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Chapter
Six
Other Mathematical
Systems
Outline
Objectives
Introduction
After completing this chapter, you should be able to
6-1 Clock Arithmetic
6-2 Modular Systems
6-3 Mathematical Systems
without Numbers
Summary
1 Identify the structure of a
mathematical system (6-1)
2 Perform addition, subtraction,
and multiplication on the 12-hour
clock (6-1)
3 Determine which properties, such as
closure, commutativity, etc., are true
for the operations on the 12-hour
clock (6-1)
6-1
4 Perform operations and determine
which properties are true for modular
systems (6-2)
5 Perform operations and determine
which properties are true for
mathematical systems without
numbers (6-3)
239
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Introduction
A
mathematical system consists of
1. A nonempty set of elements
2. Operation(s) for the elements
3. Definitions
4. Properties of the operations
The real number system, presented in Chapter 5, is an example of a mathematical system. The
elements of the real number system are the real numbers. The operations are addition, subtraction,
multiplication, and division. Definitions include factor, prime number, etc. and the properties of the
operations are closure, commutativity, associativity, etc. The real number system is an example of an
infinite mathematical system since the set of real numbers is infinite.
In mathematics, there are also finite mathematical systems. Finite mathematical systems have a
finite number of elements. This chapter explains some finite mathematical systems including clock
arithmetic systems, modular arithmetic systems, and mathematical systems without numbers. ■
6-1 Clock Arithmetic
The mathematics used in the 12-hour clock is an example of a finite mathematical system
since the elements of the system are the numbers 1 through 12. The basic operations are addition and multiplication. The symbol + will be used to denote addition on the clock. For
example, if it is 2 o’clock now, 3 hours from now it will be 5 o’clock. This is denoted as
2 + 3 = 5. If it is 9 o’clock now, 5 hours from now it will be 2 o’clock, so 9 + 5 = 2.
Addition on the Twelve-Hour Clock
The answers to addition problems on the 12-hour clock can be found by counting clockwise around the face of a clock. If it is 7 o’clock now, in 8 hours, it will be 3 o’clock, as
shown in Figure 6-1.
Mathematics in Our World
Game or Mathematical System?
When we were children, we learned how to play a game called “tic-tac-toe,” also called “Xs and Os.” The game
is played by two players using a grid consisting of nine spaces. One player uses an X and the other uses an O.
Each player takes turns placing his or her letter in a space. To win the game, one player must get either three Xs or
three Os in a vertical row, a horizontal row, or a diagonal row. Some games are shown next.
X
O
X
O
X
O
X wins
O
O
O
O
X
X
X
O wins
O
X
X
X
X wins
X
X
O
X
O
O
X
X
X
O
No winner
Actually, this game and other games such as football, baseball, Monopoly, chess, and Scrabble can be thought
of as mathematical systems. After studying this chapter, you will see why in Mathematics in Our World—Revisited.
240
6-2
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Section 6-1 Clock Arithmetic
4
5
3
241
Some other examples of addition on the 12-hour clock are:
6
7+6=1
8+7=3
7
6 + 11 = 5
4+3=7
8
2
Example 6-1
1
Using the 12-hour clock, find these sums.
Figure 6-1
(a)
(b)
(c)
(d)
(e)
9 + 12
6+5
8+8
3 + 11
5+7
Solution
(a) 9. Start at 9 and count 12 hours clockwise, ending at 9.
(b) 11. Start at 6 and count 5 hours clockwise, ending at 11.
(c) 4. Start at 8 and count 8 hours clockwise, ending at 4.
(d) 2. Start at 3 and count 11 hours clockwise, ending at 2.
(e) 12. Start at 5 and count 7 hours, ending at 12.
An addition table for the 12-hour clock can be constructed using all possible combinations of the numbers 1 through 12, as shown here.
