Lesson 1.9 the set of rational numbers

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1. Lesson 1.9 The Set of Rational Numbers

2. Rational Numbers

3. Rational Number The set of rational numbers is the set of all numbers which can be expressed in the formed: The decimal representation of a rational number either terminates or repeats .   Where a and b are integers , b ≠

4. Rational Number The set of rational numbers is the set of all numbers which can be expressed in the formed: The decimal representation of a rational number either terminates or repeats .   Where a and b are integers , b ≠     Example 1:

5. Rational Number The set of rational numbers is the set of all numbers which can be expressed in the formed: Where a and b are integers , b ≠     Example 2:

6. Rational Number The set of rational numbers is the set of all numbers which can be expressed in the formed: Where a and b are integers , b ≠     Example 3:

7. Rational Number The set of rational numbers is the set of all numbers which can be expressed in the formed: Where a and b are integers , b ≠   15 =   Example 4:

8. Rational Number The set of rational numbers is the set of all numbers which can be expressed in the formed: Where a and b are integers , b ≠   -3 =   Example 5:

9. Rational Number The set of rational numbers is the set of all numbers which can be expressed in the formed: Where a and b are integers , b ≠     = 3 Example 6:

10. Rational Number The set of rational numbers is the set of all numbers which can be expressed in the formed: Where a and b are integers , b ≠     = 3 =   Example 6:

11. Rational Number The set of rational numbers is the set of all numbers which can be expressed in the formed: Where a and b are integers , b ≠     = 5 Example 7:

12.   Rational Number The set of rational numbers is the set of all numbers which can be expressed in the formed: Where a and b are integers , b ≠   5   = = Therefore, we can say that all perfect squares can be expressed as rational numbers. Example 7:

13. Take note of this!

14.   Rational Number The set of rational numbers is the set of all numbers which can be expressed in the formed: Where a and b are integers , b ≠     Example 8:

15. Rational Number The set of rational numbers is the set of all numbers which can be expressed in the formed: The decimal representation of a rational number either terminates or repeats .   Where a and b are integers , b ≠

16. Rational Number The set of rational numbers is the set of all numbers which can be expressed in the formed: The decimal representation of a rational number either terminates or repeats .   Where a and b are integers , b ≠

17. Terminating and Repeating Decimals Terminating D ecimal  a decimal number that ends with a remainder of zero. Ex. 1.25, 0.75, 1.5 Repeating Decimal a decimal number whose answer will have one or more digits in a pattern that repeats indefinitely. Ex. 0.33, 0.166, 0.55

18. Rational Number The set of rational numbers is the set of all numbers which can be expressed in the formed: The decimal representation of a rational number either terminates or repeats .   Where a and b are integers , b ≠

19. Rational Number The set of rational numbers is the set of all numbers which can be expressed in the formed: The decimal representation of a rational number either terminates or repeats .   Where a and b are integers , b ≠ The set of rational numbers is the set of all numbers which can be expressed in the formed Where a and b are integers , b ≠   .75 =   Terminating Decimal Example 9:

20. Rational Number The decimal representation of a rational number either terminates or repeats . 0.25 =   The set of rational numbers is the set of all numbers which can be expressed in the formed Where a and b are integers , b ≠   Terminating Decimal Example 10:

21. Rational Number The decimal representation of a rational number either terminates or repeats . 0.5 =   The set of rational numbers is the set of all numbers which can be expressed in the formed Where a and b are integers , b ≠   Terminating Decimal Example 11:

22. Rational Number The decimal representation of a rational number either terminates or repeats . 1 .5 =   The set of rational numbers is the set of all numbers which can be expressed in the formed Where a and b are integers , b ≠   Terminating Decimal Example 12:

23. Rational Number The decimal representation of a rational number either terminates or repeats . .33 =   The set of rational numbers is the set of all numbers which can be expressed in the formed Where a and b are integers , b ≠   Repeating Decimal Example 13:

24. Rational Number The decimal representation of a rational number either terminates or repeats . 0.66 =   The set of rational numbers is the set of all numbers which can be expressed in the formed Where a and b are integers , b ≠   Repeating Decimal Example 14:

25. Rational Number The decimal representation of a rational number either terminates or repeats . 1.66 =   The set of rational numbers is the set of all numbers which can be expressed in the formed Where a and b are integers , b ≠   Repeating Decimal Example 15:

26. questions?

27. Changing a number as a Quotient of Two Integers

28. Changing a number as a Quotient of Two Integers Example 1:       Make the whole number as the numerator and put 1 as its denominator .  

