Zero: The Biography of a Dangerous Idea

by Charles Seife

Take any number on the plane. For the sake of argument, we’ll choose i/2. Square it. Cube it. Raise it to the fourth power. The fifth. The sixth. The seventh. Keep multiplying. It slowly spirals toward zero like water down a drain. What happens to 2i? The exact opposite. Square it. Cube it. Raise it to the fourth power. It spirals outward (Figure 41). But on the number sphere, the two curves are duplicates of each other; they are mirror images (Figure 42). All numbers in the complex plane suffer this fate. They are drawn inexorably toward 0 or toward ∞. The only numbers that escape are the ones that are equally distant from the two rivals—the numbers on the equator, like 1, –1, and i. These numbers, pulled by the tug of both zero and infinity, spiral around on the equator forever and ever, never able to escape the grasp of either. (You can see this on your calculator. Enter a number—any number. Square it. Square it again. Do it again and again; the number will quickly zoom toward infinity or toward zero, except if you entered 1 or –1 to begin with. There is no escape.)