+
1
2
3
4
5
6
7
1
2
3
4
5
6
7
2
3
4
5
6
7
8
3
4
5
6
7
8
4
5
6
7
8
9
5
6
7
8
9
6
7
8
9
10
7
8
9
10
8
9
10
8
9
10
11
12
8
9
9
10
10
11
12
1
11
12
1
9
10
2
11
12
1
2
10
3
11
12
1
2
3
4
10
11
11
12
1
2
3
4
5
12
1
2
3
4
5
11
12
6
1
2
3
4
5
6
11
12
7
1
2
3
4
5
6
7
8
9
10
11
12
1
2
3
4
5
6
7
8
9
10
11
12
1
2
3
4
5
6
7
8
9
10
11
12
1
2
3
4
5
6
7
8
9
10
11
12
1
2
3
4
5
6
7
8
9
10
11
12
Multiplication on the Twelve-Hour Clock
Multiplication can also be done on the clock. In order to multiply 6 × 3, start at 12 o’clock
and count around the clock in 3-hour sections six times. That is, 3 + 3 + 3 + 3 + 3 + 3.
You will end up on 6 since
3 + 3 = 6 o’clock
6 + 3 = 9 o’clock
9 + 3 = 12 o’clock
12 + 3 = 3 o’clock
3 + 3 = 6 o’clock
6-3
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242
Chapter 6
Other Mathematical Systems
Math Note
Any whole number
a
can be converted to
ck
clo
the
on
er
mb
nu
lock
by starting at 12 o’c
d
un
aro
and counting
le,
mp
exa
For
ck.
the clo
a
to
47
rt
nve
co
to
ur
number on the 12-ho
k
loc
o’c
12
at
clock, start
ers
mb
nu
47
t
un
and co
around the clock. You
is
will get 11. Hence, 47
the
on
11
to
equivalent
sier
12-hour clock. An ea
er
sw
an
the
find
to
y
wa
12
is to divide 47 by
and take the
3,
remainder: 47 ÷ 12 =
e,
remainder 11. Henc
47 is equivalent to 11.
is 0,
When the remainder
is
er
mb
nu
the
s
an
it me
equivalent to 0 on the
12-hour clock. For
example, 36 is
r
36 ÷ 12 = 3 remainde
0; hence, 36 is
equivalent to 0 on the
12-hour clock.
Calculator
Explorations
A
calculator can be used to
show how the whole number 47
can be converted to its 12-hour
clock equivalent.
What is the equivalent number
to 47 on the 12-hour clock? What
is the equivalent number to 23
on the 12-hour clock?
6-4
Hence, 6 × 3 = 6 on the 12-hour clock.
There is an easier way to compute products on the clock. First, multiply 3 × 6 to get
18; then divide 18 by 12 and use the remainder, 6, as your answer, as shown.
1R6
12 18
12
6
Hence, 6 × 3 = 6.
Note that 1R6 can be read as “1 revolution plus 6.”
Example 6-2
Perform these multiplications on the 12-hour clock.
(a)
(b)
(c)
(d)
(e)
5×8
12 × 9
6×4
11 × 5
10 × 9
Solution
(a) 5 × 8 = 40 and 40 ÷ 12 = 3, remainder 4. Hence, 5 × 8 = 4.
(b) 12 × 9 = 108 and 108 ÷ 12 = 9, remainder 0. Since 0 corresponds to 12 on the
12-hour clock, the answer is 12. Hence, 12 × 9 = 12.
(c) 6 × 4 = 24 and 24 ÷ 12 = 2, remainder 0. Hence, 6 × 4 = 12.
(d) 11 × 5 = 55 and 55 ÷ 12 = 4, remainder 7. Hence, 11 × 5 = 7.
(e) 10 × 9 = 90 and 90 ÷ 12 = 7, remainder 6. Hence, 10 × 9 = 6.
A table can be constructed for multiplication in the same manner as the addition table
was constructed. The multiplication table is shown here.
