Download 1.6 Perform Operations with Complex Numbers

1.6 Perform Operations with
Complex Numbers
p. 41
What is an imaginary number?
How is it defined?
What is a complex number? How is it graphed?
How do you add, subtract, multiply and divide complex
numbers?
What is a complex conjugate? When do you use it?
How do you find the absolute value of a complex
number?
Define Imaginay
•She used her imagination to write the story.
•His imagination helped him with the role in
the play.
•He was able to draw the picture of the
house he wanted to build using his
imagination.
•She had a tea party with her imaginary
friends.
•Define imagination.
•Define imaginary.
•What is the difference?
Solve:
2
x =1
2
x = -1
Not all quads. have real number
solutions. Imaginary numbers get
around this problem.
Imaginary Unit Definition
i = √−1 and i 2 = −1
Practice: Solve for x
2
2
x = -13
x = -38
x2 + 11 = 3
x2 -8 = -36
2
3x -7 = -31
2
5x + 33 = 3
Define the word:
complex
• He gave a complex explanation.
• It was a complex math problem.
• He turned a simple solution into a
complex solution.
Rube Goldberg Machine
Complex number
A complex number is made up of a real
number and an imaginary number.
Standard form for a complex number is:
a + bi
a is the real part and bi is the imaginary part.
If a = 0 and b ≠ 0, then a + bi is a pure
imaginary number (0 + bi).
Adding and Subtracting Complex
Numbers
(a + bi) + (c + di) = (a + c) + (b + d)i
(4 – i) + (3 + 2i)
(7 – 5i) – (1 – 5i)
6 – (−2 + 9i) + (−8 + 4i)
Multiplying Complex Numbers
(a + bi)(c + di) = (ac - bd) + (ad + bc)i
5i(−2 + i)
(7 – 4i)(−1 + 2i)
(6 + 3i)(6 – 3i)
Homework!
Hw 1.6a pg 45-46
#4-20 even, 22-27 all
1.6 Day 2 Menu
Divide Complex Numbers
Plot Complex Numbers
Absolute Values of Complex Numbers
Complex Conjugate
Two complex numbers of the form
a + bi and a - bi
are called Complex Conjugates.
The product of complex conjugates is
always real. (a+bi)(a-bi) = a2 + b2
Dividing Complex Numbers
Write the quotient in standard form.
1. Multiply by complex conjugate
2. FOIL
3. Use i 2 = 1 & simplify
5 + 3i
1− 2i
7+5i
1-4i
Plotting Complex Numbers
a)2 – 3i
b) -3 + 2i c) 4i
Imaginary Axis
Real Axis
Finding Absolute Values of
Complex Numbers
The absolute value of a complex number
z = a + bi is denoted as |z| is a non-negative
real number defined as
|z| = √a2 + b2
Fine the absolute value of -4+3i
Finding Absolute Values of
Complex Numbers
•
MORE PRACTICE!!!!
Plot each point in the complex plane. Then find
the absolute value for each complex number.
1. 4-i
2. -3-4i
3. 2+5i
4. -4i
Assignment 1.6b
p. 46, 30-33 all, 34-48 evens, 51-56 all