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3: Integers: Addition and Subtraction

3: Integers: Addition and Subtraction

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1.3 • Integers: Addition and Subtraction



{0, 1, 2, 3, 4, . . .}



15



Nonnegative integers



{. . . , Ϫ3, Ϫ2, Ϫ1}



Negative integers



{. . . , Ϫ3, Ϫ2, Ϫ1, 0}



Nonpositive integers



The symbol Ϫ1 can be read as “negative one,” “opposite of one,” or “additive inverse of one.”

The opposite-of and additive-inverse-of terminology is very helpful when working with variables.

The symbol Ϫx, read as “opposite of x” or “additive inverse of x,” emphasizes an important issue:

Since x can be any integer, Ϫx (the opposite of x) can be zero, positive, or negative. If x is a positive integer, then Ϫx is negative. If x is a negative integer, then Ϫx is positive. If x is zero, then Ϫx

is zero. These statements are written as follows and illustrated on the number lines in Figure 1.3.

If x ϭ 3,

then Ϫx ϭ Ϫ(3) ϭ Ϫ3.



x

−4 −3 −2 −1



If x ϭ Ϫ3,

then Ϫx ϭ Ϫ(Ϫ3) ϭ 3.



0



1



2



3



4



0



1



2



3



4



1



2



3



4



x

−4 −3 −2 −1



If x ϭ 0,

then Ϫx ϭ Ϫ(0) ϭ 0.



x

−4 −3 −2 −1



0



Figure 1.3



From this discussion we also need to recognize the following general property.

Property 1.1

If a is any integer, then

Ϫ(Ϫa) ϭ a

(The opposite of the opposite of any integer is the integer itself.)



Addition of Integers

The number line is also a convenient visual aid for interpreting the addition of integers. In

Figure 1.4 we see number line interpretations for the following examples.

Problem



Number line interpretation

3



3ϩ2



3 ϩ (Ϫ2)



Ϫ3 ϩ (Ϫ2)



Ϫ3 ϩ 2 ϭ Ϫ1



−3



−5 − 4 −3 −2 −1 0 1 2 3 4 5

Figure 1.4



3 ϩ (Ϫ2) ϭ 1



−3



−5 − 4 −3 −2 −1 0 1 2 3 4 5

−2



3ϩ2ϭ5



−2



−5 − 4 −3 −2 −1 0 1 2 3 4 5

2



Ϫ3 ϩ 2



2



−5 − 4 −3 −2 −1 0 1 2 3 4 5

3



Sum



Ϫ3 ϩ (Ϫ2) ϭ Ϫ5



16



Chapter 1 • Some Basic Concepts of Arithmetic and Algebra



Once you acquire a feeling of movement on the number line, a mental image of this movement is sufficient. Consider the following addition problems, and mentally picture the number line interpretation. Be sure that you agree with all of our answers.

5 ϩ (Ϫ2) ϭ 3



Ϫ6 ϩ 4 ϭ Ϫ2



Ϫ8 ϩ 11 ϭ 3



Ϫ7 ϩ (Ϫ4) ϭ Ϫ11



Ϫ5 ϩ 9 ϭ 4



9 ϩ (Ϫ2) ϭ 7



14 ϩ (Ϫ17) ϭ Ϫ3



0 ϩ (Ϫ4) ϭ Ϫ4



6 ϩ (Ϫ6) ϭ 0



The last example illustrates a general property that should be noted: Any integer plus its

opposite equals zero.

Remark: Profits and losses pertaining to investments also provide a good physical model for

interpreting addition of integers. A loss of $25 on one investment along with a profit of $60 on a

second investment produces an overall profit of $35. This can be expressed as Ϫ25 ϩ 60 ϭ 35.

Perhaps it would be helpful for you to check the previous examples using a profit and loss interpretation.

