Download presentation

Presentation is loading. Please wait.

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?

Similar presentations

© 2020 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google