Zero - Charles Seife [53]
Figure 29: Light rays inside an ellipse
You can see this very nicely with a flashlight. Go into a dark room, stand next to a wall, and point the flashlight directly at it. You will get a nice, round circle of light projected on the wall. Now slowly tilt the flashlight upward (Figure 31). You’ll see the circle stretch out into an ellipse that gets longer and longer as you increase the tilt. All of a sudden, the ellipse opens up and becomes a parabola. Thus, Kepler’s point at infinity proved that parabolas and ellipses are actually the same thing. This was the beginning of the discipline of projective geometry, where mathematicians look at the shadows and projections of geometric figures to uncover hidden truths even more powerful than the equivalence of parabolas and ellipses. However, it all depended upon accepting a point at infinity.
Figure 30: Stretching an ellipse yields a parabola.
Figure 31: Flashlight ellipses and parabola
Gérard Desargues, a seventeenth-century French architect, was one of the early pioneers of projective geometry. He used the point at infinity to prove a number of important new theorems, but Desargues’s colleagues couldn’t understand his terminology and concluded that Desargues was nuts. Though a few mathematicians, like Blaise Pascal, picked up on Desargues’s work, it was forgotten.
None of this mattered to Jean-Victor Poncelet. As Monge’s student, Poncelet had learned the technique of projecting diagrams onto two planes, and as a prisoner of war he had a lot of spare time on his hands. He used his stay in prison to reinvent the concept of a point at infinity, and combining it with Monge’s work, he became the first true projective geometer. Upon his return from Russia (carrying a Russian abacus, by then an archaic oddity) he raised the discipline to a high art.* However, Poncelet had no idea that projective geometry would reveal the mysterious nature of zero, because the second important advance, the complex plane, was still needed. We must turn to Germany for this piece of the puzzle.
Carl Friedrich Gauss, born in 1777, was a German prodigy, and he began his mathematical career with an investigation of imaginary numbers. His doctoral thesis was a proof of the fundamental theorem of algebra—proving that a polynomial of degree n (a quadratic has degree 2, a cubic has degree 3, a quartic has degree 4, and so on) has n roots. This is only true if you accept imaginary numbers as well as real numbers.
Throughout his life Gauss worked on an incredible variety of topics—his work on curvature would become a key component of Einstein’s general theory of relativity—but it was Gauss’s way of graphing complex numbers that revealed a whole new structure in mathematics.
In the 1830s Gauss realized that each complex number—numbers that have real and imaginary parts, like 1 - 2i—can be displayed on a Cartesian grid. The horizontal axis represents the real part of the complex number, while the vertical axis represents the imaginary part (Figure 32). This simple construction, called the complex plane, revealed a lot about the way numbers work. Take, for example, the number i. The angle between i and the x-axis is 90 degrees (Figure 33). What happens when you square i? Well, by definition, i2 =-1—a point whose angle is 180 degrees from the x-axis; the angle has doubled. The number i3 is equal to -i—270 degrees from the x-axis; the angle has tripled. The number i4 = 1; we have gone around 360 degrees—exactly four times the original angle (Figure 34). This is not a coincidence. Take any complex number and measure its angle. Raising a number to the nth power multiplies its angle by n. And as you keep raising the number to higher and higher powers, the number will spiral inward or outward, depending on whether the number is on the inside or on