Even though all problems involving the addition of integers could be done by using the

number line interpretation, it is sometimes convenient to give a more precise description

of the addition process. For this purpose we need to briefly consider the concept of

absolute value. The absolute value of a number is the distance between the number and

0 on the number line. For example, the absolute value of 6 is 6. The absolute value of Ϫ6

is also 6. The absolute value of 0 is 0. Symbolically, absolute value is denoted with vertical bars. Thus we write

͉6͉ ϭ 6



͉Ϫ6͉ ϭ 6



͉0͉ ϭ 0



Notice that the absolute value of a positive number is the number itself, but the absolute value

of a negative number is its opposite. Thus the absolute value of any number except 0 is positive, and the absolute value of 0 is 0.

We can describe precisely the addition of integers by using the concept of absolute

value as follows.

Two Positive Integers

The sum of two positive integers is the sum of their absolute values. (The sum of two

positive integers is a positive integer.)



43 ϩ 54 ϭ ͉43͉ ϩ ͉54͉ ϭ 43 ϩ 54 ϭ 97

Two Negative Integers

The sum of two negative integers is the opposite of the sum of their absolute values.

(The sum of two negative integers is a negative integer.)

(Ϫ67) ϩ (Ϫ93) ϭ Ϫ(͉Ϫ67͉ ϩ ͉Ϫ93͉)

ϭ Ϫ(67 ϩ 93)

ϭ Ϫ160

One Positive and One Negative Integer

We can find the sum of a positive and a negative integer by subtracting the smaller

absolute value from the larger absolute value and giving the result the sign of the original number that has the larger absolute value. If the integers have the same absolute

value, then their sum is 0.



1.3 • Integers: Addition and Subtraction



17



82 ϩ (Ϫ40) ϭ ͉82͉ Ϫ ͉Ϫ40͉ ϭ 82 Ϫ 40 ϭ 42

74 ϩ (Ϫ90) ϭ Ϫ(͉Ϫ90͉ Ϫ ͉74͉)

ϭ Ϫ(90 Ϫ 74) ϭ Ϫ16

(Ϫ17) ϩ 17 ϭ ͉Ϫ17͉ Ϫ ͉17͉

ϭ 17 Ϫ 17 ϭ 0



Zero and Another Integer

The sum of 0 and any integer is the integer itself.



0 ϩ (Ϫ46) ϭ Ϫ46

72 ϩ 0 ϭ 72



The following examples further demonstrate how to add integers. Be sure that you agree

with each of the results.

Ϫ18 ϩ (Ϫ56) ϭ Ϫ(0 Ϫ18 0 ϩ 0 Ϫ56 0 ) ϭ Ϫ(18 ϩ 56) ϭ Ϫ74



Ϫ71 ϩ (Ϫ32) ϭ Ϫ( 0 Ϫ71 0 ϩ 0 Ϫ32 0 ) ϭ Ϫ(71 ϩ 32) ϭ Ϫ103



64 ϩ (Ϫ49) ϭ 0 64 0 Ϫ 0 Ϫ49 0 ϭ 64 Ϫ 49 ϭ 15



Ϫ56 ϩ 93 ϭ 0 93 0 Ϫ 0 Ϫ56 0 ϭ 93 Ϫ 56 ϭ 37



Ϫ114 ϩ 48 ϭ Ϫ( 0 Ϫ114 0 Ϫ 0 48 0 ) ϭ Ϫ(114 Ϫ 48) ϭ Ϫ66



45 ϩ (Ϫ73) ϭ Ϫ(0 Ϫ73 0 Ϫ 0 45 0 ) ϭ Ϫ(73 Ϫ 45) ϭ Ϫ28

46 ϩ (Ϫ46) ϭ 0  Ϫ48 ϩ 0 ϭ Ϫ48



(Ϫ73) ϩ 73 ϭ 0  0 ϩ (Ϫ81) ϭ Ϫ81

It is true that this absolute value approach does precisely describe the process of adding integers,

but don’t forget about the number line interpretation. Included in the next problem set are other

physical models for interpreting the addition of integers. You may find these models helpful.



Subtraction of Integers

The following examples illustrate a relationship between addition and subtraction of whole

numbers.

