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CHAPTER
5
Integers
5.1
Representations of Integers
5.2
Addition and Subtraction of Integers
5.3
Multiplication and Division of Integers
5.4
Clock Arithmetic
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CHAPTER 5 Integers
Using a Calculator
For Review
21. Calculate each of these without using the M+ , M- ,
or MR key on your calculator.
24. Use the built-in constant function on a calculator to
compute the 20th term in each of these sequences.
(a) 31 - 47 + 88 + 16 - 5
(a) 5, 8, 11, 14, 17, . . .
(b) 57 + 165 , (- 11) + 17
(b) 5, 10, 20, 40, 80, . . .
(c) (47 + 81 - 56 + 9) , (67 - 31 - 9)
25. Use the colored-counter model to illustrate these operations.
22. The Diffy process described in the Cooperative
Investigation in Section 2.3 employed whole numbers.
Explain why the process is essentially unchanged if we
start with all negative or a mixture of positive and negative integers. ■
■
(a) 3 + (-5) = - 2
(b) 3 - ( -5) = 8
■
(c) (-3) + 5 = 2
■
(d) ( -3) - 5 = - 8
From State Student Assessments
■
26. Create mail-time stories that illustrate the facts given in
problem 25. ■
23. (Massachusetts, Grade 8)
Calculators are not allowed here because one can obtain
the correct answer without understanding.)
27. Draw number-line diagrams that illustrate the facts
given in problem 25. ■
Compute: (- 2)(-5)( -1) =
5.4
Clock Arithmetic
Mathematics is all around us. Making use of such naturally occurring mathematical ideas
is an important and useful teaching strategy. What, for example, is the mathematics of an
ordinary clock, as depicted in Figure 5.23?
F I G U R E 5.23
A 12-hour clock
12
11
1
10
2
9
3
8
4
5
7
6
Clock Addition and Multiplication
What is the point on the number line 9 units to the right of 7?
7
9
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
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11
0
7 + 12 9 = 4,
2
9
3
4
8
7
6
5
F I G U R E 5.24
The 12-hour clock with 12
replaced by 0
11
4 5 6
10
9
8
0
3
2
1
7
0 11 10
6
1
7
8
9
327
This is an addition problem that can be solved on the number line by “counting on.”
Starting at 0, we first count out to 7 and then count on 9 more to 16. Thus, 7 + 9 = 16.
Now, thinking of the clock, we might ask, “What time is it 9 hours after 7 o’clock?” Again,
this can be answered by counting, but this time we count clockwise around the face of the
clock. Starting at 12 on the clock diagrammed in Figure 5.23, we count to 7 and then count
on 9 more to arrive at 4. Thus, in 12-hour clock arithmetic, it is reasonable to say that 7
plus 9 is 4 and to write
1
10
5.4 Clock Arithmetic
2
3
4
5
F I G U R E 5.25
Illustrating 7 +12 9 = 4
with two rotatable discs
where we indicate 12-hour clock addition by the symbol + 12.
Proceeding in the same way, we find that 9 + 12 12 = 12 + 12 9 = 9. Indeed, since
counting on 12 steps takes us all the way around the clock, it is clear that
n + 12 12 = 12 + 12 n = n for every n on the clock. Thus, 12 plays the role of 0 in 12-hour
clock arithmetic, and it will suit our purposes to replace 12 by 0 and to number the clock
as shown in Figure 5.24. Thus, the numbers in the arithmetic are those in the set
T = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, and the arithmetic for the clock is determined as for
whole numbers except that we count around the clock instead of on the number line.
A simple, but effective, manipulative for visualizing addition (and subtraction) in clock
arithmetic is easily made from two circular discs marked as shown in Figure 5.25 and
joined at the center so that they can rotate against one another. (Paper plates, where one is
trimmed down a bit, are reasonably durable and work well.) To add 9 to 7 in 12-hour clock
arithmetic, for example, set the 0 on the inner circle under the 7 on the outer circle. Then
count around to 9 on the inner circle. Since 9 on the inner circle lies under 4 on the outer
circle, 7 + 12 9 = 4, as noted above.
Now consider the sum 7 + 12 9 + 12 11 + 12 5 + 12 10 + 12 8 + 12 6. Starting at 0 and
counting to 7, then counting on 9 more, then 11 more, then 5 more, then 10 more, then 8
more, and then 6 more, gets us to 8. Thus, we have
7 + 12 9 + 12 11 + 12 5 + 12 10 + 12 8 + 12 6 = 8.
In whole-number arithmetic this sum would be 56. Since 56 = 4 # 12 + 8, we would have
counted 4 times around the clock and then on 8 more, obtaining the result shown. In a similar way, we determine that
7 * 12 9 = 9 + 12 9 + 12 9 + 12 9 + 12 9 + 12 9 + 12 9 = 3
and, in ordinary whole-number arithmetic, 7 # 9 = 63 = 5 # 12 + 3. These considerations
lead to the following definition.
DEFINITION Computing 12-Hour Clock Sums and Products
To compute sums and products in 12-hour clock arithmetic, perform the same computation
in whole-number arithmetic, divide by 12, and take the remainder, r, as the answer.
