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Aim: How do we multiply complex
numbers?
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Write an equivalent expression for
74
62
Aim: Multiply Complex Numbers
Course: Adv. Alg. & Trig.
The Powers of i
 
1
2
 i2 = –1

 1

2
 i2 = –1
Find the product: 3(-2 + 3i)
distributive property
(3)(-2) + (3)(3i)
-6 + 9i
Find the product: i4(-2 + 3i)
distributive property
(i4)(-2) + (i4)(3i)
simplify
-2i4 + 3i5
-2 + 3i
Aim: Multiply Complex Numbers
i0 = 1
i1 = i
i2 = –1
i3 = –i
i4 = 1
i5 = i
i6 = –1
i7 = –i
i8 = 1
i9 = i
i10 = –1
i11 = –i
i12 = 1
Course: Adv. Alg. & Trig.
FOILing Complex Numbers
Multiply the binomials
(3 + 2i)(2 + i)
F-
(3 + 2i)(2 + i)
O-
(3 + 2i)(2 + i)
I-
6
+3i
(3 + 2i)(2 + i)
+4i
+ 4i
L - (3 + 2i)(2 + i)
2i2 = -2
6 + 7i
–2
4 + 7i
Aim: Multiply Complex Numbers
Course: Adv. Alg. & Trig.
Distributive Property
Multiply the binomials
(3 + 2i)(2 + i)
distribute:
3(2 + i) + 2i(2 + i)
6 + 3i + 4i + 2i2
6 + 7i + 2i2
i2 = -1
6 + 7i + 2(-1)
4 + 7i
Aim: Multiply Complex Numbers
Course: Adv. Alg. & Trig.
Conjugates
2x2 - 50 = 2(x – 5)(x + 5)
conjugates of each other
General Terms a2 – b2 = (a – b)(a + b)
When conjugates are multiplied, the result is the difference
between perfect squares.
The conjugate of a complex number a + bi is
i2 = -1
a – bi
(a + bi)(a – bi) = a2 – (bi)2 = a2 – b2i2 = a2 + b2
(5 + 2i)(5 – 2i) = 52 – (2i)2 = 25 – b2i2 = 25 + 4
= 29
The product of two complex numbers
that are conjugates is a real number.
Aim: Multiply Complex Numbers
Course: Adv. Alg. & Trig.
Model Problems
Express the number (4 – i)2 – 8i3 in simplest
form.
(4 – i)2 – 8i3 = (4 – i)(4 – i) – 8i3
= 16 – 8i + i2 – 8i3
= 16 – 8i – 1 – 8(-i)
= 15
i3 = -i
Express the product of 2  i 5 and its
conjugate in simplest form
2  i 52  i 5   4  5  9
(a + bi)(a – bi) = a2 + b2 a = 2
b= 5
Aim: Multiply Complex Numbers
Course: Adv. Alg. & Trig.
Model Problems
6  2  8i   3  5  7i 
 3  4i 
2
 6  4i  6  4i 
Aim: Multiply Complex Numbers
Course: Adv. Alg. & Trig.
Model Problems
8 

11
5  2i

 8 
11

2
 4  3i  2  5i  4  3i 
Aim: Multiply Complex Numbers
Course: Adv. Alg. & Trig.
Graph Representation
Multiply i(2 + i)
i(2 + i) = 2i + i2 = -1 + 2i
yi
5i
Draw &
4i
compare vectors
3i
2 + i & -1 + 2i
(-1 + 2i) 2i
i
-5
-4 -3 -2 -1
Multiplication by i
is equivalent to a
counterclockwise
rotation of 900 about
the origin.
-i
-2i
(2 + i)
0
1
2
3
4
R
x
5
6
(2  i) 

1  2i
0
90
-3i rotational transformation
Rotation
of 900 about the origin
-4i
R90º(x,y) = (y,-x)
-5i
-6i
Aim: Multiply Complex Numbers
i(2 + i) = -1 + 2i
Course: Adv. Alg. & Trig.
Graph Representation
Multiply by distributing (3 + 2i)(2 + i)
distributed:
3(2 + i) + 2i(2 + i) = 4 + 7i
D33
2  i 

6  3i
Multiplication by 3
is equivalent to a
dilation of 3.
-5
-4 -3 -2 -1
5i
yi
4i
(6 + 3i)
3i
2i
i
-i
(2 + i)
0
1
2
3
4
x
5
6
-2i
-3i
-4i
-5i
-6i
Aim: Multiply Complex Numbers
Course: Adv. Alg. & Trig.
Graph Representation (con’t)
Multiply by distributing (3 + 2i)(2 + i)
distributed:
3(2 + i) + 2i(2 + i) = 4 + 7i
(6 + 3i)
Rotation of 900 about
the origin
R90º(x,y) = (y,-x)
Multiplication by i
is equivalent to a
counterclockwise
rotation of 900 about
the origin.
-5
recall:
5i
(-2 + 4i)
R
2  i 

1  2i
4i
2i
i
-4 -3 -2 -1
Multiplication by 2
is equivalent to a
dilation of 2.
0
90
2•i(2 + i) = 2(-1 + 2i)
3i
(-1 + 2i)
i(2 + i) = -1 + 2i
yi
-i
-2i
(2 + i)
0
1
2
3
4
x
5
6
D2
1 2i 

2  4i
-3i
-4i
-5i
-6i
Aim: Multiply Complex Numbers
Course: Adv. Alg. & Trig.
Graph Representation (con’t)
Multiply the binomials (3 + 2i)(2 + i)
3(2 + i) + 2i(2 + i) = 4 + 7i
(6 + 3i) + (-2 + 4i)
7i
yi
(4 + 7i)
6i
5i
(-2 + 4i)
4i
(6 + 3i)
3i
2i
i
-5
-4 -3 -2 -1
-i
0
1
2
3
4
5
6
x
-2i
-3i
Aim: Multiply Complex Numbers
-4i
Course: Adv. Alg. & Trig.