The Journal of Online Mathematics and Its Applications, Volume 8 (2008)

The Most Marvelous Theorem in Mathematics, Dan Kalman

Just as real numbers are depicted visually as points on a line, complex numbers are depicted as points in a plane. For example, the complex number 3 + 4`i` is represented by the point (3, 4) in the `x`,`y`-plane. Similarly, the complex number `i` -- which can be thought of as 0 + 1`i`, corresponds to the point (0, 1).

The operations on complex numbers have geometric interpretations that are often quite useful. For example, complex number addition is the same as vector addition. Thus, for example, the addition (3 + 4`i`) and (5 + 2`i`) = (8 + 6`i`) can be represented in the complex plane as the addition of vectors (3,4) + (5,2) = (8,6), with two different geometric interpretations: either place the tails of the vectors both at the origin and complete the parallelogram (Figure 1), or place the tail of one vector at the origin and the tail of the second vector at the head of the first (Figure 2).

There is also a connection between multiplication of complex numbers and polar coordinates. As usual, polar coordinates of a point (`x`, ` y`) in the plane are defined as (`r`, `θ`) where `r` is the length of the vector (`x`,` y`) and `θ` is the angle this vector makes with the positive `x` axis. This is illustrated below in Figure 3. In general, we will follow the convention that `θ` is between -π and π, with a negative value for points with negative `y` coordinates.

The multiplication of two complex numbers has a simple interpretion in polar coordinates: multiply the `r` components and add the `θ` components (adding or subtracting 2π if necessary to keep the final value of `θ` between −π and π). This is illustrated in Figure 4 above, where the blue line represents 2 + `i`, the red line is 1 + 3`i`, and the black line is the product (2 + `i`)(1 + 3`i`) = −1 + 7`i`. As the figure suggests, the length of the black line is given by the product of the lengths of the red and blue lines, whereas the black angle is obtained by adding the red and blue angles. One way to understand this phenomenon is to write complex numbers in the *polar form* `r``e`^{iθ}. This depends on Leonhard Euler's amazing discovery that, for any real angle `θ`, `e`^{iθ} = cos `θ` + `i` sin `θ`. Thus `r``e`^{iθ} = `r`cos `θ` + `i r`sin `θ` = `x` + `iy` where `x` = `r` cos `θ` and `y` = `r` sin `θ`. These familiar equations show that the polar form `r``e`^{iθ} really does specify the complex number whose polar coordinates are `r` and `θ`. Now suppose we want to multiply two complex numbers, and we express them in polar form as `z` = `r``e`^{iθ} and `w` = `s``e`^{iφ}. Then the usual rules of algebra show `z``w` = `r``s``e`^{i(θ + φ)}. This justifies the claim that multiplying complex numbers has the effect of adding their angles and multiplying their lengths. Or, put another way, when you multiply a complex number `w` by `r``e`^{iθ}, it has the effect of scaling `w` by `r` and rotating it counter-clockwise about the origin through an angle `θ`.