EXAMPLE
5.27
Computing 12-Hour Clock Sums and Products
Perform the following computations in 12-hour clock arithmetic:
(a) 7 + 12 8
(c) 11 + 12 7 + 12 6 + 12 10
(e) 4 * 12 9
(b) 3 + 12 9
(d) 7 * 12 8
(f) 5 * 12 5 * 12 5
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CHAPTER 5 Integers
Using the division algorithm, we have the following:
Solution
(a) 7 + 8 = 15 = 1 # 12 + 3, so 7 + 12 8 = 3.
(b) 3 + 9 = 12 = 1 # 12 + 0, so 3 + 12 9 = 0.
(c) 11 + 7 + 6 + 10 = 34 = 2 # 12 + 10, so 11 + 12 7 +12 6 +12 10 = 10.
(d) 7 # 8 = 56 = 4 # 12 + 8, so 7 * 12 8 = 8.
(e) 4 # 9 = 36 = 3 # 12 + 0, so 4 * 12 9 = 0.
(f) 5 # 5 # 5 = 125 = 10 # 12 + 5, so 5 * 12 5 * 12 5 = 5.
It follows from the above definition that the properties for addition and multiplication
of whole numbers also hold in 12-hour clock arithmetic. For example, in whole-number
arithmetic, (a + b) + c = a + (b + c). Since we obtain 12-hour clock sums by taking
ordinary sums, dividing by 12, and taking the remainder as the answer, it follows that
(a + 12 b) + 12 c = a + 12 (b + 12 c). Since the other properties are treated in the same way,
it follows that, in 12-hour clock arithmetic,
•
•
•
•
the closure properties for both addition and multiplication hold,
addition and multiplication are both commutative,
addition and multiplication are both associative, and
0 is the additive identity and 1 is the multiplicative identity.
Clock Subtraction
Like whole-number subtraction, subtraction in clock arithmetic is defined in terms of addition. In whole-number subtraction, a - b = c if, and only if, a = b + c. In like manner,
we have this definition.
DEFINITION Clock Subtraction
Let T = {0, 1, 2, 3, . . . , 11}. For all a T and b T, a - 12 b = c if, and only if,
a = b +12 c.
EXAMPLE
5.28
12-Hour Clock Subtraction
Compute the following clock “differences”:
(a) 11 - 12 4
Solution
(a) Since 4
(b) Since 7
(c) Here, 8
(d) Since 8
(b) 4 - 12 7
(c) 8 - 12 5
(d) 5 - 12 8
+ 12 7 = 11, it follows that 11 - 12 4 = 7.
+ 12 9 = 4, it follows that 4 - 12 7 = 9.
= 5 + 12 3, and so 8 - 12 5 = 3.
+ 12 9 = 5, it follows that 5 - 12 8 = 9.
Since any point on the clock can be reached from any other point by counting in either
the clockwise direction or the counterclockwise direction, clock subtraction can always be
performed as clock addition. For example, 5 - 12 9 = 8, since if we start at 0 and count 5
in the clockwise direction and 9 in the counterclockwise direction, we arrive at 8.
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5.4 Clock Arithmetic
329
However, 8 can also be reached by starting at 0 and counting 5 in the clockwise direction
and then 3 more in the clockwise direction. Thus,
5 - 12 9 = 5 + 12 3 = 8,
as illustrated in Figure 5.26. Indeed, since the counts of 9 and 3 have to reach all the way
around the circle, it is clear that 9 + 3 = 12. This is always the case, and we have the following theorem.
F I G U R E 5.26
Illustrating clock subtraction as clock addition
0
11
1
10
2
9
9
3
5
8
4
5
7
6
3
5 –12 9 = 5 +12 3 = 8
THEOREM Clock Subtraction as Clock Addition
Let T = {0, 1, 2, 3, . . . , 11}. For all a and b in T, a -12 b = a +12 (12 - b).
EXAMPLE
5.29
Subtracting by Adding in Clock Arithmetic
Perform the following subtractions as additions:
(a) 3 - 12 8
Solution
(a) Since 12
(b) Since 12
(c) Since 12
(d) Since 12
(b) 11 - 12 7
-
(c) 9 - 12 11
(d) 2 - 12 6
8 = 4, 3 - 12 8 = 3 + 12 4 = 7. Check: 8 + 12 7 = 3.
7 = 5, 11 - 12 7 = 11 + 12 5 = 4. Check: 7 + 12 4 = 11.
11 = 1, 9 - 12 11 = 9 + 12 1 = 10. Check: 11 + 12 10 = 9.
6 = 6, 2 - 12 6 = 2 + 12 6 = 8. Check: 6 + 12 8 = 2.
In the preceding theorem and example, we saw that we can subtract b in 12-hour
clock arithmetic by adding 12 - b. We call 12 - b the additive inverse of b and note
that we can subtract b from a by adding the additive inverse of b to a just as we can by
adding the negative, or additive inverse, of b to a in integer arithmetic. In the terminology that was introduced in Section 5.1, we see that clock arithmetic has the additive
inverse property.
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CHAPTER 5 Integers
DEFINITION The Additive Inverse in Clock Arithmetic
Let T = {0, 1, 2, . . . , 11}. If a T and a +12 b = b +12 a = 0, then b is called the
additive inverse of a. Also, a is the additive inverse of b.
