. Feynman’s, on the other hand, are just what we have come to expect from Feynman—brash, outrageous, almost over-thetop. Feynman’s comment really goes too far, in fact, and I think it was uttered as a joke, simply to get attention. Physicist Feynman was also a highly skilled mathematician, and not for a moment do I believe he really thought mathematics to be “trivial.” (Mathematicians shouldn’t take such “Feynman put-downs” too seriously; I’m an electrical engineer and while Feynman had some jibes for EEs, too, I have never let them influence my appreciation for his genius.)

The mathematician Peter Lax, in his 2007 Gibbs Lecture to the American Mathematical Society, gave a good, concise summary of the interplay between mathematics and physics. His talk opened with these words: “Mathematics and physics are different enterprises: physics is looking for laws of nature, mathematics is trying to invent the structures and prove the theorems of mathematics. Of course these structures are not invented out of thin air but are linked, among other things, to physics” (my emphasis). a few years earlier a physicist, in his own Gibbs Lecture, gave a specific illustration of this connection: “The . . .

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