What you’ll learn to do: Make calculations with complex numbers

The study of mathematics continuously builds upon itself. Negative integers, for example, fill a void left by the set of positive integers. The set of rational numbers, in turn, fills a void left by the set of integers. The set of real numbers fills a void left by the set of rational numbers. The set of real numbers has voids as well. For example, we still have no solution to equations such as

[latex]{x}^{2}+4=0[/latex]

Our best guesses might be +2 or –2. But if we test +2 in this equation, it does not work. If we test –2, it does not work. If we want to have a solution for this equation, we will have to go farther than we have so far. After all, to this point we have described the square root of a negative number as undefined. Fortunately there is another system of numbers that provides solutions to problems such as these. In this section we will explore this number system and how to work within it.