The table below lists the additive inverses in clock arithmetic.
Additive Inverses in 12-Hour Clock Arithmetic
a
0
1
2
3
4
5
6
7
8
9
10
11
Additive
Inverse of a
0
11
10
9
8
7
6
5
4
3
2
1
Using this terminology, the preceding theorem can now be written as follows.
THEOREM Clock Subtraction as Clock Addition
To subtract b from a in clock arithmetic, add the additive inverse of b to a.
Since clock subtraction can always be performed as clock addition and clock arithmetic is closed under clock addition, it follows that clock arithmetic is also closed under
clock subtraction. However, parts (c) and (d) of Example 5.28 make it clear that subtraction is not commutative in clock arithmetic, and the fact that 8 - 12 (7 - 12 5) = 6 and
(8 - 12 7) - 12 5 = 8 makes it clear that clock subtraction is not associative. In summary,
•
•
•
•
clock arithmetic is closed under addition,
clock arithmetic is closed under subtraction,
subtraction is not commutative in clock arithmetic, and
subtraction is not associative in clock arithmetic.
Studying clock arithmetic is both interesting and informative to middle school students. The system is similar to, yet different from, ordinary integer arithmetic, and studying these similarities and differences helps students to more fully understand ordinary
arithmetic.
Division in Clock Arithmetic
Division without remainder is defined in clock arithmetic just as it is for whole numbers.
DEFINITION Clock Division
Let T = {0, 1, 2, 3, . . . , 11}. For a T and b T, we say that b divides a in clock
arithmetic, and write a ,12 b = c, if, and only if, there exists a unique c T such that
a = b *12 c.
Note: the word
"unique" is
important here.
EXAMPLE
5.30
12-Hour Clock Division
Perform these divisions, if possible.
(a) 8 , 12 5
(b) 7 , 12 8
(c) 4 , 12 10
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Solution
5.4 Clock Arithmetic
331
Division in clock arithmetic is greatly facilitated if one has a complete multiplication
table, as shown below.
12-Hour Clock Multiplication Table
12
0
1
2
3
4
5
6
7
8
9
10
11
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
2
3
4
5
6
7
8
9
10
11
2
0
2
4
6
8
10
0
2
4
6
8
10
3
0
3
6
9
0
3
6
9
0
3
6
9
4
0
4
8
0
4
8
0
4
8
0
4
8
5
0
5
10
3
8
1
6
11
4
9
2
7
6
0
6
0
6
0
6
0
6
0
6
0
6
7
0
7
2
9
4
11
6
1
8
3
10
5
8
0
8
4
0
8
4
0
8
4
0
8
4
9
0
9
6
3
0
9
6
3
0
9
6
3
10
0
10
8
6
4
2
0
10
8
6
4
2
11
0
11
10
9
8
7
6
5
4
3
2
1
(a) From the table, 4 is the only element a in T such that 8 = 5 * 12 a. Therefore,
8 , 12 5 = 4.
(b) From the table, there is no c T such that 7 = 8 * 12 c. Therefore, this division is
not defined.
(c) From the table, 10 * 12 4 = 10 * 12 10 = 4. Since there is more than one number
c T such that 10 * 12 c = 4, this division is not defined.
Into the Classroom
Clock Arithmetic
Clock arithmetic should be considered a worthwhile enrichment topic for the elementary
school classroom. Our experience is that children like mathematical ideas drawn from their
immediate surroundings and that they are intrigued and interested by this arithmetic generated by an ordinary clock, an arithmetic that is both similar to and different from ordinary
arithmetic. As usual, the arithmetic should be introduced via manipulatives (in this case, an
ordinary clock or a device like that illustrated in Figure 5.25 and, later, diagrams of n-hour
clocks for other values of n), and then students should be led to develop the resulting arithmetic pretty much on their own. The teacher should ask only occasional pertinent questions
and make occasional suggestions as the development proceeds. Of special importance is the
difference between the arithmetics generated by n-hour clocks when n is prime and when n is
composite. In particular, n-hour clock arithmetic where n is a prime number is closed under
division except for division by 0, and, as we have seen, this is not so when n is composite.
Other interesting questions and properties of clock arithmetic that can be turned into classroom activities will be seen in Problem Set 5.4.
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CHAPTER 5 Integers
332
Suppose we had a “10-hour” clock as shown in Figure 5.27 instead of a 12-hour clock.
The resulting arithmetic is like 12-hour clock arithmetic, except that to find the 10-hour
sum or product of two elements of
1
10
2
9
3
8
we find the sum or product as in ordinary arithmetic, divide the answer by 10, and use the
resulting remainder r as the answer in 10-hour clock arithmetic. Here, 10 corresponds to
0, not 12.
Subtraction and division are then defined in terms of addition and of multiplication as
above.
4
7
5
6
F I G U R E 5.27
A 10-hour clock
EXAMPLE
T = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9},
5.31
Calculating in 10-Hour Clock Arithmetic
Perform each of these 10-hour clock computations.
