The hardest thing about working with complex numbers is understanding why you might want to. Before introducing complex numbers, let’s backup and look at simpler examples of the need to deal with new numbers.
If you are like most people, initially number meant whole number, 0,1,2,3,… Whole numbers make sense. They provide a way to answer questions of the form “How many … ?” You also learned about the operations of addition and subtraction, and you found that while subtraction is a perfectly good operation, some subtraction problems, like 3 – 5, don’t have answers if we only work with whole numbers. Then you find that if you are willing to work with integers, …,-2, -1, 0, 1, 2, …, then all subtraction problems do have answers! Furthermore, by considering examples such as temperature scales, you see that negative numbers often make sense.
Now we have fixed subtraction we will deal with division. Some, in fact most, division problems do not have answers that are integers. For example, 3 ÷ 2 is not an integer. We need new numbers! Now we have rational numbers (fractions).
There is more to this story. There are problems with square roots and other operations, but we will not get into that here. The point is that you have had to expand your idea of number on several occasions, and now we are going to do that again.
The “problem” that leads to complex numbers concerns solutions of equations.
Equation 1: x2 – 1 = 0.
Equation 1 has two solutions, x = -1 and x = 1. We know that solving an equation in x is equivalent to finding the x-intercepts of a graph; and, the graph of y = x2 – 1 crosses the x-axis at (-1,0) and (1,0).
Equation 2: x2 + 1 = 0
Equation 2 has no solutions, and we can see this by looking at the graph of y = x2 + 1.
Since the graph has no x-intercepts, the equation has no solutions. When we define complex numbers, equation 2 will have two solutions.