# A Characteristic Property of the Algebra C(Q)B by Karakhanyan M.I., Khor'kova T.A. By Karakhanyan M.I., Khor'kova T.A.

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4 ϭ 0 Subtracting Real Numbers We can describe the subtraction of real numbers in terms of addition. ” In other words, every subtraction problem can be changed to an equivalent addition problem. Consider the following example. 3 7 1 7 1 7 2 5 (d) Ϫ Ϫ aϪ b ϭ Ϫ ϩ ϭ Ϫ ϩ ϭ Ϫ 8 4 8 4 8 8 8 7 1 (d) Ϫ Ϫ aϪ b 8 4 14 Chapter 1 • Basic Concepts and Properties It should be apparent that addition is a key operation. To simplify numerical expressions that involve addition and subtraction, we can first change all subtractions to additions and then perform the additions.

48. 50. 2 • Operations with Real Numbers 3 4 51. aϪ ba b 4 5 53. 3 1 Ϭ aϪ b 4 2 1 4 52. a b aϪ b 2 5 85. 6) 5 7 54. aϪ b Ϭ aϪ b 6 8 87. 6) For Problems 55 – 94, simplify each numerical expression. (Objective 7) 19 86. 8) 88. 6) 89. 9) 90. 5) 55. 9 Ϫ 12 Ϫ 8 ϩ 5 Ϫ 6 56. 6 Ϫ 9 ϩ 11 Ϫ 8 Ϫ 7 ϩ 14 57. Ϫ21 ϩ (Ϫ17) Ϫ 11 ϩ 15 Ϫ (Ϫ10) 91. 2 3 5 Ϫa Ϫ b 3 4 6 58. Ϫ16 Ϫ (Ϫ14) ϩ 16 ϩ 17 Ϫ 19 1 3 1 92. Ϫ Ϫ a ϩ b 2 8 4 1 1 7 59. 7 Ϫ a2 Ϫ 3 b 8 4 8 1 2 5 93. 3a b ϩ 4a b Ϫ 2a b 2 3 6 3 1 3 60. Ϫ4 Ϫ a1 Ϫ 2 b 5 5 10 3 1 3 94.

Finally, let’s consider the product of two negative integers. The following pattern using integers helps with the reasoning. 4(Ϫ2) ϭ Ϫ8 3(Ϫ2) ϭ Ϫ6 2(Ϫ2) ϭ Ϫ4 1(Ϫ2) ϭ Ϫ2 0(Ϫ2) ϭ 0 (Ϫ1)(Ϫ2) ϭ ? To continue this pattern, the product of Ϫ1 and Ϫ2 has to be 2. In general, this type of reasoning helps us realize that the product of any two negative real numbers is a positive real number. Using the concept of absolute value, we can describe the multiplication of real numbers as follows: Multiplication of Real Numbers 1.