5-9 Complex Numbers

Objective:Students will be able to add, subtract, multiply, and divide complex numbers.

Up until this point in our lives, every number we have seen has been a real number.

As a result, there are certain mathematical problems that cannot be simplified. Consider trying to simplify this:

Until now, we would just say “no real roots.” However, there is a concept of imaginary numbers that we can consider.

Rene Descartes proposed the concept of the number i. The number i can be defined such that:

Therefore, taking the square root of both sides of the above equation, we would deduce that:

Let’s go back to the first example and look at how we can simplify the expression using imaginary numbers.

Example 1: Simplify.1) 2)

3)

5)

Try these:6) 7)

Let’s take a minute to further explore powers of i. Let’s try and fill in this chart:

i 1

i 2

i 3

i 4

i 5

Example 2: Simplify1) 2)

3)

Try these:5) 6)

Look at the equation below. How would you solve it? What are the solution(s)?

Now look at this equation. What are the solutions?

Example 3: Solve each equation.

1) 2)

Try This:

To Solve an Equation involving Complex Numbers:

Set the real terms equal and solveSet the imaginary terms equal and solve

Example: Find the values of x and y that make the equation true.

Example 4: Find the values of x and y that make the equation true.

1)

2)

Try this:3)