×
1
2
3
4
5
6
7
8
9
10
11
12
1
1
2
3
4
5
6
7
8
9
10
11
12
2
2
4
6
8
10
12
2
4
6
8
10
12
3
3
6
9
12
3
6
9
12
3
6
9
12
4
4
8
12
4
8
12
4
8
12
4
8
12
5
5
10
3
8
1
6
11
4
9
2
7
12
6
6
12
6
12
6
12
6
12
6
12
6
12
7
7
2
9
4
11
6
1
8
3
10
5
12
8
8
4
12
8
4
12
8
4
12
8
4
12
9
9
6
3
12
9
6
3
12
9
6
3
12
10
10
8
6
4
2
12
10
8
6
4
2
12
11
11
10
9
8
7
6
5
4
3
2
1
12
12
12
12
12
12
12
12
12
12
12
12
12
12
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Section 6-1 Clock Arithmetic
243
Try This One
6-A
Perform the following operations on the 12-hour clock.
(a) 4 + 6
(b) 10 + 11
(c) 7 × 5
(d) 8 × 9
9
8
(e) 7(6 + 9)
7
10
For the answer, see page 266.
6
5
Subtraction on the Twelve-Hour Clock
4
1
Figure 6-2
2
3
Subtraction can be performed on the 12-hour clock by counting counterclockwise (backward). For example, 8 − 10 means that if it is 8 o’clock now, what time was it 10 hours
ago? Figure 6-2 shows that if you start at 8 and count counterclockwise, you will end at
10 o’clock; hence, 8 − 10 = 10.
Example 6-3
Perform these subtraction operations on the 12-hour clock.
(a)
(b)
(c)
(d)
(e)
2 − 10
12 − 7
5−9
6 − 12
4 − 11
Solution
(a)
(b)
(c)
(d)
(e)
Starting at 2 on the clock and counting backward 10 numbers, you will get 4.
Starting at 12 and counting seven numbers backward, you will get 5.
Starting at 5 and counting nine numbers backward, you will get 8.
Starting at 6 and counting 12 numbers backward, you will get 6.
Starting at 4 and counting 11 numbers backward, you will get 5.
Try This One
6-B Perform the following subtractions on the 12-hour clock.
(a) 7 − 2
(b) 3 − 10
(c) 9 − 12
For the answer, see page 266.
6-5
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Chapter 6
Other Mathematical Systems
Sidelight
Evariste Galois
Evariste Galois (1811–1832) was a brilliant young mathematician who lived a
short life with incredible bad luck. When he was 17 years old, he submitted a
manuscript on the solvability of algebraic equations to the French Academy
of Sciences. Augustin-Louis Cauchy, a famous mathematician of his time, was
appointed as a referee to read it. However, Cauchy apparently lost the
manuscript. A year later, Galois submitted a revised version to the academy.
A new referee was appointed to read it but died before he could read it, and
the manuscript was lost a second time.
A year later, Galois submitted the manuscript again for a third time. After a
6-month delay, the new referee, Simon-Dennis-Poisson, rejected it, saying it was
too vague and recommended that Galois rewrite it in more detail.
Galois at that time was considered a dangerous political radical, and he
was provoked into a duel. (Some feel that the challenger was hired by local
police to eliminate him.) Galois realized that he would probably die the next
morning, so he spent the night trying to revise his manuscript as well as some
other papers he had written, but he did not have enough time to finish everything.
The next day, he was shot and killed. Eleven years after his death, Galois’s
writings were found by Joseph Liouville, who studied them, realized their importance, and published them. At last, Galois was given credit for his work.
Properties of the Twelve-Hour Clock
Math Note
verify
In order to
using
s
ie
propert
mples, all
xa
e
c
ifi
spec
ns
m
o binatio
possible c
e
th
bers on
of all num
be shown
st
u
m
clock
. Since this
to be true
of
uire a lot
would req
a
ly
n
effort, o
time and
les
p
m
xa
ted e
few selec
an
c
e
n
o
d
an
are used,
at
th
ductively
reason in
.
e
u
rty is tr
the prope
The 11 properties for addition and multiplication of real numbers were explained in Chapter 5.