(a) 6 + 10 9
(b) 6 - 10 9
(c) 6 * 10 9
(d) 6 , 10 9
Solution
Understand the Problem
We are asked to do various calculations in 10-hour clock arithmetic where the elements are
T = {0, 1, 2, . . . , 9}.
Devise a Plan
As in 12-hour clock arithmetic, we find the results for addition and multiplication in 10hour clock arithmetic by finding the results in ordinary integer arithmetic, dividing by 10,
and taking the remainder r as the desired answer. In 10-hour clock arithmetic, we can subtract b from a by adding the additive inverse of b, 10 - b, to a. To find a , 10 b, we must
see if there is one, and only one, number c in T such that b * 10 c = a.
Carry Out the Plan
(a) Since 6 + 9 = 15 = 10 # 1 + 5, 6 + 10 9 = 5.
(b) 6 - 10 9 = 6 + 10 (10 - 9) = 6 + 10 1 = 7.
(c) Since 6 # 9 = 54 = 10 # 5 + 4, 6 * 10 9 = 4.
(d) We must find a number c in T such that 9 * 10 c = 6. If there is precisely one such
number c, then 6 , 10 9 = c. Running through the values of c, we see that 4 is the
only number in T whose product with 9 leaves a remainder of 6—that is,
9 * 10 4 = 6—and so 6 , 10 9 = 6 as required.
Look Back
Basically, all we had to do was to make sure that we understood the problem. Beyond that,
it was a matter of simple arithmetic.
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5.4 Clock Arithmetic
333
Problem Set 5.4
9.
Understanding Concepts
1.
2.
3.
4.
Compute these 12-hour clock sums.
(a) 5 +12 9
(b) 8 +12 7
(c) 8 +12 4
(d) 4 +12 7
(e) 10 +12 10
(f) 7 +12 8
Compute these n-hour clock sums, where + n denotes
n-hour clock addition; for example, 5 +7 3 indicates the
7-hour clock addition of 5 and 3.
(a) 3 +5 4
(b) 17 +26 13
(c) 2 +10 7
(d) 7 +9 4
(e) 12 +16 10
(f) 2 +7 6
In 12-hour clock arithmetic, a -12 b can be computed
by starting at a and counting b steps counterclockwise
around the clock. Use this method to compute each
difference.
(a) 9 -12 7
(b) 8 -12 11
(c) 5 -12 9
(d) 8 -12 10
(e) 2 -12 11
(f) 8 -12 8
Recall that 0 is the additive identity in 12-hour clock
arithmetic and that b is the additive inverse of a if, and
only if, a +12 b = b +12 a = 0. Compute the additive
inverses of each of these numbers in 12-hour clock
arithmetic.
(a) 7
5.
(b) 11
7.
8.
(d) 8
Compute each of these differences by adding. Be sure
to show what you are adding each time.
(a) 9 -12 7 ■
6.
(c) 9
(b) 8 -12 11 ■
(c) 5 -12 9 ■
(d) 8 -12 10 ■ (e) 2 -12 11 ■
(f) 8 -12 8 ■
Compute these products in 12-hour clock arithmetic.
(a) 5 *12 7
(b) 9 *12 11
(c) 8 *12 9
(d) 8 *12 6
(e) 4 *12 6
(f) 4 *12 9
Perform these divisions if they are defined. (Suggestion:
See the table in Example 5.30.)
(a) 5 ,12 7
(b) 7 ,12 10 ■
(c) 8 ,12 4 ■
(d) 8 ,12 5
(e) 9 ,12 5
(f) 6 ,12 11
Two numbers are said to be relatively prime if their
only common divisor is 1.
(a) List the numbers in T = {0, 1, 2, . . . , 11} that
are relatively prime to 12.
(b) List the numbers in T = {0, 1, 2, . . . , 11} that
are not relatively prime to 12.
(c) Compare the results of parts (a) and (b) with the
results of Example 5.30. What conjecture does this
comparison suggest?
Construct complete addition and multiplication tables
for 5-hour clock arithmetic. ■
10. Perform these computations in 5-hour clock arithmetic.
(a) 3 +5 4
(b) 2 +5 3
(c) 4 +5 4
(d) 3 *5 4
(e) 2 *5 3
(f) 4 *5 4
(g) 3 -5 4
(h) 2 -5 3
(i) 4 -5 4
(j) 3 ,5 4
(k) 2 ,5 3
(l) 4 ,5 4
11. (a) What is the additive identity in 5-hour clock arithmetic? Why?
(b) What is the multiplicative identity in 5-hour clock
arithmetic? Why?
12. If a +5 b = b +5 a = 0, then b is the additive inverse
of a and a is the additive inverse of b in 5-hour clock
arithmetic. Why?
(a) Compute the additive inverse of each of 1, 2, 3, 4,
and 0 in 5-hour clock arithmetic.
As in 12-hour clock arithmetic, we can subtract b from
a in 5-hour clock arithmetic by adding the additive
inverse of b to a. Perform each of these subtractions in
two ways—(i) by counting backward on a 5-hour clock
and (ii) by adding the additive inverse of the number
being subtracted.