7 Ϫ 2 ϭ 5  because 2 ϩ 5 ϭ 7

9 Ϫ 6 ϭ 3  because 6 ϩ 3 ϭ 9

5 Ϫ 1 ϭ 4  because 1 ϩ 4 ϭ 5

This same relationship between addition and subtraction holds for all integers.

5 Ϫ 6 ϭ Ϫ1  because 6 ϩ (Ϫ1) ϭ 5

Ϫ4 Ϫ 9 ϭ Ϫ13    because 9 ϩ (Ϫ13) ϭ Ϫ4

Ϫ3 Ϫ (Ϫ7) ϭ 4

8 Ϫ (Ϫ3) ϭ 11



 because Ϫ7 ϩ 4 ϭ Ϫ3

 because Ϫ3 ϩ 11 ϭ 8



Now consider a further observation:

5 Ϫ 6 ϭ Ϫ1



and



Ϫ4 Ϫ 9 ϭ Ϫ13 and

Ϫ3 Ϫ (Ϫ7) ϭ 4

8 Ϫ (Ϫ3) ϭ 11



and

and



5 ϩ (Ϫ6) ϭ Ϫ1

Ϫ4 ϩ (Ϫ9) ϭ Ϫ13

Ϫ3 ϩ 7 ϭ 4

8 ϩ 3 ϭ 11



18



Chapter 1 • Some Basic Concepts of Arithmetic and Algebra



The previous examples help us realize that we can state the subtraction of integers in terms

of the addition of integers. A general description for the subtraction of integers follows.

Subtraction of Integers

If a and b are integers, then a Ϫ b ϭ a ϩ (Ϫb).

It may be helpful for you to read a Ϫ b ϭ a ϩ (Ϫb) as “a minus b is equal to a plus the

opposite of b.” Every subtraction problem can be changed to an equivalent addition problem

as illustrated by the following examples.

6 Ϫ 13 ϭ 6 ϩ (Ϫ13) ϭ Ϫ7

9 Ϫ (Ϫ12) ϭ 9 ϩ 12 ϭ 21

Ϫ8 Ϫ 13 ϭ Ϫ8 ϩ (Ϫ13) ϭ Ϫ21

Ϫ7 Ϫ (Ϫ8) ϭ Ϫ7 ϩ 8 ϭ 1

It should be apparent that the addition of integers is a key operation. The ability to effectively

add integers is a necessary skill for further algebraic work.



Evaluating Algebraic Expressions

Let’s conclude this section by evaluating some algebraic expressions using negative and positive integers.



Classroom Example

Evaluate each algebraic expression

for the given values of the variables.

(a) m Ϫ n for m ϭ Ϫ10 and n ϭ 23

(b) Ϫ x ϩ y for x ϭ Ϫ11 and y ϭ Ϫ2

(c) Ϫc Ϫ d for c ϭ Ϫ57 and d ϭ Ϫ4



EXAMPLE 1

Evaluate each algebraic expression for the given values of the variables.

(a) x Ϫ y  for x ϭ Ϫ12 and y ϭ 20

(b) Ϫa ϩ b  for a ϭ Ϫ8 and b ϭ Ϫ6

(c) Ϫx Ϫ y  for x ϭ 14 and y ϭ Ϫ7



Solution

(a) x Ϫ y ϭ Ϫ12 Ϫ 20  when x ϭ Ϫ12 and y ϭ 20



ϭ Ϫ12 ϩ (Ϫ20)

ϭ Ϫ32



Change to addition



(b) Ϫa ϩ b ϭ Ϫ(Ϫ8) ϩ (Ϫ6)  when a ϭ Ϫ8 and b ϭ Ϫ6



ϭ 8 ϩ (Ϫ6)

ϭ2



Note the use of parentheses when substituting the values



(c) Ϫx Ϫ y ϭ Ϫ(14) Ϫ (Ϫ7)  when x ϭ 14 and y ϭ Ϫ7



ϭ Ϫ14 ϩ 7

ϭ Ϫ7



Change to addition



Concept Quiz 1.3

For Problems 1– 4, match the letters of the description with the set of numbers.

1.

2.

3.

4.