Some of these properties were the commutative property of addition, the closure property for
multiplication, etc. The system of the 12-hour clock also has some of the same basic properties for addition and multiplication as the real numbers. These properties are explained next.
The closure property for addition and multiplication can be verified by looking at the
two operation tables shown previously. The answers for every combination of addition
problems and for every combination of multiplication problems are numbers on the clock.
The commutative properties for addition and multiplication can be verified by noting that
for addition or multiplication, the order of operating on the numbers does not matter. For example, 6 + 7 gives the same answer on the clock as 7 + 6. The same is true for multiplication.
The associative properties for addition and multiplication are also true on the 12-hour
clock system. In order to verify these properties, you would need to show that (a + b) + c =
a + (b + c) and (a · b) · c = a · (b · c) is true for all the numbers on the 12-hour clock. Since
this would be very time-consuming, only an example of the associative property is shown.
?
(6 + 9) + 10 = 6 + (9 + 10)
?
3 + 10 = 6 + 7
1=1
6-6
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Section 6-1 Clock Arithmetic
Math Note
identity
that the
Recall
dition
y for ad
propert
exists a
re
at the
th
s
te
ta
s
system
r in the
numbe
n any
at w he
is
such th
ystem
in the s
r
e
b
e
m
nu
e sam
to it, th
added
ined.
ta
b
r is o
numbe
e
Math Not
an
bers c
ve num
g
in
Negati
start
nd by
be fou
unting
o
c
nd
or
at 12 a
wise. F
rclock
te
n
u
o
c
is the
le, −1
examp
2 is the
−
,
1
as 1
same
etc.
as 10,
same
245
The identity element for addition is 12 since adding 12 hours to any number on the
clock brings you back to the same number:
1 + 12 = 1
2 + 12 = 2
3 + 12 = 3
etc.
The identity property for multiplication is 1 since
1×1=1
2×1=2
3×1=3
etc.
The inverse property for addition on the 12-hour clock is also true. Recall that if a
number and its inverse are added, one gets the identity for addition. Since the identity for
addition is 12 on the 12-hour clock, the inverse for 1 is 11, the inverse for 2 is 10, the inverse
for 3 is 9, etc. In other words,
1 + 11 = 12
2 + 10 = 12
3 + 9 = 12
etc.
The inverse property for multiplication is not true for all numbers on the 12-hour
clock. Some numbers do have multiplicative inverses. For example, the multiplicative inverse for 5 is 5 since 5 × 5 = 1 (recall that if you multiply a number by its inverse, you get
the identity). The multiplicative inverse of 11 is 11 since 11 × 11 = 1. But there is no
= 1.
multiplicative inverse for 4 since there is no number such that 4 ×
The distributive property for multiplication over addition is also valid for the operations performed on the 12-hour clock. Again, all cases of a · (b + c) = a · b + a · c would
need to be checked to verify this property. Only one case is shown.
?
5 × (6 + 8) = 5 × 6 + 5 × 8
?
5 × 14 = 30 + 40
?
5 × 2 = 70
10 = 10
Recall that a mathematical system consists of a set of elements, operations for elements, definitions, and properties of each operation. The operations used on the systems
presented in this chapter are called binary operations since they are performed on two elements of the set. Addition, subtraction, multiplication, and division are binary operations.
Some mathematical systems are called groups. Besides mathematics, the theory of
groups is used in chemistry, physics, and other areas such as the secret codes that were used
in the enigma machine. In particle theory, an example of a binary operation is when particles are combined to form new particles.
A mathematical system is called a group if it has these properties:
1.
2.
3.
4.
The set of elements is closed for the binary operation.
There exists an identity element for the set.
Any three elements in the set are associative for the binary operation.
Every element has an inverse.
6-7
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Chapter 6
Other Mathematical Systems
Sidelight
Niels Henrik Abel
Niels Henrik Abel (1802–1829) was a famous Norwegian mathematician who made
many contributions to the theory of equations. At age 19, he tried to find a general
solution to a quintic equation using radicals, but he was unable to do so. Later, he
proved that it could be done in general. He tried to publish his findings, but due to
a lack of funds, he had to condense his work to a six-page pamphlet. Because the
condensation was difficult to follow, contemporary mathematicians dismissed his
work.
