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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)