Section 7.6 Complex Numbers
Objective 1: Simplify Powers of i
Definition Imaginary Unit i
The imaginary unit i is defined as
i  1 , where i 2  1 .
Consider some powers of i and look for patterns.
Simplifying i n
Step 1. Divide n by 4 and find the remainder r .
Step 2. Replace the exponent (power) on i by the remainder, i n  i r .
Step 3. Use the results i 0  1 , i1  i , i 2  1 , and i3  i to simplify if necessary.
7.6.5 Evaluate each term, then find the sum
Complex Numbers
a  bi ,
The set of all numbers of the form
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.
If b  0 , then the complex number is a purely real number. If a  0 , then the complex number is
a purely imaginary number. The figure illustrates the relationships between complex numbers.
Complex Numbers
a  bi
Pure Real Numbers
a  b i, b  0
Non-real
Complex Numbers
a  bi , b  0
Pure Imaginary
Numbers
a  bi, a  0, b  0
Other Non-real
Complex Numbers
a  b i, a  0, b  0
Objective 2: Add and subtract complex numbers
Adding and Subtracting Complex Numbers
To add complex numbers, add the real parts and add the imaginary parts.
 a  bi    c  di    a  c    b  d  i
To subtract complex numbers, subtract the real parts and subtract the imaginary parts.
 a  bi    c  di    a  c    b  d  i
7.6.13 Find the difference.
Objective 3: Multiply complex numbers
When multiplying complex numbers, use the distributive property and the FOIL method as when
multiplying polynomials. Remember that
when simplifying.
7.6.19 Perform the indicated operation.
7.6.21 Multiply.
Complex Conjugates
The complex numbers  a  bi  and  a  bi  are called complex conjugates of each other. A
complex conjugate is obtained by changing the sign of the imaginary part in a complex number. Also,
 a  bi   a  bi   a2  b2 .
7.6.22 Find the product of the complex number and its conjugate.
Objective 4: Divide complex numbers
The goal in dividing complex numbers is to eliminate the imaginary part from the denominator
and to express the quotient in standard form
. To do this, multiply the numerator and
denominator by the complex conjugate of the denominator.
7.6.29 Write the quotient in the form
.
Objective 5: Simplify radicals with negative radicands
Square Root of a Negative Number
a  1  a  i a
For any positive real number a ,
CAUTION: When simplifying or performing operations involving radicals with a negative radicand and
an even index, it is important to first write the numbers in terms of the imaginary unit i if possible.
The property a  b  ab is only true when a  0 and b  0 so that a and b are real
numbers. This property does not apply to non-real numbers. To find the correct answer if a or b are
negative, we must first write each number in terms of the imaginary unit i .
 3   12  
3  12 
36  6 False
3  12  1  3  1  12  i 3  i 12  i 2
i
7.6.34 Write the expression in the form
7.6.35 Write the expression in the form
i
i 1
2
36  6 True