Complex Numbers, p' 126130 1'5

1
Complex Numbers, p. 126-130 (1.5)

• OBJECTIVES
• Use the imaginary unit i to write complex numbers
• Add, subtract, and multiply complex numbers
• Use quadratic formula to find complex solutions
  of quadratic equations

2

• Consider the quadratic equation x2 1 0.
• What is the discriminant ?
• a 1 , b 0 , c 1 therefore the discriminant
  is
• 02 4 (1)(1) 4
• If the discriminant is negative, then the
  quadratic equation has no real solution. (p. 114)
  
• Solving for x , gives x2 1

We make the following definition
3
Note that squaring both sides yields

• Real numbers and imaginary numbers are subsets of
  the set of complex numbers.

Complex Numbers
Imaginary Numbers
Real Numbers
4
Definition of a Complex Number p. 126

• If a and b are real numbers, the number a bi is
  a complex number written in standard form.
• If b 0, the number a bi a is a real number.

If , the number a bi is called an
imaginary number. A number of the form bi, where
, is called a pure imaginary number.
Write the complex number in standard form Try p.
131 5-16
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Equality of Complex Numbers p. 126

• Two complex numbers a bi and c di, are equal
  to each other if and only if a c and b d
• Find real numbers a and b such that the equation
  ( a 6 ) 2bi 6 5i .
• a 6 6 2b 5
• a 0 b 5/2
• p. 131 1-4

6
Addition and Subtraction of Complex Numbers,p.
127

• If a bi and c di are two complex numbers
  written in standard form, their sum and
  difference are defined as follows.

Sum
Difference
7

• Perform the subtraction and write the answer in
  standard form.
• 20 ( 3 2i ) ( 6 13i )
• 3 2i 6 13i
• 3 11i
• 22

4 Try p. 131 17-26
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Properties for Complex Numbers p.126

• Associative Properties of Addition and
  Multiplication
• Commutative Properties of Addition and
  Multiplication
• Distributive Property of Multiplication
• Multiplying complex numbers is similar to
  multiplying polynomials and combining like terms.
  
• 28 Perform the operation and write the result
  in standard form. ( 6 2i )( 2 3i )
• F O I L
• 12 18i 4i 6i2
• 12 22i 6 ( -1 )
• 6 22i

9

• Consider ( 3 2i )( 3 2i )
• 9 6i 6i 4i2
• 9 4( -1 )
• 9 4
• 13
• This is a real number. The product of two
  complex numbers can be a real number.

10
Complex Conjugates and Division p. 129

• Complex conjugates-a pair of complex numbers of
  the form a bi and a bi where a and b
  are real numbers.
• ( a bi )( a bi )
• a 2 abi abi b 2 i 2
• a 2 b 2( -1 )
• a 2 b 2
• The product of a complex conjugate pair is a
  positive real number.

11

• To find the quotient of two complex numbers
  multiply the numerator and denominator by the
  conjugate of the denominator.

12
p. 131 50

• Perform the operation and write the result in
  standard form. (Try p.131 45-54)

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Principle Square Root of a Negative Number,p. 130
If a is a positive number, the principle square
root of the negative number a is defined as
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p.128 66

• Use the Quadratic Formula to solve the quadratic
  equation.
• 9x2 6x 37 0
• a 9 , b - 6 , c 37
• What is the discriminant?
• ( - 6 ) 2 4 ( 9 )( 37 )
• 36 1332
• -1296
• Therefore, the equation has no real solution.

15

• 9x2 6x 37 0
• a 9 , b - 6 , c 37

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HOMEWORK

• Work p. 131-132 1-54, 1-82, 93-103 alt. odd
• Read p. 189-196 (2.2)
• PRE QUIZ (2.2)
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