{…, Ϫ3, Ϫ2, Ϫ1}

{1, 2, 3,…}

{0, 1, 2, 3,…}

{…, Ϫ3, Ϫ2, Ϫ1, 0}



A.

B.

C.

D.



Positive integers

Negative integers

Nonnegative integers

Nonpositive integers



1.3 • Integers: Addition and Subtraction



19



For Problems 5–10, answer true or false.

5. The number zero is considered to be a positive integer.

6. The number zero is considered to be a negative integer.

7. The absolute value of a number is the distance between the number and one on the

number line.

8. The ͉Ϫ4͉ is Ϫ4.

9. The opposite of Ϫ5 is 5.

10. a minus b is equivalent to a plus the opposite of b.



Problem Set 1.3

For Problems 1–10, use the number line interpretation to

find each sum. (Objective 2)

1. 5 ϩ (Ϫ3)



2. 7 ϩ (Ϫ4)



3. Ϫ6 ϩ 2



4. Ϫ9 ϩ 4



5. Ϫ3 ϩ (Ϫ4)



6. Ϫ5 ϩ (Ϫ6)



7. 8 ϩ (Ϫ2)



8. 12 ϩ (Ϫ7)



9. 5 ϩ (Ϫ11)



10. 4 ϩ (Ϫ13)



For Problems 51– 66, add or subtract as indicated. (Objective 2)

51. 6 Ϫ 8 Ϫ 9



52. 5 Ϫ 9 Ϫ 4



53. Ϫ4 Ϫ (Ϫ6) ϩ 5 Ϫ 8



54. Ϫ3 Ϫ 8 ϩ 9 Ϫ (Ϫ6)



55. 5 ϩ 7 Ϫ 8 Ϫ 12



56. Ϫ7 ϩ 9 Ϫ 4 Ϫ 12



57. Ϫ6 Ϫ 4 Ϫ (Ϫ2) ϩ (Ϫ5)

58. Ϫ8 Ϫ 11 Ϫ (Ϫ6) ϩ (Ϫ4)

59. Ϫ6 Ϫ 5 Ϫ 9 Ϫ 8 Ϫ 7



For Problems 11–30, find each sum. (Objective 2)



60. Ϫ4 Ϫ 3 Ϫ 7 Ϫ 8 Ϫ 6



11. 17 ϩ (Ϫ9)



12. 16 ϩ (Ϫ5)



61. 7 Ϫ 12 ϩ 14 Ϫ 15 Ϫ 9



13. 8 ϩ (Ϫ19)



14. 9 ϩ (Ϫ14)



62. 8 Ϫ 13 ϩ 17 Ϫ 15 Ϫ 19



15. Ϫ7 ϩ (Ϫ8)



16. Ϫ6 ϩ (Ϫ9)



63. Ϫ11 Ϫ (Ϫ14) ϩ (Ϫ17) Ϫ 18



17. Ϫ15 ϩ 8



18. Ϫ22 ϩ 14



64. Ϫ15 ϩ 20 Ϫ 14 Ϫ 18 ϩ 9



19. Ϫ13 ϩ (Ϫ18)



20. Ϫ15 ϩ (Ϫ19)



65. 16 Ϫ 21 ϩ (Ϫ15) Ϫ (Ϫ22)



21. Ϫ27 ϩ 8



22. Ϫ29 ϩ 12



23. 32 ϩ (Ϫ23)



24. 27 ϩ (Ϫ14)



66. 17 Ϫ 23 Ϫ 14 Ϫ (Ϫ18)



25. Ϫ25 ϩ (Ϫ36)



26. Ϫ34 ϩ (Ϫ49)



27. 54 ϩ (Ϫ72)



28. 48 ϩ (Ϫ76)



29. Ϫ34 ϩ (Ϫ58)



30. Ϫ27 ϩ (Ϫ36)



For Problems 31– 50, subtract as indicated. (Objective 2)

31. 3 Ϫ 8



32. 5 Ϫ 11



33. Ϫ4 Ϫ 9



34. Ϫ7 Ϫ 8



35. 5 Ϫ (Ϫ7)



36. 9 Ϫ (Ϫ4)



37. Ϫ6 Ϫ (Ϫ12)



38. Ϫ7 Ϫ (Ϫ15)



39. Ϫ11 Ϫ (Ϫ10)



40. Ϫ14 Ϫ (Ϫ19)



41. Ϫ18 Ϫ 27



The horizontal format is used extensively in algebra, but

occasionally the vertical format shows up. Some exposure to

the vertical format is therefore needed. Find the following

sums for Problems 67–78. (Objective 2)

67.



