Another Kind of e-book

In the unlikely event that someone asks me, "Mike, in all your years of tracking computer technology and generally observing the passing show, what's the most outrageous thing you've ever heard?" I have a ready answer.

The most outrageous thing I've ever heard has only indirect relevance to computer technology, but part of its outrageousness is that it seems to have some relevance to so many different subjects. My candidate for the most outrageous statement ever uttered is:

eⁱ+1=0

You may or may not find it outrageous; it requires some mathematics to be shocked by it, but not so much mathematics that it becomes obvious. I happen to fall into that semieducated slot, and my first reaction to the equation is, "How could that possibly be true?" followed quickly by, "And even if it is, what can it possibly mean?" The equation unites five of the most fundamental constants of mathematics: e, i, , 0, and 1. It uses the three basic operations: addition, multiplication, and exponentiation -- and uses each of them exactly once. It ties together arithmetic, geometry, logarithms, algebra, integral and differential calculus, and complex numbers. It introduces two different transcendental numbers and one imaginary number on the left side of the equal sign, and through some magic that just shouldn't work, makes them all cancel each other out, leaving zero on the right side.

Nineteenth-century mathematician Benjamin Peirce said of this equation, "That is surely true, it is absolutely paradoxical; we cannot understand it, and we don't know what it means. But we have proved it, and therefore we know it must be the truth."

It's poetry. In just seven symbols, it packs in an amazingly rich collection of ideas and relationships, plus those five constants, three of which are pretty mysterious on their own:

• The letter e stands for the transcendental number 2.71828..., the base of the natural logarithms, the necessary constant in the only function that is its own derivative, and an important number in financial calculations. It pops up in the quantification of human perception, in architecture, and in the growth patterns of sunflowers, nautilus shells, and spiral galaxies.
• The Greek letter , of course, represents the ratio of the circumference to the diameter of a circle, but can be expected to show up in any formula involving curvature or cyclic repetition.
• The letter i represents the imaginary base, the square root of negative one, the foundation of the complex number system.

What connects these numbers, and how that connection was discovered, is the subject of the delightful and very readable book e: The Story of a Number (Princeton University Press, 1993, ISBN 0-691-05854-1), by Eli Maor, who teaches the history of mathematics at Loyola University in Chicago. The book was just recently released in paperback, with new material on a recent discovery regarding prime numbers. Maor tells the story of the rather odd John Napier, who changed the way scientists calculated; recounts the birth of differential and integral calculus, with all the controversy it engendered; shows how Hamilton's notation for complex numbers made them seem less "imaginary"; and shows the extraordinary chutzpah of Leonhard Euler, who plugged imaginary exponents into functions involving the number e, played fast and loose with infinities, and took the results seriously when suddenly appeared in his calculations, seemingly from out of nowhere.

By the end of the book, Maor has told some good stories; shown some remarkable properties of e, , and i; and led the reader to see how e, , and i are related and why the outrageous equation actually makes sense.

But best of all, in doing so, he doesn't make it seem any less outrageous. He doesn't destroy the poetry in analyzing it. I recommend the book to any DDJ reader.

Michael Swaine
editor-at-large
mswaine@swaine.com