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

The Most Marvelous Theorem in Mathematics, Dan Kalman

The complex number system combines real numbers with `i`, which is defined as the square root of -1. Any complex number can be expressed in the form `a` + `b``i`, where `a` and `b` are real numbers. Every real number `a` is also complex, with corresponding `b` value equal to 0. With `a` = 0 we obtain a complex number of the form `b``i`. Such a number is said to be *imaginary*, or sometimes for emphasis,

The operations of addition, subtraction, multiplication, and division are all defined for complex numbers, except of course that division by 0 is not defined. Addition, subtraction, and multiplication are performed as if `i` were a variable, except that whenever it occurs, `i`^{2} is replaced by −1. For example, adding the complex numbers (3 + 4`i`) and (5 + 2`i`) results in (8 + 6`i`). Multiplication is performed as follows:

(3 + 4`i`)(5 + 2`i`) = 15 + 6`i` + 20`i` + 8`i`^{2} = 15 + 26`i` − 8 = 7 + 26`i`

Before proceeding to an example of division, note that (5 + 2`i`)(5 − 2`i`) = 29. With that in mind, consider the following example of division