5

Ϫ9



68.



8

Ϫ13



69. Ϫ13

Ϫ18



70. Ϫ14

Ϫ28



71. Ϫ18

   9



72. Ϫ17

   9



42. Ϫ16 Ϫ 25



73. Ϫ21

39



74. Ϫ15

  32



43. 34 Ϫ 63



44. 25 Ϫ 58



75.



76.



45. 45 Ϫ 18



46. 52 Ϫ 38



27

Ϫ19



31

Ϫ18



47. Ϫ21 Ϫ 44



48. Ϫ26 Ϫ 54

50. Ϫ76 Ϫ (Ϫ39)



77. Ϫ53

  24



78.



49. Ϫ53 Ϫ (Ϫ24)



47

Ϫ28



20



Chapter 1 • Some Basic Concepts of Arithmetic and Algebra



For Problems 79 – 90, do the subtraction problems in vertical

format. (Objective 2)

79. 5

12



80.



81.



6

Ϫ9



82. 13

Ϫ7



83. Ϫ7

Ϫ8



84. Ϫ6

Ϫ5



85.



86.



17

Ϫ19



8

19



18

Ϫ14



87. Ϫ23

  16



88. Ϫ27

  15



89. Ϫ12

  12



90. Ϫ13

Ϫ13



For Problems 91–100, evaluate each algebraic expression

for the given values of the variables. (Objective 3)

91. x Ϫ y  for x ϭ Ϫ6 and y ϭ Ϫ13

92. Ϫx Ϫ y  for x ϭ Ϫ7 and y ϭ Ϫ9

93. Ϫx ϩ y Ϫ z  for x ϭ 3, y ϭ Ϫ4, and z ϭ Ϫ6

94. x Ϫ y ϩ z  for x ϭ 5, y ϭ 6, and z ϭ Ϫ9

95. Ϫx Ϫ y Ϫ z  for x ϭ Ϫ2, y ϭ 3, and z ϭ Ϫ11

96. Ϫx Ϫ y ϩ z  for x ϭ Ϫ8, y ϭ Ϫ7, and z ϭ Ϫ14

97. Ϫx ϩ y ϩ z  for x ϭ Ϫ11, y ϭ 7, and z ϭ Ϫ9

98. Ϫx Ϫ y Ϫ z  for x ϭ 12, y ϭ Ϫ6, and z ϭ Ϫ14

99. x Ϫ y Ϫ z  for x ϭ Ϫ15, y ϭ 12, and z ϭ Ϫ10

100. x ϩ y Ϫ z  for x ϭ Ϫ18, y ϭ 13, and z ϭ 8

A game such as football can be used to interpret addition

of integers. A gain of 3 yards on one play followed by a

loss of 5 yards on the next play places the ball 2 yards

behind the initial line of scrimmage; this could be

expressed as 3 ϩ (Ϫ5) ϭ Ϫ2. Use this football interpretation to find the following sums for Problems 101–110.

