1. Math is about to get imaginary!
Complex Numbers
Math is about to get imaginary!

2. Exercise
Simplify the following square roots:

3. Consider the quadratic equations:
x2-1 = 0 and x2+1= 0
Solve the equations using square roots.
Notice something weird?

4. Let’s look at their graphs to see what is going on…
f(x) = x f(x) = x2 + 1
How many x-intercepts does this graph have? What are they?
How many x-intercepts does this graph have? What are they?

5. Imaginary Numbers

6. Simplify imaginary numbers
Remember 28

8. Answer: -i

9. Complex Numbers: A little real, A little imaginary…
A complex number has the form a + bi, where a and b are real numbers.
a + bi
Real part
Imaginary part

10. Adding/Subtracting Complex Numbers
When adding or subtracting complex numbers, combine like terms.

11. Try these on your own

12. ANSWERS:

13. Multiplying Complex Numbers
To multiply complex numbers, you use the same procedure as multiplying polynomials.

14. Lets do another example.F O I L
Next

15. Answer:
21-i
Now try these:

16. Next

17. Answers:

18. Now it’s your turn!

19. Do Now What is an imaginary number? What is i7 equal to? Simplify:
√-32 *√2
(5 + 2i)(5 – 2i)

20. The Conjugate
Let z = a + bi be a complex number. Then, the conjugate of z is a – bi
Why are conjugates so helpful? Let’s find out!

21. We get Real Numbers!! The Conjugate = a2 + abi – abi –(bi)2
What happens when we multiply conjugates
(a + bi)(a – bi) F O I L
= a2 + abi – abi –(bi)2
= a2 – (bi)2
= a2 – b2i2 = a2 – b2(-1)
= a2 + b2
We get
Real
Numbers!!

22. Lets do an example:
Rationalize using the conjugate
Next

23. Reduce the fraction

24. Lets do another example
Next

25. Try these problems.

27. So why are we learning all this complex numbers stuff anyway?

28. Remember when we looked at this the other day??????
f(x) = x f(x) = x2 + 1
How many x-intercepts does this graph have? What are they?
How many x-intercepts does this graph have? What are they?

29. Quadratic Formula What does it do? It solves quadratic equations!
Do we remember it?
What does it do?
It solves quadratic equations!

30. Using the Discriminant
Quadratic Equations can have two, one, or no solutions.
Discriminant: The expression under the radical in the quadratic formula that allows you to determine how many solutions you will have before solving it.
Discriminant

31. Why is knowing the discriminant important?
Find the discriminant of the functions below:
Put the functions into your graphing calculator:
Do you notice something about the discriminant and the graph?

32. Properties of the Discriminant
2 Solutions
Discriminant is a positive number
1 Solutions
Discriminant is zero
No Solutions
Discriminant is a negative number

33. Find the number of solutions of the following.
Ex. 1

36. Now it’s your turn!

37. Exit Slip! Simplify: (-4 + 2i) (3-9i) What is the conjugate of 2 – 3i?
What type and how many solutions does the equations x2 + 2x + 5 =0 have?
What are the solution(s) to the equation in #3?