College Algebra

1

• College Algebra
• Fifth Edition
• James Stewart ? Lothar Redlin ? Saleem Watson

2
1

• Equations and Inequalities

3
Introduction

• In Section 1.3, we saw that, if the discriminant
  of a quadratic equation is negative, the equation
  has no real solution.
• For example, the equation x2 4 0
  has no real solution.

4
Introduction

• If we try to solve this equation, we get x2
  4
• So,
• However, this is impossiblesince the square of
  any real number is positive.
• For example, (2)2 4, a positive number.
• Thus, negative numbers dont have real square
  roots.

5
1.4

• Complex Numbers

6
Complex Number System

• To make it possible to solve all quadratic
  equations, mathematicians invented an expanded
  number systemcalled the complex number system.

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Complex Number

• First, they defined the new number
• This means i 2 1.
• A complex number is then a number of the form a
  bi, where a and b are real numbers.

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Complex NumberDefinition

• A complex number is an expression of the form
• a bi
• where
• a and b are real numbers.
• i 2 1.

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Real and Imaginary Parts

• The real part of this complex number is a.
• The imaginary part is b.
• Two complex numbers are equal if and only if
  their real parts are equal and their imaginary
  parts are equal.

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Real and Imaginary Parts

• Note that both the real and imaginary parts of a
  complex number are real numbers.

11
E.g. 1Complex Numbers

• Here are examples of complex numbers.
• 3 4i Real part 3, imaginary part 4
• 1/2 2/3i Real part 1/2, imaginary part 2/3
• 6i Real part 0, imaginary part 6
• 7 Real part 7, imaginary part 0

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Pure Imaginary Number

• A number such as 6i, which has real part 0, is
  called
• A pure imaginary number.

13
Complex Numbers

• A real number like 7 can be thought of as
• A complex number with imaginary part 0.

14
Complex Numbers

• In the complex number system, every quadratic
  equation has solutions.
• The numbers 2i and 2i are solutions of x2 4
  because (2i)2 22i2 4(1) 4 and
  (2i)2 (2)2i2 4(1) 4

15
Complex Numbers

• Though we use the term imaginary here, imaginary
  numbers should not be thought of as any less
  realin the ordinary rather than the
  mathematical sense of that wordthan negative
  numbers or irrational numbers.
• All numbers (except possibly the positive
  integers) are creations of the human mindthe
  numbers 1 and as well as the number i.

16
Complex Numbers

• We study complex numbers as they completein a
  useful and elegant fashionour study of the
  solutions of equations.
• In fact, imaginary numbers are useful not only in
  algebra and mathematics, but in the other
  sciences too.
• To give just one example, in electrical theory,
  the reactance of a circuit is a quantity whose
  measure is an imaginary number.

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• Arithmetic Operations on Complex Numbers

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Arithmetic Operations on Complex Numbers

• Complex numbers are added, subtracted,
  multiplied, and divided just as we would any
  number of the form a b
• The only difference we need to keep in mind is
  that i2 1.

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Arithmetic Operations on Complex Numbers

• Thus, the following calculations are valid.
• (a bi)(c di)
• ac (ad bc)i bdi2 (Multiply and collect
  all terms)
• ac (ad bc)i bd(1) (i2 1)
• (ac bd) (ad bc)i (Combine real and
  imaginary parts)

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Arithmetic Operations on Complex Numbers

• We therefore define the sum, difference, and
  product of complex numbers as follows.

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Adding Complex Numbers

• (a bi) (c di) (a c) (b d)i
• To add complex numbers, add the real parts and
  the imaginary parts.

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Subtracting Complex Numbers

• (a bi) (c di) (a c) (b d)i
• To subtract complex numbers, subtract the real
  parts and the imaginary parts.

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Multiplying Complex Numbers

• (a bi) . (c di) (ac bd) (ad bc)i
• Multiply complex numbers like binomials, using i
  2 1.

