Complex Numbers
CCSS objective: Use complex numbers in
polynomial identities
 N-CN.1 Know there is a complex number i
such that i2 = −1, and every complex
number has the form a + bi with a and b
real.
N-CN.2 Use the relation i2 = –1 and the
commutative, associative, and distributive
properties to add, subtract, and multiply
complex numbers.
Cycle of "i"
i  1 i  i i  1
0
i  i
3
i 1
4
i i
5
1
2
Imaginary Numbers:
2
2
b  b  1  bi
where i is the imaginary unit
i is not a variable it is a symbol for a
specific number
Simplify each expression.
1. √-81
= √81 √-1
2. √-100 = √100 √-1
3. √-121
 9i
=
Simplify each expression.
2
4. 8i  3i  24i  24 1  24
2
Remember i  1
Leave space in your notes for #5
6. 4i ∙ 3i = 12i2 = 12 ∙ -1
7. 19i ∙ 17i =
5. 5  20 i  100 10
2
Do separately ? Must combine first ?
8. √-100 ∙ √-81 =
Application: Imaginary Numbers
Monday: (all students) textbook p. 278, even only, # 2-10 &
even only, 42-46; copy problem & circle your answer to be
graded.
Do at the beginning of next class
Simplify: Enrichment
i
12
To figure out where we
are in the “i”cycle divide
the exponent by 4 and
look at the remainder.
12 4 = 3 with remainder 0
So i  i  1
12
0
Simplify:Enrichment
i
1 7 Divide the exponent by 4
and look at the remainder.
17 4 = 4 with remainder 1
So i  i  i
17
1
Definition of Equal Complex Numbers
Two complex numbers are equal if
their real parts are equal and their
imaginary parts are equal.
If a + bi = c + di,
then a = c
and b = d
Simplify:
When adding or subtracting complex
numbers, combine like terms.
Ex: 8  3i  2  5i 
8  2  3i  5i
10  2i
Simplify.
8 7i 12 11i
8 12 7i  11i
4 18i
Simplify.
9 6i 12 2i 
9 – 6i -12 – 2i
3 8i
Application
Textbook p. 278 #29-34
– Non-Enrichment: p.278 #50-52
– Enrichment: p/ 278 #14-16



Must show work up to point of calculator entry
Must write question
Circle answer you want graded
Due next class
Multiplying Complex
Numbers.
To multiply complex
numbers, you use the same
procedure as multiplying
polynomials.
Multiplying: (a + b)(c + d)
Multiplying Polynomials
 (a + b)(c + d)
Multiplying Complex Numbers
 (a + bi)(c + di)
 (a – b)(c – d)
 (a – bi)(c – di)
 (a + b)(c – d)
 (a + bi)(c – di)
Simplify.
3i
F8O5i2
I
L
16 24i 10i 15i
16 – 24i + 10i -15(-1)
16 14i 15
31 14i
2
Simplify.
6 2i 5 3i 
F
O
I
L
3018i  10i  6i
30 28i  6
24 28i
2
Group work
Teams of no more than 2; both names on one
paper if fine. Everyone in team must understand
how to do if asked.
look up and define the vocabulary term
Conjugate
Textbook page 279 # 57-66
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

Due next class
Circle answer
Write question
hint: Order of Operations