July 4, 2004 -- Specific numbers sometimes play a role in fiction. Witness the novel The Da Vinci Code, where the number is the Golden Ratio symbolized by the Greek letter phi, or the movie Pi, where the number is pi, of course.
The Da Vinci Code, a thriller offering an alternative view of various conundrums in Western history ranging from the Holy Grail to Mona Lisa's smile, is dependent on the decoding power of phi and the Fibonacci numbers. Pi is about a numerologically obsessed mathematician who thinks he's found the secret to just about everything in the decimal expansion of pi and is pursued by religious zealots, greedy financiers, and others.
Reflecting on the use of these numbers in fiction, I wondered how a number that doesn't get as much attention as phi or pi might serve as a plot element in a mystery. The number does not have a Greek name, but must make do with a simple moniker — e. The base of the natural logarithm and truly one of the most important numbers in all of mathematics, e is approximately 2.71828182845904 … (approximately because its decimal expansion continues without repetition).
The first part of an e-based story might briefly sketch the theoretical importance of e, its role in finance, number theory, physics, geometry, and so on. The number might then pop up inexplicably. Here are some possibilities.
I. A thriller about outer space. The physicist Robert Matthews has recently written that, looked at in the right way, the night sky contains the signature of the number pi. Looked at in a different way, the sky also reflects the number e.
Here's the mathematical telescope that allows us to see it: Divide up a square portion of the night sky into a very large number, N, of equal smaller squares. That is, imagine a celestial checkerboard. Then search for the N brightest stars in this portion of the sky and count how many of the N smaller squares contain none of these N brightest stars. Call this number U. (We're assuming the stars are distributed randomly so by chance some of the smaller squares will contain one or more of the brightest stars, others none.)
If one knows some probability theory, it's not hard to prove that the ratio of N to U (N divided by U, that is) is very close to e and approaches it more and more closely as N gets large. If one doesn't know probability, the appearance of the ratio could seem quite portentous. Before trying to come up with a plot twist that links the number e, this celestial map of the night sky, and some cosmic event, check out the claim. Find a regular 8x8 checkerboard, a random number generator, and 64 checkers placed randomly according to the dictates of the generator.
II. A gambling mystery. How might e arise in such a story? A somewhat unusual appearance of the number involves two decks of cards. Shuffle each deck thoroughly, turn over a card from each, and note if it's the same card (both 7s of diamonds, for example, or both jacks of clubs). Then turn over another card from each deck, and note if it's the same card. Continue doing this until all 52 cards in the decks are turned over. It can be shown that the probability of no matches at all between the two decks during this sequence of turnovers is extremely close to one chance in e; that is, the probability is 1/e or about 37 percent.
Equivalently, about 63 percent of the time there will be at least one match between the two decks sometime during this turnover process. Again try it yourself and then figure out a plot element that depends on this appearance of e and the surprising frequency of matches during this process.
III. The number e might also pop up when we are interested in record-breaking events. To illustrate, imagine this year's high school graduates running a quarter-mile race. Runners are randomly selected and sequentially over a period of months each of them runs a quarter-mile and we keep track of the number of record times that they establish. The first runner would surely establish a record time and perhaps the fourth runner would be faster than the first three and establish the second record time.
We might then have to wait until the 17th runner who runs faster than each of the previous 16 runners to establish the third record time. If we were to continue recording times for, say, ten thousand runners, we would find that there would have been only about nine record times. If we were to keep measuring the times of one million runners, we would probably note only about 14 record times. It is no coincidence that the 9th root of 10,000 and the 14th root of 1,000,000 are approximately equal to e.
If the Nth runner sets the Rth record, it can be proved that the Rth root of N will be an approximation to e, and this approximation approaches e more and more closely as N increases without bound. This is harder to verify empirically than the previous examples, but you can try.
IV. Idly picking numbers at random can also give rise to e. Using a calculator, pick a random whole number between 1 and 1,000. (Say you pick 381.) Pick another random number (Say 191) and add it to the first (which, in this case, results in 572). Continue picking random numbers between 1 and 1,000 and adding them to the sum of the previously picked random numbers. Stop only when the sum exceeds 1,000. (If the third number were 613, for example, the sum would exceed 1,000 after three picks.)
How many random numbers, on average, will you need to pick? In other words if a large group of people went through this procedure, generated numbers between 1 and 1,000, kept adding them until the sum exceeded 1,000 and recorded the number of picks needed, the average number of picks would be — you guessed it — very close to e. One could be excused for thinking that e stood for everywhere.
The number e plays a critical role in all of mathematics, and there are many more beautiful, surprising, and cryptic manifestations of the number in everyday situations (including the process of selecting a spouse). A mystery novel about some of them, perhaps entitled E-erie E-ncounters with E-nigma, might even be a bestseller, perhaps with a list price of 10e dollars — $27.18.
Professor of mathematics at Temple University and winner of the 2003 American Association for the Advancement of Science award for the promotion of public understanding of science, John Allen Paulos is the author of several best-selling books, including Innumeracy and A Mathematician Plays the Stock Market. His Who’s Counting? column on ABCNEWS.com appears the first weekend of every month.