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Copyright © 2014, 2010, 2007 Pearson Education, Inc. 3 Complex Numbers and Imaginary Numbers The set of all numbers in the form a + bi with real numbers a and b, and i, the imaginary unit, is called the set of complex numbers. The imaginary unit i is defined as The standard form of a complex number is a + bi.
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Chapter 1Equations andInequalities
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1
1.4 Complex Numbers
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Objectives:
• Add and subtract complex numbers.
• Multiply complex numbers.
• Divide complex numbers.
• Perform operations with square roots of negative numbers.
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Complex Numbers and Imaginary Numbers
The set of all numbers in the form a + bi
with real numbers a and b, and i, the imaginary unit, is called the set of complex numbers.
The imaginary unit i is defined as 21, where 1.i i
The standard form of a complex number isa + bi.
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Operations on Complex Numbers
The form of a complex number a + bi is like the binomiala + bx. To add, subtract, and multiply complex numbers, we use the same methods that we use for binomials.
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Example: Adding and Subtracting Complex Numbers
Perform the indicated operations, writing the resultin standard form:
a. (5 2 ) (3 3 )i i
(5 3) ( 2 3)i i 8 i
b. (2 6 ) (12 )i i
2 6 12i i (2 12) (6 1)i
10 7i
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Example: Multiplying Complex Numbers
Find the product:
a. 7 (2 9 )i i214 63i i
14 63( 1)i
14 63i
63 14i
b. (5 4 )(6 7 )i i
5(6) 5( 7 ) 4 (6) 4 ( 7 )i i i i 230 35 24 28i i i
30 11 28( 1)i
30 11 28i
58 11i
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Conjugate of a Complex Number
For the complex number a + bi, we define its complex conjugate to be a – bi.
The product of a complex number and its conjugateis a real number.( )( )a bi a bi
( ) ( ) ( ) ( )a a a bi bi a bi bi 2 2 2a abi abi b i 2 2( 1)a b 2 2a b
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Complex Number Division
The goal of complex number division is to obtain a realnumber in the denominator. We multiply the numeratorand denominator of a complex number quotient by theconjugate of the denominator to obtain this real number.
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Example: Using Complex Conjugates to Divide Complex Numbers
Divide and express the result in standard form:5 44
ii
5 4 44 4
i ii i
2
220 5 16 416 4 4
i i ii i i
20 21 ( 4)16 ( 1)
i
16 2117
i
16 21 17 17
i
In standard form, the result is
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Principal Square Root of a Negative Number
For any positive real number b, the principal square root ofthe negative number – b is defined by
b i b
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Example: Operations Involving Square Roots of Negative Numbers
Perform the indicated operations and write the result instandard form:a. 27 48
3 3 4 3i i
7 3i
2b. ( 2 3)
22 3
2 3 2 3i i 24 2 3 2 3 3i i i
4 4 3 3( 1)i
1 4 3i