Later, he was given a chance to publish his work in a series of articles in a
journal, and after his death, he was given credit for his findings. The term abelian
is derived from his name.
Notice that the definition of a group does not include the commutative property. When
the elements of the set satisfy the commutative property, the group is said to be a commutative or abelian group.
The 12-hour clock system is an abelian group under the binary operation of addition.
However, it is not a group under multiplication since there is not an inverse element for
each given element.
The 12-hour clock is an example of a finite mathematical system. In Section 6-2, other
clock systems, called modular systems, will be shown.
Exercise Set 6-1
Computational Exercises
For Exercises 1–12, find the equivalent number on the 12-hour clock.
1.
4.
7.
10.
27
334
259
−10
2.
5.
8.
11.
92
18
3230
−3
3.
6.
9.
12.
155
42
−5
−20
15.
18.
21.
24.
11 + 11
4+8
For Exercises 13–24, perform the additions on the 12-hour clock.
13.
16.
19.
22.
6-8
5+9
9+7
10 + 20
8 + (10 + 9)
14.
17.
20.
23.
10 + 8
12 + 3
9+6
3 + (11 + 8)
(6 + 5) + 12
(5 + 7) + 2
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Section 6-1 Clock Arithmetic
247
For Exercises 25–36, perform the subtractions on the 12-hour clock.
25.
28.
31.
34.
8−6
10 − 12
3 − 12
12 − 8
26.
29.
32.
35.
12 − 10
0−6
0−8
3 − 11
27.
30.
33.
36.
9 − 11
6 − 10
4−5
2−7
For Exercises 37–48, perform the multiplications on the 12-hour clock.
37.
40.
43.
46.
3×2
9×7
3×7
3 × (2 × 9)
38.
41.
44.
47.
10 × 10
2×5
4×5
(6 × 4) × 7
39.
42.
45.
48.
8×6
12 × 6
5 × (6 × 9)
(8 × 3) × 5
For Exercises 49–58, find the additive inverse for each number.
49.
52.
55.
58.
12
8
7
−6
50. 3
53. 2
56. 4
51. 5
54. 9
57. −5
For Exercises 59–64, find the multiplicative inverse if it exists for each number.
59. 4
62. 9
60. 7
63. 1
61. 12
64. 10
For Exercises 65–70, give an example of each using the 12-hour clock.
65.
66.
67.
68.
69.
70.
Associative property of addition
Commutative property of multiplication
Identity property of addition
Inverse property of multiplication
Distributive property
Commutative property of addition
For Exercises 71–80, find the value of y using the 12-hour clock.
71.
74.
77.
80.
5+y =3
y+6=2
6×9= y
5 × (6 − 11) = y
72. 9 + y = 2
75. 4 × (2 + y) = 4
78. 9 × 4 = y
73. y − 5 = 8
76. 8 × 2 = y
79. 3 × (4 + 10) = y
Real World Applications
81. A computer simulation takes 3 hours to run. If it is now 11 o’clock A.M., what time
will it be when the simulation has run 10 times?
Time in the military is based on a 24-hour clock. From midnight to noon is designated as
0000 to 1200. From noon until midnight, time is designated as 1200 to 2359 where the
first two digits indicate the hour and the last two digits represent the minutes. For
6-9
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248
Chapter 6
Other Mathematical Systems
example, 1824 means 6:24 P.M. For Exercises 82–88, translate military time into
standard time.
82. 0948
85. 1542
88. 2000
83. 0311
86. 1938
84. 0500
87. 2218
For Exercises 89–96, change the standard times into military times.
89. 6:56 A.M.
92. 11:56 A.M.
95. 11:42 P.M.
90. 3:52 A.M.
93. 5:27 P.M.
96. 9:36 P.M.
91. 4:00 A.M.
