Zero: The Biography of a Dangerous Idea

by Charles Seife

doesn’t exist. Instead, the Greeks just saw the terms as simply getting smaller and smaller, meandering outside the realm of numbers. Modern mathematicians know that the terms have a limit; the numbers 1, 1/2, 1/4, 1/8, 1/16, and so forth are approaching zero as their limit. The journey has a destination. Once the journey has a destination, it is easy to ask how far away that destination is and how long it will take to get there. It is not that difficult to sum up the distances that Achilles runs: 1 + 1/2 + 1/4 + 1/8 + 1/16 + . . . + 1/2n + .... In the same way that the steps that Achilles takes get smaller and smaller, and closer and closer to zero, the sum of those steps gets closer and closer to 2. How do we know this? Well, let’s start off with 2, and subtract the terms of the sum, one by one. We begin with 2 – 1, which is, of course, 1. Next, we subtract 1/2, leaving 1/2. Then remove the next term: subtract 1/4, leaving 1/4 behind. Subtracting 1/8 leaves 1/8 behind. We’re back to our familiar sequence. We already know that 1, 1/2, 1/4, 1/8, and so forth has a limit of zero; thus, as we subtract the terms from 2, we have nothing left. The limit of the sum 1 + 1/2 + 1/4 + 1/8 + 1/16 + . . . is 2 (Figure 11). Achilles runs 2 feet in catching up to the tortoise, even though he takes an infinite number of steps to do it. Better yet, look at the time it takes Achilles to overtake the tortoise: 1 + 1/2 + 1/4 + 1/8 + 1/16 + . . . — 2 seconds. Not only does Achilles take an in...