Section 1.6

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Hawkes Learning Systems:
College Algebra
Section 1.6: The Complex Number System
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Objectives
o The imaginary unit i and its properties.
o The algebra of complex numbers.
o Roots and complex numbers.
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The Imaginary Unit i and its Properties
Consider the following question:
For a given non-zero real number a, how many solutions
does the equation x 2  a have?
As we have noted before, this equation has two solutions:
x  a and x   a if a is positive. However, if a is
negative, it has no (real) solutions . The following
definitions of complex numbers explain this situation in
more detail.
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The Imaginary Unit i and its Properties
The Imaginary Unit i
The imaginary unit i is defined as i  1. In other
words, i has the property that its square is 1: i 2  1 .
Square Roots of Negative Numbers
If a is a positive real number, a  i a . Note that by
this definition, and by a logical extension of
2
2
2
exponentiation, i a  i
a   a.
 
 
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Example 1:The Imaginary Unit i and its
Properties
Write the following radicals in simplified form in terms of i.
a. 9  i 9  3i
We write a constant, such as 3, before letters in
algebraic expressions, even if the letter is not a
2
variable. Remember: i is NOT a variable:i  1 .
b. 18  i 18  3i 2
We write the radical factor last.
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Example 2: The Imaginary Unit i and its Properties
Write the following expressions in simplified form.
c. i 5  i 2  i 2  i   1   1  i  i
6
2
2
2
i

i

i

i
d.
  1   1   1  1
4
2
2

i


1
i

i

i

i
1
e.      
4
4
4
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The Imaginary Unit i and its Properties
Complex Numbers
For any two real numbers a and b, the sum a  bi is a
complex number. The collection
 a  bi | a and b are both real
is called the set of complex numbers.
Ex: The number 2  3i is a complex number,
where a = 2 and b = –3.
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The Imaginary Unit i and its Properties
Complex Numbers
The number a is called the real part of a  bi , and the
number b is called the imaginary part. If the imaginary
part of a given complex number is 0, the number is
simply a real number.
Ex. 4   0  i  4 (4 is a real number)
If the real part of a given complex number is 0, the
number is a pure imaginary number.
Ex. 0   7  i  7i (7i is a pure imaginary number)
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The Algebra of Complex Numbers
Some notes about Complex Numbers:
Every real number is a complex number with 0 as the
imaginary part. Furthermore, the set of real numbers is
a subset of the complex numbers.
Real Numbers ( )
Complex
Numbers
( )
Rational Numbers ( )
Integers ( )
Whole Numbers
Natural Numbers ( )
Irrational
Numbers
The field properties (closure, commutate, associative, distributive and the existence
of an identity and inverse), discussed in Section 1.2, still apply to complex numbers.
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The Algebra of Complex Numbers
When faced with a complex number, the goal is to
write it in the form a  bi .
Simplifying Complex Expressions
Step 1: Add, subtract, or multiply the complex
numbers, as required, by treating every complex
number a  bi as a polynomial expression. Remember,
though, that i is not actually a variable. Treating a  bi
as a binomial in i is just a handy device.
Step 2: Complete the simplification by using the fact
that i 2  1.
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Example 3: The Algebra of Complex Numbers
Simplify the following complex number expression.
a.  6  2i    3  5i 
  6  3   5i  2i 
 3  7i
b. 1  4i    2  4i 
  1   2     4i  4i 
 1  0i
1
Treating the two complex numbers as
polynomials, we combine the real and
then the imaginary parts and simplify.
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Example 3: The Algebra of Complex Numbers
Simplify the following complex number expressions.
 2  i  1  2i 
 2  i  4i  2i 2
 2   1  4  i  2  1
 2  3i   2 
 4  3i
The product of two complex numbers leads
to four products via the distributive
property.
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Example 4: The Algebra of Complex Numbers
Simplify the following complex number expression.
 3  2i 
2
  3  2i  3  2i 
 9  6i  6i  4i 2
 9   6  6  i  4  1
 5   12  i
 5  12i
Squaring a complex number leads to four
products via the distributive property.
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The Algebra of Complex Numbers
Division of Complex Numbers:
In order to rewrite a quotient in the standard form a  bi,
we make use of the following observation:
2
2 2
2
2
a

bi
a

bi

a

abi

abi

b
i

a

b



Given any complex number a  bi , the complex number
a  bi is called its complex conjugate.
We simplify the quotient of two complex numbers by
multiplying the numerator and denominator of the
fraction by the complex conjugate of the denominator.
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Example 4: The Algebra of Complex Numbers
Simplify the quotient.
Find the complex conjugate of the denominator and
1  2i  2  3i 

a.
multiply that by both the numerator and the

2  3i  2  3i 
denominator.
2
2
2
a

bi
a

bi

a

b
Remember
that
, which



2  3i  4i  6i
makes solving the product in the denominator

49
much easier than using the distributive property.
2  7i  6

49
This would be an acceptable answer however, we
4  7i

have stated that all simplified complex numbers
13
should be in the form a  bi .
4 7

 i
13 13
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Example 5: The Algebra of Complex Numbers
Simplify the quotients.
a.  2  i 
1
1  2i 



2i  2i 
2i

4 1
2i

5
2 1
  i
5 5
Find the complex
conjugate of the
denominator.
1 i
b. 
i i
i
 2
i
i

1
i
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Roots and Complex Numbers
We earlier said that if a and b are real numbers, then
a
a
a b  ab and

. If a and b are not real
b
b
numbers, then these properties do not necessarily hold. For
instance:
9 4
 9  4 
  3i  2i 

 36
6
 6i 2
 6
In order to apply either of these two properties, first simplify any
square roots of negative numbers by rewriting them as pure
imaginary numbers.
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Example 6: Roots and Complex Numbers
Simplify the following expressions.
2
a. 3  2
b. 9
9
 3  2 3  2
3

3i
 9  6 2  2
2
1
2

 9  6i 2  i 2
i




 
 
 9  6i 2  2
 7  6i 2


 i

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