(b) 2 -5 4 ■
(c) 3 -5 2 ■
(d) 1 -5 3 ■
13. If a *5 b = b *5 a = 1, then a is called the multiplicative inverse of b and b is called the multiplicative
inverse of a in 5-hour clock arithmetic, and we write
a = b-1 and b = a-1.
(a) Which numbers in 5-hour clock arithmetic possess
multiplicative inverses? (Hint: Check the results of
problem 9 above.)
(b) Which numbers in 12-hour clock arithmetic possess
multiplicative inverses? (Hint: Check the table in
Example 5.30.)
14. Just as we can subtract in clock arithmetic by adding
the additive inverse, we can divide by multiplying by
the multiplicative inverse. Use the definition of clock
division to compute each of the following:
(a) 4 ,12 7 ■
(b) 3 ,12 11 ■
(c) 3 ,5 2 ■
(d) 2 ,5 2 ■
(e) 2 ,5 4 ■
(f) 4 ,5 3 ■
Compute each of the quantities in parts (a) through (f)
by multiplying by the multiplicative inverse; that is,
compute each of these products.
(g) 4 *12 7-1 ■
Copyright Pearson. All rights reserved.
(h) 3 *12 11-1 ■
(i) 3 *5 2-1
■
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CHAPTER 5 Integers
(j) 2 *5 2-1 ■
(k) 2 *5 4-1 ■
(l) 4 *5 3-1 ■
15. Using the table in Example 5.30, solve the following
equations in 12-hour clock arithmetic. The first one is
done for you.
(a) (3 *12 y) +12 7 = 4
Solution: ■
(b) (7 *12 y) -12 4 = 8
3 *12 y = 4 +12 5
(c) (y +12 2) ,12 11 = 3
3 *12 y = 9
(d) (2 ,12 y) -12 4 = 3
y = 3, 7, or 11
■
■
■
Teaching Concepts
16. In the military, the hours of the day are numbered from
1 through 24, starting at 1 A.M.
result required in the next part. (Note: To compute 35,
for example, using your calculator, enter the string 3 ^
5 = . Then reenter the answer and use INT , to
determine the result in 12-hour clock arithmetic.)
(a) 33
(b) 34
(c) 33 # 34
(d) 33 + 4
(e) 42
(f) (42)5
2#5
(g) 4
(h) 410
21. (a) Complete the following table. The first two rows
have been completed for you. Note: In using your calculator to compute 2n - 2 in n-hour clock arithmetic,
you may have to enter the string 2 ^ n - 2 = , record
the result in the second column of the table, and then
reenter the result and use INT , to determine the
correct result for column three. ■
(a) What set of numbers would you be inclined to consider in the arithmetic associated with military time?
(b) Determine each of the following in military clock
arithmetic:
(i) 17 +24 8
(ii) 8 +24 16
(iii) 2 -24 14
(iv) 23 -24 5
(v) 5 *24 7
(vi) 23 *24 23
(vii) 3 ,24 11
(viii) 11 ,24 3
2n 2
2n 2 in n-hour
clock arithmetic
2
2
0
3
6
0
n
4
5
17. What other notions from ordinary life might you use to
illustrate other clock arithmetics?
6
18. (a) What ordinary arithmetic operations must be performed to compute 7 *12 11? ■
8
(b) Compare the likely attractiveness to school students
of computing 7 *12 11 with that of solving the ordinary corresponding drill problem. Compute 7 * 11.
Compute the quotient and remainder when 77 is
divided by 12. ■
19. Falla claims that 8 ,12 2 = 4 since 8 = 4 *12 2. How
would you respond to Falla? ■
Thinking Critically
20. Powers in n-Hour Clock Arithmetic. Since
as = a # a Á a, with s factors of a, we can compute as
in n-hour clock arithmetic in the usual way. That is,
as = r in n-hour clock arithmetic, where r is the
remainder when as is divided by n. Moreover, it follows
that the usual rules
and
9
10
11
12
13
(b) What seems to be the case when n is a prime in the
table of part (a)? Make a conjecture. ■
Responding to Students
asat = as + t
7
(as)t = ast
(c) Compute 223 - 2 in 23-hour arithmetic. Note that
you cannot use INT , the key on some calculators to determine the remainder when dividing by
23 since the quotient is too large. Instead, enter the
sequence 2 ^ 23 - 2 = , 23 = - 364722 =
* 23 = . (Did you actually have to perform the
last six steps in this sequence to know what the
remainder was in this case?) Does this result
strengthen your belief in the conjecture you made
in part (b)? Does it prove that it is correct?
22. (a) Repeat problem 21(a) with each 2 replaced by a 3
in the top line of column headings. ■
(b) Make a conjecture on the basis of part (a).
hold in clock arithmetic just as they do in ordinary
arithmetic. Compute the given powers and products in
12-hour clock arithmetic. Do parts (e), (f), (g), and (h)
in order, using the result of each part to compute the
■
23. You have little to go on, but make a conjecture generalizing the conjectures for problems 21 and 22. If you
need more data, repeat problem 21 (a) and (b) with 2
replaced by 4, by 5, by 6, etc. ■
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Epilogue
335
Thinking Cooperatively
For Review
In the next three problems, divide the work among the members of your group, discuss the results obtained, and determine conjectures agreed upon by the group.