(Objective 4)



101. 4 ϩ (Ϫ7)



102. 3 ϩ (Ϫ5)



103. Ϫ4 ϩ (Ϫ6)



104. Ϫ2 ϩ (Ϫ5)



105. Ϫ5 ϩ 2



106. Ϫ10 ϩ 6



107. Ϫ4 ϩ 15



108. Ϫ3 ϩ 22



109. Ϫ12 ϩ 17



110. Ϫ9 ϩ 21



For Problems 111–120, refer to the Remark on page 16 and

use the profit and loss interpretation for the addition of

integers. (Objective 4)

111. 60 ϩ (Ϫ125)



112. 50 ϩ (Ϫ85)



113. Ϫ55 ϩ (Ϫ45)

114. Ϫ120 ϩ (Ϫ220)

115. Ϫ70 ϩ 45

116. Ϫ125 ϩ 45

117. Ϫ120 ϩ 250

118. Ϫ75 ϩ 165

119. 145 ϩ (Ϫ65)

120. 275 ϩ (Ϫ195)

121. The temperature at 5 A.M. was Ϫ17ЊF.

By noon the temperature had increased by 14ЊF. Use the addition of

integers to describe this situation and

to determine the temperature at noon

(see Figure 1.5).



120°

100°

80°

60°

40°

20°



−20°

− 40°



122. The temperature at 6 P.M. was Ϫ6ЊF,

and by 11 P.M. the temperature had

dropped 5ЊF. Use the subtraction of

integers to describe this situation and

to determine the temperature at 11 P.M.

Figure 1.5

(see Figure 1.5).

123. Megan shot rounds of 3 over par, 2 under par, 3 under

par, and 5 under par for a four-day golf tournament.

Use the addition of integers to describe this situation

and to determine how much over or under par she was

for the tournament.

124. The annual report of a company contained the following figures: a loss of $615,000 for 2007, a loss of

$275,000 for 2008, a loss of $70,000 for 2009, and a

profit of $115,000 for 2010. Use the addition of integers to describe this situation and to determine the company’s total loss or profit for the four-year period.

125. Suppose that during a five-day period, a share of Dell’s

stock recorded the following gains and losses:

Monday

lost $2



Tuesday

gained $1



Thursday

gained $1



Friday

lost $2



Wednesday

gained $3



Use the addition of integers to describe this situation

and to determine the amount of gain or loss for the fiveday period.

126. The Dead Sea is approximately thirteen hundred

eighty-five feet below sea level. Suppose that you

are standing eight hundred five feet above the Dead

Sea. Use the addition of integers to describe this situation and to determine your elevation.

127. Use your calculator to check your answers for

Problems 51– 66.



1.4 • Integers: Multiplication and Division



21



Thoughts Into Words

128. The statement Ϫ6 Ϫ (Ϫ2) ϭ Ϫ6 ϩ 2 ϭ Ϫ4 can be

read as “negative six minus negative two equals negative six plus two, which equals negative four.” Express

in words each of the following.

(a)

(b)

(c)

(d)



129. The algebraic expression Ϫx Ϫ y can be read as “the

opposite of x minus y.” Express in words each of the

following.

(a) Ϫx ϩ y

(b) x Ϫ y

(c) Ϫx Ϫ y ϩ z



8 ϩ (Ϫ10) ϭ Ϫ2

Ϫ7 Ϫ 4 ϭ Ϫ7 ϩ (Ϫ4) ϭ Ϫ11

9 Ϫ (Ϫ12) ϭ 9 ϩ 12 ϭ 21

Ϫ5 ϩ (Ϫ6) ϭ Ϫ11



Answers to the Concept Quiz

1. B

2. A

3. C

4. D

5. False



1.4



6. False



7. False



8. False



9. True



10. True



Integers: Multiplication and Division



OBJECTIVES



1



Multiply and divide integers



2



Evaluate algebraic expressions involving the multiplication and division of integers



3



Apply the concepts of multiplying and dividing integers to model problems



Multiplication of whole numbers may be interpreted as repeated addition. For example, 3 и 4

means the sum of three 4s; thus, 3 и 4 ϭ 4 ϩ 4 ϩ 4 ϭ 12. Consider the following examples

that use the repeated addition idea to find the product of a positive integer and a negative integer:

3(Ϫ2) ϭ Ϫ2 ϩ (Ϫ2) ϩ (Ϫ2) ϭ Ϫ6

2(Ϫ4) ϭ Ϫ4 ϩ (Ϫ4) ϭ Ϫ8

4(Ϫ1) ϭ Ϫ1 ϩ (Ϫ1) ϩ (Ϫ1) ϩ (Ϫ1) ϭ Ϫ4

Note the use of parentheses to indicate multiplication. Sometimes both numbers are enclosed

in parentheses so that we have (3)(Ϫ2) .

