Download Section 8.7 Complex Numbers

11/14/2012
Section 8.7
Complex Numbers
The set of complex numbers include all real numbers Th
t f
l
b i l d ll
l
b
as well as numbers that are even roots of negative numbers.
Objective 1: Simplify numbers of the form −b, where b > 0.
Imaginary Unit i
The imaginary unit i is defined as
i = , where i
= −1 where i2 = = ‐1
1.
In words, i is the principal square root of ‐1.
−b
For any positive number b, -b = i b.
You must change −b to the form i b
before performing any multiplications or divisons.
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Write each number as a product of a real number and i. Simplify all radical expressions.
−225
- -144
-96
Multiply or divide as indicated.
-7 • −15
-5 i 13
-40
40
-10
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-160
10
Objective 2: Recognize complex numbers.
Objective
2: Recognize complex numbers
If a and b are real numbers then any number of the form a + bi is call a complex number. The number a is called the real part and b is called the imaginary
part.
Objective 3: Add and subtract complex numbers.
Add the real parts and add the imaginary parts.
Add or subtract as indicated. Write your answers in standard form. Standard form is a + bi.
(7 + 15i) + (‐11
(7 + 15i) + (
11 + 14i)
+ 14i)
(‐2 + 6i) + (2 – 6i)
(‐2 – 30i) – (‐5 – 3i)
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(‐1 + i) + (2 + 5i) + (3 + 2i)
Objective 4: Multiply complex numbers.
Multiply just like you multiply polynomials.
(5i)(125i)
(3i)(4 + 9i)
( 2i)(3 + i)
(7 –
)(
)
(3 + 2i)2
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Objective 5: Divide complex numbers.
The quotient of two complex numbers must be a complex number. This means you must get rid of the i in the denominator. Do this by using conjugates just as you do to rationalize a denominator.
Write each quotient in the from a + bi.
29
5 + 2i
-38 - 8i
7 + 3i
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-8i
1+i
-1 + 5i
3 + 2i
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