Operations with Complex
Numbers
Unit 1 Lesson 2
Make Copies of:
• Comparing Polynomials and Complex
Numbers Graphic Organizer
• Kuta-Operations with Complex Numbers WS
GPS Standard
• MM2N1b- Write complex numbers in the form
a + bi
• MM2N1c- Add, subtract, multiply, and divide
complex numbers
• MM2N1d- Simplify expressions involving
complex numbers
Essential Questions
• How do I add and subtract complex numbers?
• How do I multiply complex numbers?
COMPLEX NUMBERS
Real Numbers
Rational
Numbers
Irrational
Numbers
Imaginary
Numbers
Standard Form
of a Complex Number
a + bi
REAL PART
IMAGINARY
PART
Adding/Subtracting Complex
Numbers
• Adding and subtracting complex numbers is
just like any adding/subtracting you have ever
done with variables.
• Simply combine like terms.
• (6 + 8i) + (2 – 12i) = 8 – 4i
• (7 + 4i) – (10 + 9i) = 7 + 4i – 10 – 9i = -3 –
5i
To Add Complex Numbers
•
•
•
•
(a + bi) + (a + bi)
Drop the parentheses
Combine like terms
Remember: the real number comes first, then
the imaginary number
Examples
•
•
•
•
•
(3 + 5i) + (2 – 7i)
(12 – 3i) + (2 + 4i)
(13 +24i) + (17+ 5i)
(3 – 6i) + (5 – 2i)
(8 – 3i) + (4 – 11i)
Test Prep Example
• What is (5 – 2i) + (6 + 4i)?
A)
B)
C)
D)
-3i
3i
11 + 2i
11 + 6i
Test Prep Example
• Perform the indicated operation.
(2 + 3i) +(13 – 2i) =
• A) 15 + 5i
B) 15 + i
C) 11 – 5i
D) -11 – i
To Subtract Complex Numbers:
• (a + bi) – (a + bi)
• Change the minus sign to plus
• Change the sign of each term in the second
set of parentheses
• Drop parentheses
• Combine like terms
• Remember: real number comes first, then
imaginary number
Examples
•
•
•
•
(6 + 7i) – (4 + 3i)
(8 + 2i) – (3 – 7i)
(12 – 7i) – (2 + 6i)
(3 – 8i) – (7 – 11i)
Test Prep Example
• Perform the indicated operation.
(-9 + 2i) – (-12 + 4i) =
A)
B)
C)
D)
-21 – 6i
-3 + 6i
3 – 2i
21 + 2i
Multiplying Complex Numbers
• This will be FOIL method with a slight twist at
the end.
• An i2 will ALWAYS show up. You will have to
adjust for this.
• (4 + 9i)(2 + 3i) = 8 + 12i + 18i + 27i2 = 8 +
30i – 27 = -19 + 30i
• (7 – 3i)(6 + 8i) = 42 + 56i – 18i – 24i2 = 42 +
38i + 24 = 66 + 38i
Examples
Test Prep Examples
• 1.
(5 – 3i)(6 + 2i)
Multiply and simplify.
• A) 24 – 8i
• B) 36 – 8i
• C) 36 + 8i
• D) 24 + 8i
Binomial Squares
and Complex Numbers
• You can still do the five-step shortcut, or you
can continue to do FOIL.
• You will still have to adjust for the i2 that will
show up.
• (7 + 3i)2 = 49 + 42i + 9i2 = 49 + 42i – 9 = 40 +
21i
• (8 – 9i)2 = 64 – 144i + 81i2 =
64 –
144i – 81= -17 – 144i
Example
Test Prep Example
• Which has the same value as (4 + 3i)2 ?
A)
B)
C)
D)
7
7 + 24i
25
25 + 24i
D2S and Complex Numbers
• Situations that in the real numbers would
have been differences of two squares (D2S)
demonstrate in the complex numbers what
are known as conjugates.
• (3 + 4i)(3 – 4i) = (3)2 – (4i)2 = 9 – 16i2 = 9 + 16
= 25
• When conjugates are used, there will be no i
in the answer.
Examples
Test Prep Example
• 2.) Perform the indicated operation.
(4 – 7i)(4 + 7i) =
A)
B)
C)
D)
-33
16 – 49i
16 – 105i
65
Test Prep Example
•
•
•
•
•
What is the square of 4 – 7i?
A) 33 – 56i
B) -33 – 56i
C) -33 + 56i
D) 33 + 56i
Test Prep Example
•
•
•
•
Which is equivalent to (3 + 2i)(2 + 5i)?
A) -4 + 19i
B) 16 + 19i
C) 6 + 29i
Test Prep Example
•
•
•
•
•
What is a if a + bi = (2 – i)2
A) a = 3
B) a = 5
C) a = 2
D) a = 1
Test Prep Example
• Simplify: -10 + √-16
2
• A) -5 + 2i
• B) -5 – 4i
• C) 20 + 4i
• D) 30 + 2i
Test Prep Example
• Perform the indicated operation.
• (3 – 8i)(4 + i) =
•
•
•
•
A) 4
B) 20 – 29i
C) 12 – 8i
D) 5 + 35i
Test Prep Example
• Multiply 2i(i – 2) over the set of complex
numbers.
• A) 0
• B) 2 – 4i
• C) -2 – 4i
• D) 2 + 4i
Graphic Organizer
• Comparing Polynomials and Complex
Numbers.doc
Assignment
• Kuta-Operations with Complex Numbers.pdf
Support Assignment
• Pg 8: 1-27
• Pg. 13: 1 - 26