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Irrational and Complex Numbers

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Chapter 6 Irrational and Complex Numbers Example 3 Solve (2x + 5) =2 2x + 1 Answer X = 2/9 Section 6-6 Rational and Irrational Numbers Completeness Property of ... – PowerPoint PPT presentation

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Title: Irrational and Complex Numbers


1
Chapter 6
  • Irrational and Complex Numbers

2
Section 6-1
  • Roots of Real Numbers

3
Square Root
  • A square root of a number b is a solution of the
    equation x2 b. Every positive number b has two
    square roots, denoted vb and -vb.

4
Principal Square Root
  • The positive square root of b is the principal
    square root
  • The principal square root of 25 is 5

5
Examples Square Root
  • Simplify
  • x2 9
  • x2 4 0
  • 5x2 15

6
Cube Root
  • A cube root of b is a solution of the equation
  • x3 b.

7
Examples Cube Root
  • Simplify
  • 3v8
  • 3v27
  • 3v106
  • 3va9

8
nth root
  • is the solution of xn b
  • If n is even, there could be two, one or no nth
    root
  • If n is odd, there is exactly one nth root

9
Examples nth root
  • Simplify
  • 4v81
  • 5v32
  • 5v-32
  • 6v-1

10
Radical
  • The symbol nvb is called a radical
  • Each symbol has a name
  • n index
  • v radical
  • b radicand

11
Section 6-2
  • Properties of Radicals

12
Product and Quotient Properties of Radicals
  • 1. nvab nva nvb
  • 2. nvab nva nvb

13
Examples
  • Simplify
  • 3v25 3v10
  • 3v(81/8)
  • v2a2b
  • v36w3

14
Rationalizing the Denominator
  • Create a perfect square, cube or other power in
    the denominator in order to simplify the answer
    without a radical in the denominator

15
Examples
  • Simplify
  • v(5/3)
  • 4
  • 3vc

16
Theorems
  • 1. If each radical represents a real number, then
  • nqvb nv(qvb).
  • 2. If nvb represents a real number, then
  • nvbm (nvb)m

17
Examples
  • Give the decimal approximation to the nearest
    hundredth.
  • 4v100
  • 3v1702

18
Section 6-3
  • Sums of Radicals

19
Like Radicals
  • Two radicals with the same index and same
    radicand
  • You add and subtract like radicals in the same
    way you combine like terms

20
Examples
  • Simplify
  • v8 v98
  • 3v81 - 3v24
  • v32/3 v2/3

21
Examples
  • Simplify
  • v12x5 - xv3x3 5x2v3x
  • Answer
  • 6x2v3x

22
Section 6-4
  • Binomials Containing Radicals

23
Multiplying Binomials
  • You multiply binomials with radicals just like
    you would multiply any binomials.
  • Use the FOIL method to multiply binomials

24
Examples
  • Simplify
  • (4 v7)(3 2v7)
  • Answer
  • 26 11v7

25
Conjugate
  • Expressions of the form avb cvd and avb - cvd
  • Conjugates can be used to rationalize denominators

26
Example - Conjugate
  • Simplify
  • 3 v5
  • 3 - v5
  • Answer
  • 7 3v5
  • 2

27
Example - Conjugate
  • Simplify
  • 1
  • 4 - v15
  • Answer
  • 4 v15

28
Section 6-5
  • Equations Containing Radicals

29
Radical Equation
  • An equation which contains a radical with a
    variable in the radicand.
  • 40 v22d

30
Solving a Radical Equation
  • First isolate the radical term on one side of the
    equation

31
Solving a Radical Equation - Continued
  • If the radical term is a square root, square both
    sides
  • If the radical term is a cube root, cube both
    sides

32
Example 1
  • Solve
  • v(2x 1) 3
  • Answer
  • X 5

33
Example 2
  • Solve
  • 23vx 1 3
  • Answer
  • X 8

34
Example 3
  • Solve
  • v(2x 5) 2v2x 1
  • Answer
  • X 2/9

35
Section 6-6
  • Rational and Irrational Numbers

36
Completeness Property of Real Numbers
  • Every real number has a decimal representation,
    and every decimal represents a real number

37
Remember
  • A rational number is any number that can be
    expressed as the ratio or quotient of two integers

38
Decimal Representation
  • Every rational number can be represented by a
    terminating decimal or a repeating decimal

39
Example 1
  • Write each terminating decimal as a fraction in
    lowest terms.
  • 2.571
  • 0.0036

40
Example 2
  • Write each repeating decimal as a fraction in
    lowest terms.
  • 0.32727
  • 1.89189189

41
Remember
  • An irrational number is a real number that is not
    rational

42
Decimal Representation
  • Every irrational number is represented by an
    infinite and nonrepeating decimal
  • Every infinite and nonrepeating decimal
    represents an irrational number

43
Example 3
  • Classify each number as either rational or
    irrational
  • v2 v4/9
  • 2.0303 2.030030003

44
Section 6-7
  • The Imaginary Number i

45
Definition
  • i v-1
  • and
  • i2 -1

46
Definition
  • If r is a positive real number, then
  • v-r ivr

47
Example 1
  • Simplify
  • v-5
  • v-25
  • v-50

48
Combining imaginary Numbers
  • Combine the same way you combine like terms
  • v-16 - v-49
  • iv2 3iv2

49
Multiply - Example
  • Simplify
  • v-4 ? v-25
  • iv2 ? iv3

50
Divide - Example
  • Simplify
  • 2
  • 3i
  • 6
  • v-2

51
Example
  • Simplify
  • v-9x2 v-x2
  • v-6y ? v-2y

52
Section 6-8
  • The Complex Number

53
Complex Numbers
  • Real numbers and imaginary numbers together form
    the set of complex numbers
  • The form a bi, represents a complex number

54
Equality of Complex Numbers
  • a bi c di
  • if and only if
  • a c and b d

55
Sum of Complex Numbers
  • (a bi ) (c di ) (a c) (b d)i

56
Product of Complex Numbers
  • (a bi )?(c di ) (ac bd) (ad bc)i

57
Example 1
  • Simplify
  • (3 6i) (4 2i)
  • (3 6i) - (4 2i)

58
Example 2
  • Simplify
  • (3 4i)(5 2i)
  • (3 4i)2
  • (3 4i)(3 - 4i)

59
Using Conjugates
  • Simplify using conjugates
  • 5 i
  • 2 3i

60
Reciprocals
  • Find the reciprocal of
  • 3 i
  • Remember
  • the reciprocal of x 1/x

61
THE END!
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