29. Changing a number as a Quotient of Two Integers Example 2:       Make the whole number as the numerator and put 1 as its denominator .  

30. Changing a number as a Quotient of Two Integers Example 3:       Make the whole number as the numerator and put 1 as its denominator .  

31. Changing a number as a Quotient of Two Integers Example 4 :       Make the whole number as the numerator and put 1 as its denominator .  

32. Changing a number as a Quotient of Two Integers Example 5 :       Make the whole number as the numerator and put 1 as its denominator .  

33. questions?

34. Changing a number as a Quotient of Two Integers   =   =   =   =   =   14   =   =   =   =   =  

35. Converting a Terminating Decimal to a Rational Number

36. Use the place value of the last digit in the number to determine what the denominator of the fraction will be. Converting a Terminating Decimal to a Rational Number

37. Write the repeating decimal number as the numerator . Converting a Terminating Decimal to a Rational Number Example 1:     Use the place value of the last digit in the number to determine what the denominator of the fraction will be.     25 100

38. Converting a Terminating Decimal to a Rational Number Example 1:       Simplify.       1 4

39. Converting a Terminating Decimal to a Rational Number Example 1:      

40. Write the repeating decimal number as the numerator . Converting a Terminating Decimal to a Rational Number Example 2 :     Use the place value of the last digit in the number to determine what the denominator of the fraction will be.   5 10

41. Converting a Terminating Decimal to a Rational Number Example 2 :       Simplify.       1 2

42. Converting a Terminating Decimal to a Rational Number Example 2:      

43. Write the repeating decimal number as the numerator . Converting a Terminating Decimal to a Rational Number Example 3:     Use the place value of the last digit in the number to determine what the denominator of the fraction will be.     125 1000  

44. Converting a Terminating Decimal to a Rational Number Example 3:       Simplify.       1 8

45. Converting a Terminating Decimal to a Rational Number Example 3:      

46. Write the repeating decimal number as the numerator . Converting a Terminating Decimal to a Rational Number Example 4 :     Use the place value of the last digit in the number to determine what the denominator of the fraction will be.       100

47. Converting a Terminating Decimal to a Rational Number Example 4 :       Simplify.       21 25

48. Converting a Terminating Decimal to a Rational Number Example 4 :      

49. Write the repeating decimal number as the numerator . Converting a Terminating Decimal to a Rational Number Example 5:     Use the place value of the last digit in the number to determine what the denominator of the fraction will be.       100   Put the whole number as is.

50. Converting a Terminating Decimal to a Rational Number Example 5:       Simplify.       1 4    

51. Converting a Terminating Decimal to a Rational Number Example 5:     Change to improper fraction .   5 Multiply the denominator with the whole number then add the numerator with the result . This will be your new numerator .      

52. Converting a Terminating Decimal to a Rational Number Example 5:       5 Copy the denominator . 4

53. Converting a Terminating Decimal to a Rational Number Example 5:      

54. Write the repeating decimal number as the numerator . Converting a Terminating Decimal to a Rational Number Example 6 :     Use the place value of the last digit in the number to determine what the denominator of the fraction will be.       100   Put the whole number as is.

55. Converting a Terminating Decimal to a Rational Number Example 6 :       Simplify.       3 4    

56. Converting a Terminating Decimal to a Rational Number Example 6 :     Change to improper fraction .   11 Multiply the denominator with the whole number then add the numerator with the result . This will be your new numerator .      

57. Converting a Terminating Decimal to a Rational Number Example 6 :       Copy the denominator . 4 11

58. Converting a Terminating Decimal to a Rational Number Example 6 :      

59. questions?

60. Converting a Terminating Decimal to a Rational Number     =   =   =   =  

61. questions?

62. Converting a Repeating Decimal to a Rational Number

63. Converting a Repeating Decimal to a Rational Number Example 1:   Let n = 0.3 Multiply both sides by the power of 10 , where the exponent is determined by the number of digits in the block of repeating digits .     3       Subtract the result by  

64. Converting a Repeating Decimal to a Rational Number Example 1:         Subtract the result by  

65.   Converting a Repeating Decimal to a Rational Number Example 1:       Subtract the result by                                

66. Converting a Repeating Decimal to a Rational Number Example 1:      

67. Converting a Repeating Decimal to a Rational Number Example 2:   Let n = 3.21 Multiply both sides by the power of 10 , where the exponent is determined by the number of digits in the block of repeating digits .               Subtract the result by  

