Magazine article National Forum

Serendipit-E

Magazine article National Forum

Serendipit-E

Article excerpt

An Inventor and a Meteorologist Make Mathematical Discoveries

A year ago in this column I described a modest meteorological research collaboration of mine that seemed to capture the essence of everyday work in science. While I was writing those words, I was embarking on a far more unusual collaboration with an inventor in Connecticut. The two of us are not inventing a better mousetrap, or a new way to chase tornadoes. Instead, we are performing research in number theory, one of the more arcane areas of mathematics - and a subject in which neither one of us has had any formal training. Even so, our work on calculating the number e will soon be published in two mathematical journals! This tale is so improbable, a perfect "who wudda thunk it?" complement to my earlier column, that I just have to share it with you.

ALL ABOUT e

The number e is one of the universal constants of nature; it plays a fundamental role in problems ranging from the growth of fruit flies to radioactive decay. The classical (circa 1600s) formula for e is similar to the one you use to calculate how much money you have in an interestbearing account at the bank:

You can try this formula out with a calculator. If you insert 1000 for x, then the formula gives you 2.7169239322. This value is very close to the "exact" value of e, which is 2.718281828459.... But in fact, we do not know what e is exactly. This is not for lack of effort: mathematicians since Isaac Newton and Leonhard Euler have given it their best shots. As it turns out, e, like pi, is a "transcendental" number whose exact value is not quite knowable. This lack of certainty is no cause for day-to-day alarm - for example, we know e well enough to calibrate radiation exposures. Even so, the determination of classical constants to high precision is a subject of considerable interest to mathematical specialists known as "number theorists."

WE MET BY MAIL

Harlan Brothers is an inventor and musician, not a number theorist. Despite his lack of formal mathematical training, he likes to dabble in math with a pencil and pad late at night. One night he became intrigued with a particular sequence of numbers. After some calculations, Harlan realized he had found an improvement to the formula for e above.

Harlan wrote up his work in a short paper and in March 1997 mailed it to the inventor-host of "Science Friday," a National Public Radio program based in New York City. In a remarkable coincidence, my wife Pam was working at "Science Friday" as an unpaid intern during my post-doctoral scientist days at Columbia University. Pam's job was to sort through the show's mail and respond to the inquiries, which ran the gamut from fan letters to ridiculous "scientific" discoveries - UFO sightings and conspiracy theories. One day she found Harlan's paper in the middle of this muddle and mulled it over.

What are the odds that a novice has discovered something new in a centuries-old mathematical specialty? Roughly zero. However, both Pam and I root for the underdog, and so she mentioned Harlan's paper to me. I had majored in math in college, and I agreed to look over the paper to see whether the amateur had done his math right and whether he had found anything new. Fat chance, I thought.

To my growing astonishment, I determined that Harlan's calculations were correct; also, I could not find his formula in any of my textbooks. I wrote him a carefully worded letter saying that there was a possibility that he was on to something, and I suggested we work together. A collaboration was born!

RESEARCH IS A TWO-WAY STREET

Over the next several months, Harlan and I pooled our collective wits and talents so that we could flesh out his ideas. …

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