27. Draw rectangular diagrams to illustrate all the factorings of 12, taking order into account; for example, think
of 1 # 12 as different from 12 # 1. ■
24. Make a conjecture concerning the value of
1 # 2 # 3 Á (p - 1) + 1 in p-hour clock arithmetic,
where p is a prime. (Suggestion: Consider a number of
examples using the first few primes.) ■
28. Use the method of intersection of sets to determine the
following:
25. Make a conjecture concerning the value of ap - 1 - 1 in
p-hour clock arithmetic if p is a prime and p does not
divide a. ■
26. Fill in the following table for a number of choices of a
and b in T = {0, 1, 2, 3, . . . , 11}, and endeavor to
discover when a * 12 y = b is solvable and how many
solutions this equation has. (Hint: Use the table in
Example 5.30.) ■
a
b
GCD
(a, 12)
b/GCD
(a, 12)
12/GCD
(a, 12)
No. of solutions
of a *12 y = b
(a) GCD(60, 150)
■
(b) LCM(60, 150)
■
29. (a) Use a factor tree to determine all the prime divisors
of 540. ■
(b) How can you tell at a glance if 540 is or is not
divisible by 9? ■
30. (a) Write the prime-power representations of 540
and 600. ■
(b) Use part (a) to determine GCD(540, 600).
■
(c) Use part (a) to determine LCM(540, 600).
■
31. Does 23 # 35 # 72 # 116 evenly divide 24 # 37 # 52 # 71 # 118?
Explain briefly. ■
32. To determine whether 427 is a prime, which primes
must you check as possible divisors? ■
Epilogue
Developing the Number System
In the struggle to develop numbers, history shows that it took a very long time for people
to develop the natural numbers, and even longer to develop the notion of zero. Even then,
zero was first introduced only as a placeholder in positional systems like the Mayan, and
it was not until much later that it was considered to be a number.
Somewhat surprisingly, negative numbers made their appearance on the mathematical
scene before zero did. No evidence of the recognition of negative numbers, as distinct from
subtrahends, appears in ancient Egyptian, Babylonian, Hindu, Chinese, or Greek mathematics. Still, the rules of signs, considered at length in this chapter, were established early
on by considering such products as (8 - 4) # (7 - 5). The Chinese made use of such subtractions at least as early as 200 B.C., but the rules of signs in Chinese mathematics were
not stated explicitly until A.D. 1299. The first mention of negative numbers in Western
mathematics occurred in Arithmetica by the Greek mathematician Diophantus, in about
A.D. 275, though spoken of there in disparaging terms. Diophantus called the equation
4x + 20 = 4 absurd, since it would require that x = - 4. The first substantial use of negative numbers occurred in the work of the Hindu mathematician Brahmagupta, in about
A.D. 628, and after that time they appear in all Indian works on the subject.
In this chapter, we have used various manipulatives and pictorial representations to
introduce the notions of negative numbers and the rules for operating with them. The fact
that these numbers were considered absurd or nonnumbers by early mathematicians
notwithstanding, negative numbers, like all the other numbers, are simply ideas in our
minds. They are in constant use to solve problems that occur daily in modern society.
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CHAPTER 5 Integers
We have also introduced clock arithmetic in this chapter, not only because it is intrinsically interesting to school students but also because it is similar to, and yet different
from, integer arithmetic. Thus, studying clock arithmetic not only helps students more
fully understand integer arithmetic but also provides interesting (and disguised) drill in that
arithmetic. Moreover, the concept of the multiplicative inverse in clock arithmetic (see
problems 13 and 14 in Problem Set 5.4) is a precursor of the same notion that will reappear in the study of fractions and rational numbers in Chapter 6.
Chapter 5 Summary
Key Concepts
In what follows, all symbols represent integers.
Section 5.1
Representations of Integers
•
•
•
•
•
•
•
•
Section 5.2
Addition and Subtraction of Integers
•
•
•
•
•
•
•
Section 5.3
-n. The opposite, or negative, of n, defined by n + ( -n) = (-n) + n = 0.
Integers. . . . , -3, -2, -1, 0, 1, 2, 3, . . . .
Additive inverse, negative. An integer m such that n + m = m + n = 0; that is,
m = - n.
Colored counters. Representations of integers with sets of black and red markers.
Mail-time stories. Bringing a check for n dollars represents n. Bringing a bill for n
dollars represents - n.
Number-line representations. If n is positive, an arrow of length n oriented from left
to right and placed anywhere on the number line represents n. If the arrow is oriented
from right to left, it represents - n. Each integer is also represented as a point along
the number line, with positive integers to the right of 0 and negative integers to the
left of 0.
The negative of the negative of an integer. - (- n) = n.
Absolute value. The absolute value of a number r is the distance of r from 0 on the
number line. Also, | r | = r if r Ú 0, and | r | = - r if r 6 0.
Addition of integers. If r and s are integers, then (-r) + ( -s) = - (r + s). If
r 7 s 7 0, then r + ( -s) = r - s. If 0 6 r 6 s, then r + ( -s) = - (s - r).
Law of trichotomy. If r and s are integers, then either r 6 s, r = s, or r 7 s.