68. Converting a Repeating Decimal to a Rational Number Example 2:           Subtract the result by  

69. Converting a Repeating Decimal to a Rational Number Example 2:         Subtract the result by                                  

70. Converting a Repeating Decimal to a Rational Number Example 2:          

71. Converting a Repeating Decimal to a Rational Number Example 3:   Let n = 0.34 Multiply both sides by the power of 10 , where the exponent is determined by the number of digits in the block of repeating digits .               Subtract the result by  

72. Converting a Repeating Decimal to a Rational Number Example 3:           Subtract the result by  

73. Converting a Repeating Decimal to a Rational Number Example 3:         Subtract the result by                            

74. Converting a Repeating Decimal to a Rational Number Example 3:      

75. questions?

76. Converting a Repeating Decimal to a Rational Number   =   =   =   =   =  

77. questions?

78. E-Math 7 Practice and Application Test I Page 57

79. Expressing a Rational Number as a Terminating or Repeating Decimal

80. Expressing a Rational Number as a Terminating or Repeating Decimal In writing a rational number as a Terminating or Repeating Decimals, we simply divide the numerator with its denominator .

81. Expressing a Rational Number as a Terminating or Repeating Decimal Example 1: In writing a rational number as a Terminating or Repeating Decimals, we simply divide the numerator with its denominator .   3 2 .0 . 6 1 8 2 .6

82. Expressing a Rational Number as a Terminating or Repeating Decimal Example 2:   6 5 .0 . 8 4 8 2 3 18 2 .83 In writing a rational number as a Terminating or Repeating Decimals, we simply divide the numerator with its denominator .

83. Example 3: Express as a decimal. . 3 -24 8 3 .00 6 Begin long division Add decimal and zeros as needed Align decimal and divide * Terminate * Repeat OR 7 -56 4 5 -40 Answer will:

84. Express as a decimal. . 7 -28 4 3 .00 2 5 -20 Example 4:

85. Example 5: Convert to a decimal. . 5 1 - 8 8 1 .00 2 * Set up long division on the fraction part and complete the division. 2 -16 4 5 -40 . * Place whole number in front of decimal * This is the final answer

86. . Convert to a decimal. Example 6: . 12 6 -48 8 5 .00 2 2 -16 4 5 -40

87. questions?

88. Converting a Repeating Decimal to a Rational Number 3              

89. E-Math 7 Practice and Application Test II Page 57

90. Comparing and Ordering Rational Numbers

91. Comparing and Ordering Rational Numbers A number line can be used to order rational numbers . If a rational number lies to the right of another, then it is greater than the other rational number v ice versa. -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 Number Line

92. Comparing and Ordering Rational Numbers -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 A number line can be used to order rational numbers . If a rational number lies to the right of another, then it is greater than the other rational number vice versa. Example 1 : Arrange the following rational numbers in ascending order.              

93. Comparing and Ordering Rational Numbers -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 A number line can be used to order rational numbers . If a rational number lies to the right of another, then it is greater than the other rational number vice versa. Example 1 : Arrange the following rational numbers in ascending order.              

94. Comparing and Ordering Rational Numbers -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 A number line can be used to order rational numbers . If a rational number lies to the right of another, then it is greater than the other rational number vice versa. Example 1 : Arrange the following rational numbers in ascending order.     You can express a rational number to its decimal form to make it more easier to locate its place on the number line . 0.75  

95. Comparing and Ordering Rational Numbers -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 A number line can be used to order rational numbers . If a rational number lies to the right of another, then it is greater than the other rational number vice versa. Example 1 : Arrange the following rational numbers in ascending order.       You can express a rational number to its decimal form to make it more easier to locate its place on the number line .  

96. Comparing and Ordering Rational Numbers -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 A number line can be used to order rational numbers . If a rational number lies to the right of another, then it is greater than the other rational number vice versa. Example 1 : Arrange the following rational numbers in ascending order.       You can express a rational number to its decimal form to make it more easier to locate its place on the number line .    

97. Comparing and Ordering Rational Numbers -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 A number line can be used to order rational numbers . If a rational number lies to the right of another, then it is greater than the other rational number vice versa. Example 1 : Arrange the following rational numbers in ascending order.       You can express a rational number to its decimal form to make it more easier to locate its place on the number line .      

98. Comparing and Ordering Rational Numbers -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 A number line can be used to order rational numbers . If a rational number lies to the right of another, then it is greater than the other rational number vice versa. Example 1 : Arrange the following rational numbers in ascending order.       You can express a rational number to its decimal form to make it more easier to locate its place on the number line .        

