Zero - Charles Seife [54]
Figure 32: The complex plane
Figure 33: i is at 90 degrees
Figure 34: Different powers of i
The person who combined these two ideas was a student of Gauss’s: Georg Friedrich Bernhard Riemann. Riemann merged projective geometry with the complex numbers, and all of a sudden lines became circles, circles became lines, and zero and infinity became the poles on a globe full of numbers.
Figure 35: Spirals inside and outside of the unit circle
Riemann imagined a translucent ball sitting atop the complex plane, with the south pole of the ball touching zero. If there were a tiny light at the north pole of the ball, any figures that are marked on the ball would cast shadows on the plane below. The shadow of the equator would be a circle around the origin. The shadow of the southern hemisphere is inside the circle and the shadow of the northern hemisphere is outside (Figure 36). The origin—zero—corresponds to the south pole. Every point on the ball has a shadow on the complex plane; in a sense, every point on the ball is equivalent to its shadow on the plane and vice versa. Every circle on the plane is the shadow of a circle on the ball, and a circle on the ball corresponds to a circle on the plane…with one exception.
If you’ve got a circle that goes through the north pole of the ball, the shadow is no longer a circle. It is a line. The north pole is like the point at infinity that Kepler and Poncelet imagined. Lines on the plane are simply circles on the sphere that go through the north pole—the point at infinity (Figure 37).
Figure 36: Stereographic projection of the globe
Once Riemann saw that the complex plane (with a point at infinity) was the same thing as a sphere, mathematicians could see multiplication, division, and other, more difficult operations by analyzing the way the sphere deformed and rotated. 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.
Figure 37: Lines and circles are the same.
Take all of the numbers in the complex plane and multiply them by two. That is like putting your hands on the south pole and stretching a rubber cover on the sphere away from the south pole and toward the north pole. Multiplying by one-half has the opposite effect. It is like stretching the rubber cover away from the north pole and toward the south pole. Multiplying by infinity is like sticking a needle in the south pole; the rubber sheet all flings upward toward the north pole: anything times infinity is infinity. Multiplying by zero is like sticking a needle on the north pole and everything winds up at zero: anything times zero is zero. Infinity and zero are equal and opposite—and equally destructive.
Figure 38: Riemann sphere
Figure 39: Riemann sphere transformed by i
Figure 40: Riemann sphere transformed by (x - 1)/(x + 1)
Zero and infinity are eternally locked in a struggle to engulf all the numbers. Like a Manichaean nightmare, the two sit on opposite poles of the number sphere, sucking numbers in like tiny black holes. Take any number on the plane. For the sake of argument, we’ll choose i/2. Square it. Cube