The Journal of Online Mathematics and Its Applications, Volume 8 (2008)
The Most Marvelous Theorem in Mathematics, Dan Kalman

## Complex numbers: Polar form and geometric interpretations

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 + 4i is represented by the point (3, 4) in the x,y-plane. Similarly, the complex number i -- which can be thought of as 0 + 1i, 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 + 4i) and (5 + 2i) = (8 + 6i) 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).

Figure 1. Parallelogram law.