Download 4.4 Complex Numbers

4.4 Complex
Numbers
New kinds of numbers
• In previous classes, you have heard “all real
numbers” What are unreal numbers?
• The “imaginary Unit” i is defined to be i2 = -1
• Or i = −1
• Numbers such as 6i, -4i, or i 3 are called pure
imaginary numbers. They represent square roots of
negative numbers. Ie −4 = 4 · −1 = 2i
Simplify Radicals
A.
B.
Equations with Pure
Imaginary Solutions
• Solve 5y 2 + 20 = 0.
Operations with Pure
Imaginary Numbers
• A. Simplify –3i ● 2i.
B.
Properties with Imaginary
Numbers
• Commutative Property : a + b = b + a
• Associative Property : a + (b + c) = (a + b) + c
• Powers of i
i1 = i
i2 = -1
i3 = i2 · i = -i
i4 = i2· i2 = 1
i5 = i4 ·i = i
i6 = i4 ·i2= -1
i7 =
i8 =
Day 2: Complex
Numbers
Equate Complex
Numbers
• Find the values of x and y that make the equation
2x + yi = –14 – 3i true.
Operations with complex
Numbers: Add/Subtract
• Combine Real parts, combine imaginary parts
• A. Simplify (3 + 5i) + (2 – 4i).
B. Simplify (4 – 6i) – (3 – 7i).
Operations with complex
Numbers: Multiply
• Multiply using distributive property (FOIL)
• (5 + 2i)(4 – 6i)
• (8 – 3i)2
Conjugates
• a + bi and a – bi
• When you multiply conjugates
together all imaginary numbers go
away.
Conjugates
• Find the product of (8 + 2i) and (8 – 2i)
• Find the product of (3 – 5i) and (3 + 5i)
Dividing Complex
Numbers
3
5i
2i
4  3i
8
7  5i
2i
1 i
Writing Equations
• Write a quadratic equation in standard form with
5i and -5i as solutions.
• Write a quadratic equation in standard form with
3+i and 3 – i
Application
• ELECTRICITY In an AC circuit, the voltage E, current
I, and impedance Z are related by the formula
E = I ● Z.
Find the voltage in a circuit with current 1 + 4j amps
and impedance 3 – 6j ohms.
• Find the values of x and y that make the equation
3x – yi = 15 + 2i true.
A. x = 15
y=2
B. x = 5
y=2
C. x = 15
y = –2
D. x = 5
y = –2