Math 2 Honors
Lesson 1-7 Complex Numbers
Name ________________________
Learning Goals:

Coming soon!
Before you begin, on your N spire go to the home screen enter 5, then 2 to get into the settings. Change the “Real or Complex” to “Rectangular” and make default.
Part I. Introduction to Imaginary Numbers
1.
Graph the following functions on your calculator: ( )
 x
2 and ( )

25
Set your window to XMin = -10, XMax = 10, YMin = 0 and YMax = 50 a.
How many times did your functions intersect? _________ b.
What would solving x
2 
25 accomplish in this situation?
__________________________________________________________________________ c.
How many solutions should you get? _________ d.
Solve the following equation: x
2 
25 e.
What is/are your solution(s)? ________________
2.
Now let’s look at x
2  
25.
When solving x
2  
25, a student would first square root both sides. a.
When you plug

25 into your calculator, what does the calculator say is the solution? _______ b.
Using the idea that

25 can be written as 25
 
1, what must the value of

1 be? _______ c.
Going back to the equation x
2  
25, what are the solutions? __________ and _________
3.
What do you think the learning goal for this lesson so far is?
______________________________________________________________________________
______________________________________________________________________________
4.
Solve the following: a.
x 2  
225 b.

4 x 2 
200 c.

3 x 2  
90

5.
What do you think i
2
equals? __________ a.
Since i
2
can be written as the product of i i , what is another way to express i
2
using what you learned in number 2 part b ? ________ b.
Simplify your product from part a completely. ____________ c.
i
2
= ________
6.
What is i 3 
? _______
Show your work to explain your answer.
7.
What is i
4 
?_______
Show your work to explain your answer.
Fill in the table to find a pattern for i raised to different exponents. i
 i 5  i
2  i
3  i 4  i i i
6
7
8


 i 9  i
10  i
11  i 12 
8.
For what values of n is i n
a real number? _____________
9.
What must be true about n if i n
= 1? __________
What expression would give all possible values of n that would make i n
=1? ___________
10.
What must be true about n if i n
= -1? _________
What expression would give all possible values of n that would make i n = -1? ___________
11.
For what values of n is i n
an imaginary number? _______________
12.
What must be true about n if i n
=

1 ? _______________
What expression would give all possible values of n that would make i n
=

1 ? ___________
13.
What must be true about n if i n
=
 
1 ? _______________
What expression would give all possible values of n that would make i n =
 
1 ? ___________

14.
Solve the following: a. i
40  b. i
66 
c. i
33  d. i
115  e. i
11  i
38 
Part II. Introduction to Complex Numbers
Complex numbers are made up of two parts, a real part and an imaginary part.
When combining like terms, you cannot combine 5 and 6 .
When combining like terms, treat i as you would any other variable.
Example: When simplifying an expression like 4 7 i 3 5 i you would first combine all real parts, to get 1. Then combine the imaginary parts to get 2 i . Your simplified form needs to be written in a bi form, where a is the real part, and bi is the imaginary part. Simplifying 4 7 i 3 5 i will result in

15.
Simplify the following: Put answers in the form a
 bi , where a and b are both real numbers. a. 3 i 6 5 7 i b. 6 2 i i (7

5 ) c.

49
  
12 d. (3
 i )(2

5 ) e.

2 (6 3 4) f. (2 i

5)( 3 7 )
Part III. Simplifying complex expressions with i in the denominator:
When simplifying expressions or equations with a complex denominator, you must multiply both the numerator and denominator by the complex conjugate of the denominator.
 i
Look at the following example:
4

2 i
Since the denominator is a complex number, we will multiply both the numerator and denominator by

You will need to simplify the following:
 i 4

2 i
4 2 i

4

2 i

Why doesn’t multiplying both the numerator and denominator by the complex conjugate of the denominator change the value of the original expression?
16.
Simplify the following: a.
7
 i
 i
7 b.
5 3 i
c.
 i
7 i
d.
6 i

1
 i e.
8
 
72
3 i 32
f.
6
 
 i
18
54 g.
15
 
45

15
Learning Goals for this lesson:

I can identify that i is a complex number where i
2  
1 and

I can recognize that i
4 i
8 i
12  i
16
...
i
4 k
(where k i
 
1.
is a positive integer) = 1; i 2 i i 10  i 14 ...
i 4 k

2 (where k is a positive integer) = -1; i 3 i i 11  i 15 ...
i 4 k

1 (where k is a positive integer) = -i ; and 5 9 i i i i
13
...
i
4 k

3 (where k is a positive integer) = i , and use the relation i
2  
1 to justify this fact.

I can identify that a complex number is written in the form a + bi , where a and b are both real numbers.

I can use the commutative and associative properties to add, subtract, and multiply complex numbers, substituting -1 for i
2
.