Section 76 Complex Numbers
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1
2
3Section 7.6 Complex Numbers
Objective 1: Simplify Powers of i
Definition Imaginary Unit i
i , where
i .
2
The imaginary unit i is defined as
1
1
Consider some powers of i and look for patterns.
n
i
Simplifying
Step 1. Divide n by 4 and find the remainder r .
i .
n
r
Step 2. Replace the exponent (power) on i by the remainder,
i
i ,
i ,
i , and
to simplify if necessary.
0
1
2
3
Step 3. Use the results
1
i
1
i
i
7.6.5 Evaluate each term, then find the sum
Complex Numbers
The set of all numbers of the form
a bi
,
where a and b are real numbers and i is the imaginary unit, is called the set of complex
numbers. The number a is called the real part, and the number b is called the imaginary
part.
b , then the complex number is a purely real number. If
a , then the complex number is
If
0
0
a purely imaginary number. The figure illustrates the relationships between complex numbers.
Complex Numbers
a bi
Non-real
Pure Real Numbers
Complex Numbers
a
b i b
,
0
a
b i
,
b
0
Other Non-real
Pure Imaginary
Complex Numbers
Numbers
a
b i a
,
0 ,
b
0
a bi a
,
0,
b
0
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