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Name: _______________________________ Period ______
Section 7.5 – Complex Numbers
Big Idea: You have explored several ways to solve quadratic equations. You can find the x-intercepts
on a graph, you can solve by completing the square, or you can use the quadratic formula. What
happens if you try to use the quadratic formula on an equation whose graph has no x-intercepts?
Example A: Solve
x  _________ or x  _________
x2  4 x  5  0
The solution to the problem requires taking the square root of a negative number. The solutions are
unlike any of the numbers you have worked with this year. They are non-real, but they are still
numbers. Numbers that include the real numbers as well as the square roots of negative numbers
are called complex numbers.
To express the square root of a negative number we use an imaginary unit called i , defined by
i 2  1 or i  1 . You can rewrite
4
=
 4 

1 = 2i
Therefore, you can write the two solutions to the quadratic equation above as the complex numbers:
4  2i

2
 2  i and
4  2i

2
 2  i
These two solutions are a conjugate pair. That is, one is
or
2  i and  2  i
a  bi and the other is a  bi . The two
numbers in a conjugate pair are called complex conjugates.
Example B: Solve x  3  0
by using undo operations
2
Example C: Solve x  2 x  4  0
by using the quadratic formula.
2
Powers of i:
i = ______
i 5 = _____________
i 9 = _____________
i 2 =______
i 6 =_____________
i10 =_____________
i 3 =______
i 7 =_____________
i11 =_____________
i 4 =______
i 8 =_____________
i12 =_____________
What pattern do you notice?
How can we use that pattern to find the value of i
How about
i 401
How about
i 402
400
?
How about
i 403
Can a number be both a real number and also an imaginary number? _________
Can a number be both a real number and a rational number? __________
Can a number be both a rational number and an irrational number? ________
Are all integers real numbers?________ Are all rational and irrational numbers real numbers? _____
Give an example of a rational number that is not an integer or natural number. _________
Give an example of an complex number that is not a real number. __________
How do we do arithmetic with complex numbers?
Adding and Subtracting: Add or Subtract Real Parts and Add or Subtract Imaginary Part
(a)
(2  4i)  (3  5i)
You try:
(7  2i)  (2  i)
(b)
(2  4i)  (3  5i)
You try:
(4  4i)  (1  3i)
Multiplying: Multiply like you would any binomial treating i like the variable. Then substitute
(a)
(2  4i )(3  5i )
You try:
(7  3i)(2  i)
(b)
i 2  1
(2  5i )(2  5i )
You try:
(4  3i )(4  3i)
What do you notice about the answer from part b?
We say that
(2  5i) and (2  5i) are complex conjugates. Notice that when a complex
number and it’s complex conjugate are multiplied together the product is a real number.
Complex Conjugate: Verify that a complex number and it’s complex conjugate always give you a
real number.
Multiply:
(a  bi )(a  bi )
Dividing: We do not divide complex numbers. Rather, we simplify the fraction by multiplying the
numerator and denominator by the complex conjugate of the denominator.
(a)
7  2i
1 i
What is the complex conjugate of the denominator?
 7  2i 


 1  i 
(b)
2i
4  6i



What is the complex conjugate of the denominator?
 2  i 


 4  6i 
You try:
Multiply the top and bottom by this complex conjugate.
2  4i
2  4i
 2  4i 


 2  4i 



Multiply the top and bottom by this complex conjugate.
What is the complex conjugate of the denominator?



Multiply the top and bottom by this complex conjugate.