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7.7
Complex Numbers
Imaginary Numbers
Previously, when we encountered square roots of negative numbers in solving equations, we would say “no real solution” or “not a real number”.
That’s still true. However, we will now introduce a new set of numbers.
Imaginary numbers which includes the imaginary unit i.
Imaginary Numbers
The imaginary unit, written i, is the number whose square is ‒1. That is,
2 1 and 1i i
Write using i notation.
a.
b.
c.
25
32
121
Examples
Multiply or divide as indicated.
a.
b.
3 5
125
5
Example
A complex number is a number that can be written in the form a + bi where a and b are real numbers.
a is a real number and bi would be an imaginary number.
If b = 0, a + bi is a real number.
If a = 0, a + bi is an imaginary number.
Standard Form of Complex Numbers
Adding and Subtracting Complex Numbers
Add or subtract as indicated.
a.
b.
(8 + 2i) – (4i)
(4 + 6i) + (3 – 2i)
Example
To multiply two complex numbers of the form a + bi, we multiply as though they were binomials. Then we use the relationshipi2 = – 1 to simplify.
Multiplying Complex Numbers
Multiply:
a. 8i · 7i b. -4i · 7
c. 3i ·3i ·3i d. 2i ·3i ·i ·5i
Example
Multiply.
a. 5i(4 – 7i) b. 4i(3i + 5)
Example
Multiply.
(6 – 3i)(7 + 4i)
Example
In the previous chapter, when trying to rationalize the denominator of a rational expression containing radicals, we used the conjugate of the denominator.
Similarly, to divide complex numbers, we need to use the complex conjugate of the number we are dividing by.
Complex Conjugate
The complex numbers a + bi and a – bi are called complex conjugates of each other.
(a + bi)(a – bi) = a2 + b2
Complex Conjugate
Divide.
Example
6 2
4 3
i
i
Divide.
Example
5
6i
i
2i
3i4i
5i
6i
7i
8i
Patterns of i
Find each power of i.
a. b.
c. d.
21i
32i
Example
53i
17i