Objectives:
1. To simplify powers
of complex numbers
2. To add, subtract,
multiply, and divide
complex numbers
3. To find complex
solutions to
quadratic equations
•
•
•
•
•
•
Assignment:
P. 167: 17-26 (Some)
P. 167: 27-36 (Some)
P. 167: 45-54 (Some)
P. 168: 65-74 (Some)
P. 168: 86
Binomial Theorem
Presentation and
Supplement
Explain the humor (if any) in this sassy t-shirt.
Which of the following do not belong?
A. x2 – 1 = 0
B. x2 + 1 = 0
C. −x2 + 1 = 0
D. −(x2 – 1) = 0
Find a number that does not fit into any nook or
cranny of the Venn Diagram below.
Real Number
Complex Number
Imaginary Unit
Imaginary Number
Pure Imaginary Number
Complex Plane
Complex Conjugates
You will be able to
simplify powers of
complex numbers
Graph y = x2 + 1.
What are the xintercepts?
Solve the quadratic equation x2 + 1 = 0.
The problem here is that −1 is not a real
number since there is no real number that you
can square to get −1.
– This does not mean there is no solution; it’s more
complex than that.
The imaginary unit i can be used to find the
square roots of negative numbers.
𝑖 = −1
𝑖 2 = −1
A pattern exists as a
result of raising i, an
imaginary number, to
n, an integer greater
than or equal to 1.
According to the table,
what is the value of i
raised to the 16th
power?
in
Solution
i1
−1
−1
i2
i3
i4
i5
i6
− −1
1
−1
−1
As the previous example shows, there are only 4
possible values for in.
i1  1  i
2
2
i  1  1
All other powers of 4 just
repeat this pattern.
i 3  i 2  i  1 i  i
i 4  i 2  i 2   1 1  1
So, how do you think you
would evaluate i101?


As the previous example shows, there are only 4
possible values for in.
i1  1  i
2
2
i  1  1


i  i  i  1 i  i
i 4  i 2  i 2   1 1  1
3
2
Divide the exponent by 4,
then use the remainder
to find the result:
Remainder
Result
1
i
2
−1
3
−i
0
1
Evaluate each of the following.
1. i54
2. i120
3. i89
4. i39
How can you tell if a number is divisible by 4?
Any multiple of 100 is divisible by 4.
Note that 100𝑛 is multiple of 4, where 𝑛 ∈ ℤ
100𝑛
100
=
𝑛 = 25𝑛
4
4
Since this is an integer with
no remainder, any multiple
of 100 is divisible by 4.
How can you tell if a number is divisible by 4?
If the last two digits of a number are divisible
by 4, the whole number is divisible by 4.
132
100 + 32
100 32
=
=
+
= 25 + 8 = 33
4
4
4
4
Since this is an integer with no remainder,
as long as the last two digits are divisible
by 4, the whole number is divisible by 4.
How can you tell if a number is divisible by 4?
If the last two digits of a number are divisible
by 4, the whole number is divisible by 4.
228
200 + 28
200 28
=
=
+
= 50 + 7 = 57
4
4
4
4
Since this is an integer with no remainder,
as long as the last two digits are divisible
by 4, the whole number is divisible by 4.
How can you tell if a number is divisible by 4?
If the last two digits of a number are divisible
by 4, the whole number is divisible by 4.
316
300 + 16
300 16
=
=
+
= 75 + 4 = 79
4
4
4
4
Since this is an integer with no remainder,
as long as the last two digits are divisible
by 4, the whole number is divisible by 4.
Simplify each of the following.
1.
−36
2.
−13
3.
𝑖 5
2
The Square Root of a Negative Number
Property
Example
1. If r is a positive number, then
−𝑟 = 𝑖 𝑟
−5 = 𝑖 5
2
2.
𝑖 𝑟
2
= −𝑟
𝑖 5 = 𝑖2 ∙ 5
= −5
Find the roots of each quadratic equation.
1. x2 + 11 = 3
2. 2x2 + 18 = -72
A complex number in standard form is written
a  bi
Real Part
𝑎, 𝑏 ∈ ℝ
Imaginary Part
• All real numbers are complex numbers
– This happens when b = 0
• For imaginary numbers, b  0.
• For a pure imaginary number, a = 0.
Draw a Venn Diagram
that represents the
set of complex
numbers and includes
real, imaginary, and
pure imaginary
numbers.
ℂ: The set of all complex numbers
All complex numbers
are essentially
2-dimensional.
– When you graph a
real number, it
appears on a 1-D
number line
All complex numbers
are essentially
2-dimensional.
– However, complex
numbers have
both a real and an
imaginary part.
Imaginary
Axis
Real Axis
You will be able to add,
subtract, multiply, and
divide complex numbers
To add or subtract two complex numbers, simply
add or subtract their real and imaginary parts
separately.
Sum
(a + bi) + (c + di) = (a + c) + (b + d)i
Difference
(a + bi) – (c + di) = (a – c) + (b – d)i
Write the expression as a complex number in
standard form.
1. (12 – 11i) + (−8 + 3i)
2. (15 – 9i) – (24 – 9i)
3. 35 – (13 + 4i) + i
Write the expression as a complex number in
standard form.
1. (9 – i) + (−6 + 7i)
2. (3 + 7i) – (8 – 2i)
3. −4 – (1 + i) – (5 + 9i)
To multiply complex numbers, you have to use a
combination of the distributive property and
properties of the imaginary unit.
2i  3  5i   6i  10i 2  6i  10  1  6i  10  10  6i
Write the expression as a complex number in
standard from.
1. −5i(8 – 9i)
2. (−8 + 2i)(4 – 7i)
Write the expression as a complex number in
standard from.
1. i(9 – i)
2. (3 + i)(5 – i)
Multiply and classify the product.
1. (5i)(−5i)
2. (3 + 6i)(3 – 6i)
To “divide” complex numbers, you have to
multiply by a complex conjugate
The complex numbers
a + bi and a – bi are
complex conjugates
The product of
complex conjugates is
always a real number
2  3i 1  i  2  3i 1  i  2  2i  3i  3i 2
1  5i




1 i 1 i
2
2
1  i 1  i 
To “divide” complex numbers, you have to
multiply by a complex conjugate
The complex numbers
a + bi and a – bi are
complex conjugates
The product of
complex conjugates is
always a real number
2  3i 1  i  2  3i 1  i  2  2i  3i  3i 2
1 5



  i
1 i 1 i
2 2
2
1  i 1  i 
Write the quotient in standard form.
3  4i
5i
Write each quotient in standard form.
5
1.
1 i
5  2i
2.
3  2i
Evaluate each of the following.
1. i−5
2. i−12
3. i−31
4. i−102
Simplify each of the following:
1. 1 + 𝑖
3
2. 1 + 𝑖
−3
You will be able to
find complex
solutions to
quadratic
equations
Solve each quadratic equation.
2
1. x  6 x  12  0
2. 2 x  4 x  14  0
2
Objectives:
1. To simplify powers
of complex numbers
2. To add, subtract,
multiply, and divide
complex numbers
3. To find complex
solutions to
quadratic equations
•
•
•
•
•
•
Assignment:
P. 167: 17-26 (Some)
P. 167: 27-36 (Some)
P. 167: 45-54 (Some)
P. 168: 65-74 (Some)
P. 168: 86
Binomial Theorem
Presentation and
Supplement