Properties of addition. For integers r, s, and t, r + s is an integer, r + s = s + r,
(r + s) + t = r + (s + t), and 0 is the additive identity, that is, 0 + r =
r + 0 = r for all integers r.
Less than. r is less than s provided there is a positive integer t such that r + t = s.
Greater than. r is greater than s provided s is less than r.
Subtraction of integers. For integers r, s, and t, r - s = t if, and only if, r = s + t.
Also, r - s = r + (- s).
Closure property for subtraction. r - s is always an integer.
Multiplication and Division of Integers
•
•
Multiplication of integers. If r and s are integers, then r # (- s) = - (r # s),
(-r) # s = - (r # s), and (-r) # (- s) = r # s.
Properties of multiplication. For integers r, s, and t, r # s is an integer, r # s = s # r,
(r # s) # t = r # (s # t), r # (s + t) = r # s + r # t, 1 # r = r # 1 = r, and 0 # r = r # 0 = 0.
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•
Section 5.4
Chapter Review Exercises
337
Division of integers. For integers r and s, r , s = t if, and only if, r = s # t for some
integer t. r , (- s) = - (r , s), (-r) , s = - (r , s), and ( -r) , (- s) = r , s.
Clock Arithmetic
•
•
•
•
•
Clock arithmetic. The arithmetic inherent in a 12-hour clock.
Clock addition and multiplication. Do the corresponding ordinary arithmetic, divide
by 12, and take the remainder as the result.
Clock subtraction. r -12 s = t if, and only if, r = s +12 t. Also,
r -12 s = r +12 (12 - s). 12 - s is the additive inverse of s for every s in the
arithmetic.
Clock division. r ,12 s = t if, and only if, s is the only element in the arithmetic for
which r = s * 12 t. No such t may exist.
n-hour clock arithmetic. The same as 12-hour clock arithmetic, but with 12 replaced by n.
Vocabulary and Notation
Section 5.1
Law of trichotomy
Integer
Negative integer
Additive inverse or opposite
Absolute value, |n|
Section 5.3
Section 5.2
12-hour clock arithmetic
+ n, - n, * n, , n
Additive identity
Additive inverse
Rule of signs
Section 5.4
Less than, 6
Less than or equal, …
Greater than, 7
Greater than or equal, Ú
Chapter Review Exercises
4.
Section 5.1
1.
You have 15 counters colored black on one side and red
on the other.
(a) If you drop them on your desktop and 7 come up
black and 8 come up red, what integer is represented?
(b) If you drop them on your desktop and twice as
many come up black as red, what number is being
represented?
(b) List five different loops of colored counters that
represent the integer 6. ■
5.
(a) If the mail carrier brings you a check for $12, are
you richer or poorer, and by how much? What integer does this situation illustrate?
(a) Give a mail-time story that illustrates - 85.
(b) Give a mail-time story that illustrates 47.
6.
(c) What numbers are represented by all possible drops
of the 15 counters?
2.
(a) List five different loops of colored counters that
represent the integer - 5. ■
■
(a) What number must you add to 44 to obtain 0?
(b) What number must you add to - 61 to obtain 0?
Section 5.2
7.
What addition is represented by this diagram?
(b) If the mail carrier brings you a bill for $37, are you
richer or poorer, and by how much? What integer
does this situation illustrate?
3.
■
(a) 12° above 0 illustrates what integer?
(b) 24° below 0 illustrates what integer?
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CHAPTER 5 Integers
338
8.
9/25/07
What subtraction is represented by this diagram?
(c) (- 27) + (-5)
(d) 5 - (- 7)
(e) 8 - ( -12)
(f) 8 - 12
12. (a) If it is 15° below 0 and the temperature falls 12°,
what temperature is it?
(b) What arithmetic does this situation illustrate?
9.
13. (a) Dina’s bank account was overdrawn by $12. What
was her balance after she deposited $37 she earned
working at a local pizza parlor?
What additions and subtractions are represented by
these mail-time stories?
(a) At mail time, the letter carrier brings you a check
for $45 and a bill for $68. Are you richer or poorer,
and by how much?
(b) At mail time, the letter carrier brings you a check
for $45 and takes away a bill for $68 left previously.
Are you richer or poorer, and by how much?
10. What additions and/or subtractions do these numberline diagrams represent?
(b) What arithmetic does this situation illustrate?
14. (a) Plot these numbers on a number line: - 2, 7, 0, - 5,
-9, and 2. ■
(b) List the numbers in part (a) in increasing order.
(c) Determine what integer must be added to each
number in your list from part (b) to obtain the next. ■
Section 5.3
(a)
15. What products do these number-line diagrams
represent?
