Zero: The Biography of a Dangerous Idea

by Charles Seife

For instance, multiplying by the number i was equivalent to spinning the sphere 90 degrees clockwise. If you take a number x and replace it with (x – 1)/(x + 1), that is equivalent to rotating the whole globe by 90 degrees so that the north and south poles lie on the equator (Figures 38, 39, 40). Most interesting of all, if you take a number x and replace it with its reciprocal 1/x, that is equivalent to flipping the sphere upside down and reflecting it in a mirror. The north pole becomes the south pole and the south pole becomes the north pole: zero becomes infinity and infinity becomes zero. It’s all built into the geometry of the sphere; 1/0=∞ and 1/∞=0. Infinity and zero are simply opposite poles on the Riemann sphere, and they can switch places in a blink. And they have equal and opposite powers.