94. 8:06 P.M.
Find the standard time for each.
97.
98.
99.
100.
0627 + 3 hours and 42 minutes
2342 + 5 hours and 6 minutes
1540 − 1 hour and 4 minutes
1242 − 2 hours and 20 minutes
Writing Exercises
101. Define a mathematical system.
102. What is the difference between a finite mathematical system and an infinite
mathematical system?
103. Explain how to find an inverse for addition for a number on the 12-hour clock.
104. Explain how you can tell if the commutative property for an operation is valid for a
mathematical system.
Critical Thinking
105. Write a short paragraph and explain how the operation of division might be per= 8)
formed on the clock. (Hint: 8 ÷ 4 =
can be rewritten as 4 ×
106. Explain how division can be performed using the multiplication table.
107. What are the answers to 4 × 4, 4 × 7, and 4 × 10? Explain why these answers are
all the same.
6-2 Modular Systems
In Section 6-1, the mathematics of the 12-hour clock was explained. This section will explain modular systems. Modular systems with a specific number of elements are analogous to the 12-hour clock system. For example, a modular 5 system, denoted as mod 5
would have five elements: 0, 1, 2, 3, and 4 and use the clock shown in Figure 6-3(a),
whereas a mod 3 system would have the elements 0, 1, and 2 and use the clock shown in
Figure 6-3(b). A mod 8 system contains the elements 0, 1, 2, 3, 4, 5, 6, and 7 and uses the
6-10
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Section 6-2 Modular Systems
0
0
4
7
1
3
(a) Mod 5
2
0
1
6
2
1
(b) Mod 3
249
2
5
4
3
(c) Mod 8
Figure 6-3
Math Note
stems,
ular sy
of
In mod
place
sed in
For
r.
a 0 is u
e
b
m
tem nu
s
y
0
s
e
,
th
od 5 a
le, in m
5
e
th
f
examp
o
place
in
d
2,
e
1
s
is u
In mod
clock.
e
th
in
n
o
used
uld be
a 0 wo
2.
1
e
of th
place
Calculator
Explorations
A
calculator can be used
to perform modular system
operations. Here, 19 is converted
to its congruent value in the
mod 5 system.
What number is congruent to
19 in the mod 5 system? What
number is congruent to 9 in the
mod 5 system? 18?
clock shown in Figure 6-3(c). In general, a mod m system consists of n numbers starting
with zero and concluding with m − 1. The number n is called the modulus.
Operations for the Modular Systems
Operations in modular systems can be performed using their corresponding clocks, as was
shown for the 12-hour clock system. For example, 4 + 2 = 1 in mod 5 since starting at
4 on the 5-hour clock and counting clockwise two numbers gives you an answer of 1.
Performing operations this way can be very time-consuming since a different clock would
be needed for every modular system; however, some rules can be formulated and used with
any modular system.
Rule 1: Any number a can be changed to a number b in a specific modular system m
by dividing a by m and taking the remainder, b. This is written as a ≡ b (mod m), and we
say a is congruent to b in the modulus m.
Example 6-4
Find the number that is congruent to 19 in the mod 5 system.
Solution
Divide 19 by 5 and take the remainder, as shown.
3R4
5 19
15
4
Hence, 19 = 4 (mod 5). This answer can be verified by starting at 0 on the mod 5
clock and counting around 19 numbers.
Example 6-5
Find the number that is congruent to 25 in the mod 3 system.
Solution
Divide 25 by 3 and take the remainder as shown.
8R1
3 25
24
1
Hence, 25 = 1 (mod 3).
6-11
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Rule 2: The operations of addition and multiplication in modular systems can be performed by adding or multiplying the numbers as usual, then converting the answers to
equivalent numbers in the specified system using rule 1.
Example 6-6
Evaluate 4 × 4 in the mod 5 system.
Solution
4 × 4 = 16 and 16 ÷ 5 = 3 remainder 1; hence, 4 × 4 = 1 (mod 5).
