Transcript 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
49
much easier than using the distributive property.
2 7i 6
49
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 2i
2i 2i
2i
4 1
2i
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