Objectives:
1.
To simplify square roots of negative numbers
2.
To plot complex numbers in the complex plane and find their absolute value
3.
To add, subtract, multiply, and divide complex numbers
Assignment:
• P. 135-136: 1-18, 25
• Challenge Problems

This is a highly complex shape know as the
Mandelbrot Set.
A set in math is just a group of numbers, like the set of real numbers.

The Mandelbrot Set,
M , is group of numbers that when graphed form a fractal.
To see if a number is part of the set, you start with the recursive formula z n +1
= z n
2 + c.

The Mandelbrot Set,
M , is group of numbers that when graphed form a fractal.
To see if a number is part of the set, you start with the recursive formula z n +1
= z n
2 + c.
In the formula, c is the number you’re testing and z
0
= 0.
Let’s say we wanted to test if c = 1 is a member of the set: z

0
0 z
1

0
2  
1

z n +1
= z n
2 + c.
In the formula, c is the number you’re testing and z
0
= 0.
Let’s say we wanted to test if c = 1 is a member of the set: z

0
0 z
1

0
2  
1
To get z
2
, you plug z
1
= 1 back into the formula, where c will remain 1, the number we’re trying to test. This is called recursion: z
2

1
2  
2 z
3

2
2  
5 z
4

5
2  
26

As you can probably see, these numbers are increasing in value, getting further away from zero.
If we were to continue iterating with the formula, the resulting values would approach infinity.
To get z
2
, you plug z
1
= 1 back into the formula, where c will remain 1, the number we’re trying to test. This is called recursion: z
2

1
2  
2 z
3

2
2  
5 z
4

5
2  
26

To be a member of the M set, these values must be bounded; that is, they cannot approach infinity.
Therefore, the number c = 1 is not a part of M .
Is c = − 1 a member of the Mandelbrot
Set?

You might be thinking that there aren't a whole lot of members in M since membership is so exclusive, but you'd be wrong as this image clearly indicates. To find more members, you just have to know where to look: within the set of complex numbers.
Click me for a Mandelbrot Zoom!

You will be able to simplify square roots of negative numbers

Use a graphing calculator to graph y = x 2 + 1.
What are the x intercepts?

Solve the quadratic equation x 2 + 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 2 𝑖 = −1
= −1
Girolamo Cardano c. 1545
Just because a number is imaginary doesn’t mean that it doesn’t exist!

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 16 th power?
i 4 i 5 i 2 i 3 i 6 i n i 1
Solution
−1
−1
− −1
1
−1
−1

i
As the previous example shows, there are only 4 possible values for i n .
i
1 i
2 i
3 i
4
 
1



1


2
 i
2  i i
 
1
1 i
  i
 i
2  i
2 
1
All other powers of 4 just repeat this pattern.
So, how do you think you would evaluate i 101 ?

i
As the previous example shows, there are only 4 possible values for i n .
i
1 i
2 i
3 i
4
 
1



1


2
 i
2  i i
 
1
1 i
  i
 i
2  i
2 
1
Divide the exponent by 4, then use the remainder to find the result:
Remainder Result
1
2 i
−1
3 − i
0 1

Evaluate each of the following.
1.
i 54
2.
i 120
3.
i 89
4.
i 39

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
=
4
100 𝑛 = 25𝑛
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
=
4
100 + 32
=
4
100
+
4
32
4
= 25 + 8 = 33
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
=
4
200 + 28
=
4
200
+
4
28
4
= 50 + 7 = 57
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
=
4
300 + 16
=
4
300
+
4
16
4
= 75 + 4 = 79
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
1.
If 𝑟 is a positive number, then −𝑟 = 𝑖 𝑟
2.
𝑖 𝑟 2 = −𝑟
Example
−5 = 𝑖 5 𝑖 5
= −5
2
= 𝑖 2 ∙ 5

Find the roots of each quadratic equation.
1.
x 2 + 1 = 0 2.
2 x 2 + 18 = − 72

You will be able to plot complex numbers in the complex plane and find their absolute value

A complex number in standard form is written a
 bi 𝑎, 𝑏 ∈ ℝ
Real Part 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

Plot the complex number 4 + 3 i in the complex plane.
How could we find its distance from the origin? In other words, how could we find its absolute value?

Absolute Value of a Complex Number
The absolute value of a complex number z = a + bi , denoted | z |, is a nonnegative real number defined as z
 a
2  b
2

Find the absolute value of each complex number.
1.
4 – i
2.
−3 – 4 i
3.
2 + 5 i
4.
−4 i

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 – 11 i ) + (-8 + 3 i )
2.
(15 – 9 i ) – (24 – 9 i )
3.
35 – (13 + 4 i ) + i

To multiply complex numbers, you have to use a combination of the distributive property and properties of the imaginary unit.
2 i

3

5 i
 
6 i

10 i
2 
6 i

10
  
6 i

10

10

6 i

Write the expression as a complex number in standard from.
1.
−5 i (8 – 9 i )
2.
(-8 + 2 i )(4 – 7 i )

Multiply and classify the product.
1.
(5 i )( −5 i )
2.
(3 + 6 i )(3 – 6 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

3 i
1
 i

1
1

 i i



2

3 i 1
1
 i


1

 i i



2

2 i

3 i

3 i
2
2

 
5 i
2

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

3 i
1
 i

1
1

 i i



2

3 i 1
1
 i


1

 i i



2

2 i

3 i

3 i
2
2
 
1

5
2 2 i

Write the quotient in standard form.
3

4 i
5
 i

A number c is a member of the
Mandelbrot Set if
| z | < 2 after each iteration.
Determine whether c = i /2 is a member of the Mandelbrot
Set.

Objectives:
1.
To simplify square roots of negative numbers
2.
To plot complex numbers in the complex plane and find their absolute value
3.
To add, subtract, multiply, and divide complex numbers
Assignment:
• P. 135-136: 1-18, 25
• Challenge Problems