1. Complex Numbers
Lesson 3.3

2. It's any number you can imagine
The Imaginary Number i
By definition
Consider powers if i
It's any number you can imagine

3. Using i
Now we can handle quantities that occasionally show up in mathematical solutions
What about

4. Complex Numbers Combine real numbers with imaginary numbers Examples
a + bi
Examples
Real part
Imaginary part

5. Try It Out
Write these complex numbers in standard form a + bi

6. Operations on Complex Numbers
Complex numbers can be combined with
addition
subtraction
multiplication
division
Consider

7. Operations on Complex Numbers
Division technique
Multiply numerator and denominator by the conjugate of the denominator

8. Complex Numbers on the Calculator
Possible result
Reset mode Complex format to Rectangular
Now calculator does desired result

9. Complex Numbers on the Calculator
Operations with complex on calculator
Make sure to use the correct character for i. Use 2nd-i

10. Warning Consider It is tempting to combine them
The multiplicative property of radicals only works for positive values under the radical sign
Instead use imaginary numbers

11. Try It Out
Use the correct principles to simplify the following:

12. The Discriminant Return of the
Consider the expression under the radical in the quadratic formula
This is known as the discriminant
What happens when it is
Positive and a perfect square?
Positive and not a perfect square?
Zero
Negative ?
Complex roots

13. Example Consider the solution to Note the graph
No intersections with x-axis
Using the solve and csolve functions

14. Fundamental Theorem of Algebra
A polynomial f(x) of degree n ≥ 1 has at least one complex zero
Remember that complex includes reals
Number of Zeros theorem
A polynomial of degree n has at most n distinct zeros
Explain how theorems apply to these graphs

15. Conjugate Zeroes Theorem
Given a polynomial with real coefficients
If a + bi is a zero, then a – bi will also be a zero

16. Assignment
Lesson 3.3
Page 211
Exercises 1 – 78 EOO