Example 6-7
Suppose that you need to take
antibiotics three times a day for
10 days. Your first pill was a
morning dose. You can’t
remember if you took your
second pill today or not. To find
out, count the remaining pills and
convert to mod 3: if the answer
is 1, you took your pill. (The one
remaining pill is your evening
dose.) If the answer is 2, you
missed a dose. If the answer is 0,
you missed two doses.
Evaluate 6 + 5 in the mod 7 system.
Solution
6 + 5 = 11 and 11 ÷ 7 = 1 remainder 4; hence, 6 + 5 = 4 (mod 7).
Example 6-8
Evaluate each.
(a) 5 × 6 in mod 9
(b) 9 + 7 in mod 11
(c) 3 × 7 in mod 8
Solution
(a) 5 × 6 = 30 and 30 ÷ 9 = 3 remainder 3; hence, 5 × 6 = 3 (mod 9).
(b) 9 + 7 = 16 and 16 ÷ 11 = 1 remainder 5; hence, 9 + 7 = 5 (mod 11).
(c) 3 × 7 = 21 and 21 ÷ 8 = 2 remainder 5; hence, 3 × 7 = 5 (mod 8).
Try This One
6-C
Evaluate each in the mod 6 system:
(a) 2 + 4
(b) 5 + 3
(c) 4 × 5
(d) 2 × 4
(e) 3 × (4 + 4)
For the answer, see page 266.
6-12
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251
Sidelight
Emmy Noether (1882–1935)
Emmy Noether was born in Erlangen, Germany, in 1882.
Her father, Max Noether, was a professor at Erlangen
University and was a distinguished mathematician who
pioneered work in algebraic functions. Her brother
became a professor of applied mathematics at Breslaw
University. At first, she selected languages in secondary
school and had planned to teach French and English.
She enrolled at the University of Erlangen and continued
to study languages; she also audited mathematics
lectures since female students were not permitted to
enroll in mathematics and sciences courses. She did,
however, take the final examination and received a
degree. In 1903, she attended the University of Göttingen
to study mathematics. While there, she taught courses
and collaborated with other mathematicians of the time.
Unfortunately, she never received a paying position.
After one semester, she returned to Erlangen
and received her doctoral degree in mathematics.
The policy had changed, and women were able
to register and receive credit for mathematics courses.
During the interim years from the time she received
her doctoral degree to the time she was forced to leave
Germany in 1933, she published mathematical papers,
substituted for male colleagues, and taught classes,
all without title or remuneration. She made contributions
to what is called modern algebra, which uses axioms,
theorems, and formal proofs.
After leaving Germany, she came to the United
States and received a faculty appointment at Bryn
Mawr College. She also lectured at Princeton University.
She died suddenly in 1935 after surgery. In her New York
Times obituary, Albert Einstein praised her as “the most
significant creative mathematical genius thus far
produced since the higher education of women began.”
Subtraction is performed by counting backward (counterclockwise) in the same manner as shown for the 12-hour clock. For example, 2 − 7 in the mod 9 system would be −5,
which would be equivalent to 4 (mod 9).
Properties of the Modular Systems
Using these two rules (or using a clock), addition and multiplication tables for specific
modular systems can be constructed. For example, the addition and multiplication tables
for mod 3 are shown here.
+
0
1
2
×
0
1
2
0
0
1
2
0
0
0
0
1
1
2
0
1
0
1
2
2
2
0
1
2
0
2
1
From the tables, the various properties such as closure, commutativity, associativity,
etc., can be checked to see if they are true for the operations in the specific modular systems.
6-13
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For example, addition in the mod 3 system is closed, commutative, and associative. The
identity for addition is 0, and each number has an inverse for addition.
Math Note
e
an also b
Closure c
king
o
lo
ed by
determin
le. If
b
ta
n
ditio
at the ad
e
ment in th
every ele
is
the table
body of
s,
e margin
th
also in
sed.
lo
c
is
ition
then add
o
ere are n
That is, th
ts
n
e
m
new ele
ody
g in the b
appearin
e
th
in
ot
that are n
s.
in
rg
a
m
Example 6-9
Show that the closure property of addition is true for the mod 3 system.