–3
–2
–1
0
1
2
3
4
5
6
7
8
9
10
(a)
(b)
–2 –1 0
–4
–3
–2
–1
0
1
2
3
4
5
6
7
8
9
(c)
1
3
2
4
5
6
7
8
9 10 11 12 13 14
(b)
–14 –13 –12 –11 –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0
1
2
1
2
(c)
–4
–3
–2
–1
0
1
2
3
4
5
6
7
8
9
–14 –13 –12 –11 –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0
(d)
–9
–8
(d)
–7
–6
–5
–4
–3
–2
–1
0
1
2
3
4
–8
–7
–6
–5
–4
–3
–2
–1
0
1
2
3
4
(f)
–6
–5
1
2
3
4
5
6
7
8
9 10 11 12 13 14
16. (a) Show that 3 # 5 + 3 # (- 5) = 0. (Suggestion: Begin
with the fact that 3 # 0 = 0.) ■
(e)
–9
–2 –1 0
–4
–3
–2
–1
0
1
2
3
4
5
6
7
(c) Show that -(3 # 5) + (- 3) # (- 5) = 0. (Suggestion:
Begin with the fact that 0 # (- 5) = 0, and use the
result of part (a).) ■
(d) If -(3 # 5) + (- 3) # (- 5) = 0, what can you conclude about the product (-3) # (- 5)? ■
(g)
–12 –11 –10 –9
(b) If 3 # 5 + 3 # (- 5) = 0, what can you conclude
about the product 3 # (- 5)? ■
–8
–7
–6
–5
–4
–3
–2
–1
0
1
17. Perform each of these computations.
(a) (-8) # (- 7)
11. Perform these additions and subtractions.
(a) 5 + ( -7)
(b) (- 27) - (- 5)
(b) 8 # (- 7)
(c) (- 8) # 7
(d) 84 , ( - 12)
(e) (- 84) , 7
(f) (-84) , (- 7)
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18. Write a mail-time story to illustrate each of these
products.
(a) 7 # (12) ■
(b) ( - 7) # (13) ■
Chapter Test
Section 5.4
(c) (- 7) # (- 13) ■
22. Perform the indicated clock calculations if they are defined.
(a) 4 +12 9
(b) 9 -12 4
19. If d divides n, prove that d divides -n, - d divides n,
and - d divides - n. ■
(c) 4 *12 9
(d) 4 ,12 9
(e) 9 +12 8
(f) 9 *12 0
20. If a and b are integers, the greatest common divisor
(GCD) of a and b is the largest positive integer dividing
both a and b. Compute each of the following:
(g) 4 ,12 0
(h) 9 ,12 7
(a) GCD(255, - 39)
339
(b) GCD( -1001, 2651)
21. If n is an integer not divisible by 2 or 3, show that
n2 - 1 is divisible by 24. (Hint: By the division algorithm, n must be of one of these forms: 6q, 6q + 1,
6q + 2, 6q + 3, 6q + 4, or 6q + 5.) ■
(i) 9 *12 7
23. Perform these clock calculations.
(b) 6 -7 5
(d) 6 ,7 5
(a) 5 +7 6
(c) 6 *7 5
24. List the numbers in 10-hour clock arithmetic for which
10-hour clock division is not defined.
Chapter Test
1.
Mary Lou’s checkbook balance was $129. What was it
after she deposited $341 and then wrote checks for $13,
$47, and $29? What arithmetic does this illustrate?
2.
At mail time, if the mail carrier took away five bills for
$27 each, are you richer or poorer, and by how much?
What calculation does this illustrate?
3.
(a) Use your calculator to compute these sums.
1 =
1 - 4 =
1 - 4 + 9 =
1 - 4 + 9 - 16 =
1 - 4 + 9 - 16 + 25 =
6.
Write a mail-time story to illustrate the subtraction
7 - (- 4) = 11. ■
7.
The Fibonacci sequence is formed by adding any two
consecutive numbers in the sequence to obtain the next
number. If the same rule is followed in each of these
sequences, correctly fill in the blanks.
(a) -5, -3, _____, _____, _____, _____ ■
■
(b) 7, _____, 2, _____, _____, _____ ■
(c) 6, _____, _____, _____, -12, _____ ■
8.
Tammie, Jody, and Nora formed a small club. After a
pizza party celebrating the first anniversary of the club’s
existence, they owed the local pizzeria $27. The bill
was paid and shared equally by the three girls. Was
each one richer or poorer, and by how much? What
arithmetic does this illustrate?
9.
Perform the indicated clock calculations.
(b) Make a conjecture suggested by the pattern of
part (a). ■
4.
(a) What sums can be obtained using only the numbers -10, -5, - 2, - 1, 1, 2, 5, and 10, each at
most once and without using any number with its
double (that is, you can’t use 5 with 10, 1 with 2,
- 1 with - 2, or - 5 with - 10)? ■
(b) Do the representations in part (a) appear to be
unique?
HM
5.
Perform each of these computations.
(a) ( -7) + (- 19)
(b) (-7) - (-19)
(c) 7 - (- 19)
(d) 7 + ( -19)
(e) (-6859) , 19
(f) (-24) # 17
(g) 36 # (- 24)
(h) ( -1155) , ( -11)
(a) 7 +8 5
(b) 7 +12 5
(c) 5 -7 7
(d) 7 *8 5
(e) 7 ,8 5
(f) 75 (in 8-hour clock arithmetic)
10. Draw a number-line diagram to illustrate each of these
calculations.
(a) ( -7) + 10 ■
(b) 10 - (- 7) ■
(c) 7 # ( -5)
11. The least common multiple (LCM) of integers a and b
is the least positive integer divisible by both a and b.
Compute LCM( - 240, 54).
(i) 0 , (- 27)
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