# Imagining a Hit Thriller With Number 'e'

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.

#### Four Mysterious Appearances of e

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.

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