Zero: The Biography of a Dangerous Idea

by Charles Seife

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 the outside of the unit circle, a circle centered at the origin with radius 1 (Figure 35). Multiplication and exponentiation in the complex plane became geometric ideas; you could actually see them happening.