perform computations involving complex numbers. copyright © 2012 pearson education, inc. publishing...
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Perform computations involving complex numbers.
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
3.1 The Complex Numbers
Complex Numbers
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A complex number is a number of the form a + bi, where a and b are real numbers. The number a is said to be the real part of a + bi and the number b is said to be the imaginary part of a + bi.
The symbol i represents .
Imaginary Number a + bi, a ≠ 0, b ≠ 0
Pure Imaginary Number a + bi, a = 0, b ≠ 0
The Complex Number System
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The Complex-Number System
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Some functions have zeros that are not real numbers.
The complex-number system is used to find zeros of functions that are not real numbers.
When looking at a graph of a function, if the graph does not cross the x-axis, then it has no x-intercepts, and thus it has no real-number zeros.
Example
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Express each number in terms of i.
a. 7 b. 16 c. 13
d. 64 e. 48
Addition and Subtraction
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Complex numbers obey the commutative, associative, and distributive laws.
We add or subtract them as we do binomials.
We collect the real parts and the imaginary parts of complex numbers just as we collect like terms in binomials.
Example
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Add or subtract and simplify each of the following.
a. (8 + 6i) + (3 + 2i) b. (4 + 5i) – (6 – 3i)
Multiplication
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When and are real numbers,
This is not true when and are not real numbers.
Note: Remember i2 = –1
a b a b ab.
a b
Example
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Multiply and simplify each of the following.
a. 16 25 b. 1 2i 1 3i c. 3 7i 2
Example
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Simplify each of the following37a. i 58b. i 75c. i 80d. i
Conjugates
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The conjugate of a complex number a + bi is a bi. The numbers a + bi and a bi are complex conjugates.
Examples: 3 + 7i and 3 7i 14 5i and 14 + 5i 8i and 8i
The product of a complex number and its conjugate is a real number.
Multiplying Conjugates - Example
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Multiply each of the following.
a. (5 + 7i)(5 – 7i) b. (8i)(–8i)
Dividing Using Conjugates - Example
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Divide 2 5i by 1 6i.