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

## Basic definitions and properties complex numbers

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 + bi, 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 bi. Such a number is said to be imaginary, or sometimes for emphasis, pure imaginary.

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, i2 is replaced by −1. For example, adding the complex numbers (3 + 4i) and (5 + 2i) results in (8 + 6i). Multiplication is performed as follows:

(3 + 4i)(5 + 2i) = 15 + 6i + 20i + 8i2 = 15 + 26i − 8 = 7 + 26i

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