99. Comparing and Ordering Rational Numbers -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 A number line can be used to order rational numbers . If a rational number lies to the right of another, then it is greater than the other rational number vice versa. Example 1 : Arrange the following rational numbers in ascending order.             You can express a rational number to its decimal form to make it more easier to locate its place on the number line . 0.75  

100. Comparing and Ordering Rational Numbers -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 A number line can be used to order rational numbers . If a rational number lies to the right of another, then it is greater than the other rational number vice versa. Example 1 : Arrange the following rational numbers in ascending order.             You can express a rational number to its decimal form to make it more easier to locate its place on the number line . 0.75    

101. Comparing and Ordering Rational Numbers -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 A number line can be used to order rational numbers . If a rational number lies to the right of another, then it is greater than the other rational number vice versa. Example 1 : Arrange the following rational numbers in ascending order.             You can express a rational number to its decimal form to make it more easier to locate its place on the number line . 0.75    

102. questions?

103. E-Math 7 Practice and Application Test III – Nos. 23, 24, 25 Page 57

104. Ordering Rational Numbers When fractions have the same denominators , the greater the numerator , the greater the value of a fraction . Let’s try arranging this given example:  

105. Ordering Rational Numbers Example 1: Arrange the rational numbers in ascending order/increasing .   When fractions have the same denominators , the greater the numerator , the greater the value of a fraction .                     Descending order/ decreasing                      

106. Ordering Rational Numbers Example 2: Arrange the given rational numbers in ascending order/increasing .   When fractions have the same denominators , the greater the numerator , the greater the value of a fraction .                     Descending order/ decreasing                      

107. questions?

108. Arrange the given rational numbers in ascending order . 1.   Arrange the given rational numbers in descending order . 2.   3.   1.   2.   3.   =   =   =   =   =   =  

109. questions?

110. Ordering Rational Numbers When fractions have the same numerators , the greater the denominator , the smaller the value of a fraction . Let’s try arranging this given example:  

111. Ordering Rational Numbers Example 1: Arrange the rational numbers in ascending order/increasing .   When fractions have the same numerators , the greater the denominator , the smaller the value of a fraction .                     Descending order/ decreasing                      

112. Ordering Rational Numbers Example 2: Arrange the given rational numbers in ascending order/increasing .   When fractions have the same numerators , the greater the denominator , the smaller the value of a fraction .                     Descending order/ decreasing                      

113. questions?

114. Arrange the given rational numbers in ascending order . 1.   Arrange the given rational numbers in descending order . 2.   3.   1.   2.   3.   =   =   =   =   =   =  

115. questions?

116. Comparing Rational Numbers Which is greater? To find which is greater of the two. Multiply the numerator of the other to the denominator of the other.   6(10) 60

117. Comparing Rational Numbers Which is greater? To find which is greater of the two. Multiply the numerator of the other to the denominator of the other.   60 9(7) 63

118. Comparing Rational Numbers Which is greater? To find which is greater of the two. Multiply the numerator of the other to the denominator of the other.   60 63  

119. Comparing Rational Numbers Which is greater? To find which is greater of the two. Multiply the numerator of the other to the denominator of the other.   3(5) 15

120. Comparing Rational Numbers Which is greater? To find which is greater of the two. Multiply the numerator of the other to the denominator of the other.   15 2(5) 10

121. Comparing Rational Numbers Which is greater? To find which is greater of the two. Multiply the numerator of the other to the denominator of the other.   15 10  

122. Comparing Rational Numbers Which is greater? To find which is greater of the two. Multiply the numerator of the other to the denominator of the other.   7(3) 21

123. Comparing Rational Numbers Which is greater? To find which is greater of the two. Multiply the numerator of the other to the denominator of the other.   21 1(4) 4

124. Comparing Rational Numbers Which is greater? To find which is greater of the two. Multiply the numerator of the other to the denominator of the other.   21 4  

125. Comparing Rational Numbers Which is greater? To find which is greater of the two. Multiply the numerator of the other to the denominator of the other.   9 (13) 117

126. Comparing Rational Numbers Which is greater? To find which is greater of the two. Multiply the numerator of the other to the denominator of the other.   12(10) 120 117

127. Comparing Rational Numbers Which is greater? To find which is greater of the two. Multiply the numerator of the other to the denominator of the other.   120   117

128. questions?

129. Math Time Practice 1.2.6 Test A – Nos.1,3,5,7,9 Test B – Nos. 1,3,5,7,9 Page 28-29