When multiplying whole numbers, the order in which we multiply two factors does not

change the product: 2(3) ϭ 6 and 3(2) ϭ 6. Using this idea we can now handle a negative

number times a positive integer as follows:

(Ϫ2)(3) ϭ (3)(Ϫ2) ϭ (Ϫ2) ϩ (Ϫ2) ϩ (Ϫ2) ϭ Ϫ6

(Ϫ3)(2) ϭ (2)(Ϫ3) ϭ (Ϫ3) ϩ (Ϫ3) ϭ Ϫ6

(Ϫ4)(3) ϭ (3)(Ϫ4) ϭ (Ϫ4) ϩ (Ϫ4) ϩ (Ϫ4) ϭ Ϫ12

Finally, let’s consider the product of two negative integers. The following pattern helps

us with the reasoning for this situation:

4(Ϫ3) ϭ Ϫ12

3(Ϫ3) ϭ Ϫ9

2(Ϫ3) ϭ Ϫ6

1(Ϫ3) ϭ Ϫ3

0(Ϫ3) ϭ 0

(Ϫ1)(Ϫ3) ϭ ?



The product of 0 and any integer is 0



22



Chapter 1 • Some Basic Concepts of Arithmetic and Algebra



Certainly, to continue this pattern, the product of Ϫ1 and Ϫ3 has to be 3. In general, this type of

reasoning helps us to realize that the product of any two negative integers is a positive integer.

Using the concept of absolute value, these three facts precisely describe the multiplication of integers:

1. The product of two positive integers or two negative integers is the product of their

absolute values.

2. The product of a positive and a negative integer (either order) is the opposite of the

product of their absolute values.

3. The product of zero and any integer is zero.



The following are examples of the multiplication of integers:

(Ϫ5)(Ϫ2) ϭ 0 Ϫ5 0 и 0 Ϫ2 0 ϭ 5 и 2 ϭ 10

(7)(Ϫ6) ϭ Ϫ( 0 7 0 и 0 Ϫ6 0 ) ϭ Ϫ(7 и 6) ϭ Ϫ42

(Ϫ8)(9) ϭ Ϫ(0 Ϫ8 0 и 0 9 0 ) ϭ Ϫ(8 и 9) ϭ Ϫ72

(Ϫ14)(0) ϭ 0

(0)(Ϫ28) ϭ 0

These examples show a step-by-step process for multiplying integers. In reality, however,

the key issue is to remember whether the product is positive or negative. In other words, we

need to remember that the product of two positive integers or two negative integers is a

positive integer; and the product of a positive integer and a negative integer (in either

order) is a negative integer. Then we can avoid the step-by-step analysis and simply write

the results as follows:

(7)(Ϫ9) ϭ Ϫ63

(8)(7) ϭ 56

(Ϫ5)(Ϫ6) ϭ 30

(Ϫ4)(12) ϭ Ϫ48



Division of Integers

By looking back at our knowledge of whole numbers, we can get some guidance for our work

8

with integers. We know, for example, that ϭ 4, because 2 и 4 ϭ 8. In other words, we can

2

find the quotient of two whole numbers by looking at a related multiplication problem. In

the following examples we use this same link between multiplication and division to determine the quotients.

8

ϭ Ϫ4  because (Ϫ2)(Ϫ4) ϭ 8

Ϫ2

Ϫ10

ϭ Ϫ2  because (5)(Ϫ2) ϭ Ϫ10

5

Ϫ12

ϭ 3  because (Ϫ4)(3) ϭ Ϫ12

Ϫ4

0

ϭ 0  because (Ϫ6)(0) ϭ 0

Ϫ6

Ϫ9

0

0

0



is undefined because no number times 0 produces Ϫ9

is indeterminate because any number times 0 equals 0



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3: Integers: Addition and Subtraction

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