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E.g. 2Adding, Subtracting, and Multiplying

• Express the following in the form a bi.
• (3 5i) (4 2i)
• (3 5i) (4 2i)
• (3 5i)(4 2i)
• i 23

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E.g. 2Adding
Example (a)

• According to the definition, we add the real
  parts and we add the imaginary parts.
• (3 5i) (4 2i)
• (3 4) (5 2)i 7 3i

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E.g. 2Subtracting
Example (b)

• (3 5i) (4 2i) (3 4) 5 ( 2)i
  1 7i

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E.g. 2Multiplying
Example (c)

• (3 5i)(4 2i) 3 . 4 5( 2) 3(2)
  5 . 4i
• 22 14i

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E.g. 2Power
Example (d)

• i 23 i 22 1 (i 2)11i (1)11i
  (1)i i

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Dividing Complex Numbers

• Division of complex numbers is much like
  rationalizing the denominator of a radical
  expressionwhich we considered in Section P.8.

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Complex Conjugates

• For the complex number z a bi, we define its
  complex conjugate to be

31
Dividing Complex Numbers

• Note that
• So, the product of a complex number and its
  conjugate is always a nonnegative real number.
• We use this property to divide complex numbers.

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Dividing Complex NumbersFormula

• To simplify the quotient multiply the numerator
  and the denominator by the complex conjugate of
  the denominator

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Dividing Complex Numbers

• Rather than memorize this entire formula, its
  easier to just remember the first step and then
  multiply out the numerator and the denominator
  as usual.

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E.g. 3Dividing Complex Numbers

• Express the following in the form a bi.
• We multiply both the numerator and denominator by
  the complex conjugate of the denominator to make
  the new denominator a real number.

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E.g. 3Dividing Complex Numbers
Example (a)

• The complex conjugate of 1 2i is

36
E.g. 3Dividing Complex Numbers
Example (b)

• The complex conjugate of 4i is 4i.

37

• Square Roots of Negative Numbers

38
Square Roots of Negative Numbers

• Just as every positive real number r has two
  square roots , every negative number has
  two square roots as well.
• If -r is a negative number, then its square roots
  are , because and

39
Square Roots of Negative Numbers

• If r is negative, then the principal square root
  of r is
• The two square roots of r are
• We usually write instead of to avoid
  confusion with .

40
E.g. 4Square Roots of Negative Numbers

41
Square Roots of Negative Numbers

• Special care must be taken when performing
  calculations involving square roots of negative
  numbers.
• Although when a and b are positive, this is
  not true when both are negative.

42
Square Roots of Negative Numbers

• For example,
• However,
• Thus,

43
Square Roots of Negative Numbers

• When multiplying radicals of negative numbers,
  express them first in the form (where r gt
  0) to avoid possible errors of this type.

44
E.g. 5Using Square Roots of Negative Numbers

• Evaluate and express in the form a bi.

45

• Complex Solutions of Quadratic Equations

46
Complex Roots of Quadratic Equations

• We have already seen that, if a ? 0, then the
  solutions of the quadratic equation ax2 bx c
  0 are

47
Complex Roots of Quadratic Equations

• If b2 4ac lt 0, the equation has no real
  solution.
• However, in the complex number system, the
  equation will always have solutions.
• This is because negative numbers have square
  roots in this expanded setting.

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E.g. 6Quadratic Equations with Complex Solutions

• Solve each equation.
• x2 9 0
• x2 4x 5 0

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E.g. 6Complex Solutions
Example (a)

• The equation x2 9 0 means x2 9.
• So,
• The solutions are therefore 3i and 3i.

50
E.g. 6Complex Solutions
Example (b)

• By the quadratic formula, we have
• So, the solutions are 2 i and 2 i.

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E.g. 7Complex Conjugates as Solutions of a
Quadratic

• Show that the solutions of the equation
  4x2 24x 37 0 are complex conjugates of
  each other.

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E.g. 7Complex Conjugates as Solutions of a
Quadratic

• We use the quadratic formula to get
• So, the solutions are 3 ½i and 3 ½i.
• These are complex conjugates.