1. Complex Numbers
What you’ll learn
To identify, graph, and perform operations
with complex numbers. To find complex numbers
solutions of quadratic equations.
Vocabulary
Imaginary unit number, complex number
pure imaginary number, complex number plane,
absolute value of a complex number,
complex conjugates.

2. Note
Did you remember how different sets of numbers
are subset of real numbers?
The set of real numbers itself is a subset of a larger
set of numbers, the complex numbers and the unit is
Complex Numbers and imaginary numbers are not
synonymous; the set of complex numbers consists of
both real and imaginary numbers. By convention is
written before the radical.

3. The Complex Numbers are based on a number whose square is -1
The Complex Numbers are based on a number whose square is -1. The imaginary unit i is the complex number whose square is -1.
The square root of a Negative Number Real Number
For any positive number a.

4. Problem 1:
How do you write by using the imaginary unit ?
Property of the square roots
Definition of
Simplify
Your turn a) b) c)

5. Note
An imaginary number is any number of the form
where a and b are real numbers and
Imaginary numbers make up the set of complex
numbers.
You can write complex number in the form where
a and b are real numbers. If b=0, then the number
is a real number. If a=0 and then the number
is a pure imaginary number.

6. In the complex plane, the point (a,b) represents the
complex number To graph a complex number
,locate the real part on the horizontal axis and
the imaginary part on the vertical axis.
The absolute value of a complex number is its distance
from the origin in the complex plane.

7. Problem 2
What is the graph and the absolute value of each number? A. B.

8. Your turn
What is the graph and the absolute value of each number? C A. B. C. A B

9. Adding and Subtracting Complex Numbers
Problem 3:What is the sum or difference?
A.(4-3i)+(-4+3i) B. (5-3i)-(-2+4i)
4+(-4)+(-3i)+3i
5+2-3i-4i
0+0=0
7-7i
This means that
4-3i and -4+3i
are additive inverses
The associative and commutative
properties apply to Complex Numbers
as well

10. Your turn
What is each sum or difference?
(7-2i)+(-3+i)
(1+5i)-(3-2i)
(8+6i)-(8-6i)
(-3+9i)+(3+9i)

11. Multiplying and Dividing Complex Numbers
Problem 4:What is each product?
(3i)(-5+2i)
B. (4+3i)(-1-2i)
You multiply complex numbers as you would multiply binomials.
For the imaginary parts
Your turn

12. Problem 5: What is each quotient?

13. Your turn
What is each quotient?

14. Finding Pure Imaginary Solutions. Finding Imaginary Solutions
What are the solutions of

15. Your turn
What are the solutions of

16. Classwork odd Homework even
TB pg 253 Exercises 8-55