Solution
To determine if a system is closed, one must perform all possible additions using
two numbers and verifying that the solution is indeed a number in the modular
system.
0+0=0
0+1=1
0+2=2
1+0=1
1+1=2
1+2=0
2+0=2
2+1=0
2+2=1
Since all the solutions are elements of the mod 3 system, we can say the system is
closed under addition.
Example 6-10
Show that addition in the mod 3 system is commutative.
Solution
To show that addition is commutative, you need to perform all possible operations
of the form a + b = b + a for any two numbers in the system. For example,
0 + 1 = 1 + 0, 2 + 1 = 1 + 2, etc.
Example 6-11
What is the identity for addition in the mod 3 system?
Solution
Zero is the identity for addition in the mod 3 system, since when 0 is added to any
number a, 0 + a = a for a = 0, 1, 2.
Example 6-12
A lamp with a four-way switch
(off, low, medium, and high) can
be represented by addition in the
mod 4 system (0, 1, 2, 3). Turning
the knob is like doing addition—
two turns moves you up two
notches, and so on. What is the
inverse of 1 in the system? (That is,
if the lamp is on low, how many
turns does it take to turn it off?)
6-14
What are the inverses for addition of 0, 1, and 2 in the mod 3 system?
Solution
Recall that the inverse property states that an element + inverse = identity for the
operation, and 0 is the identity for addition. Hence, the inverse of 0 is 0 since
0 + 0 = 0. The inverse of 1 is 2 since 1 + 2 = 0, and the inverse of 2 is 1.
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253
Properties for multiplication in the mod 3 system and for other modular systems can
be verified in the same manner as Examples 6-9 through 6-12 have shown.
Try This One
6-D
Construct an addition table for a mod 2 system and verify these properties.
(a) Closure
(b) Commutativity
(c) Identity
(d) Inverse
For the answer, see page 266.
Application of Modular Systems
Money orders, checks, Federal Express bills, and other items that require legal transactions
contain tracking or identification numbers. These numbers contain “checking digits.” The
checking digit provides for the security of the documents and keeps people from altering
the documents. For example, if a person decides to forge a driver’s license, he needs to create
an identification number. When the number is entered into a computer, the computer
checks it to see if it is a valid driver’s license number. This is done by using modular arithmetic. Here’s how.
A security system based on
modular systems is used for this
check.
6-15
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Calculator
Explorations
W
hen applying modular
systems to large numbers using
a calculator, rounding must be
considered.
What number is congruent to
320476566 in the mod 7 system?
5547721?
Suppose a driver’s license contained the identification number 3204765664. In
order to verify that this is a valid driver’s license number, the computer would divide the
number 320476566 (the last digit is omitted) by a certain number and compare the remainder with the checking digit. In this case, the checking digit is the last digit of the
original number, which is 4. The divisor is only known by the state that issues the driver’s
license.
For this example, assume that the divisor is 7. Hence 320476566 ÷ 7 = 45782366
with a remainder of 4. When the remainder matches the check digit, the identification number on the driver’s license is valid. In other words, 320476566 ≡ 4 (mod 7).
If the checking digit does not match the remainder or the congruent number in the
mod 7 system, the driver’s license is a forgery. Since only the state knows which digit is the
checking digit and which modulus is being used, it becomes very difficult to select a valid
license number by chance alone.
Modular systems are only a few types of systems that are found in mathematics. In
Section 6-3, you will see mathematical systems that do not use numbers.
Exercise Set 6-2
Computational Exercises
For Exercises 1–30, perform the following operations in the specified mod system.
1.
3.
5.
7.
9.
11.
13.
15.
17.
19.
21.
23.
25.
27.
29.
4+3=
3+3=
5×8=
(mod 5)
(mod 4)
(mod 9)
3 × 3 = (mod 4)
3 − 8 = (mod 9)
2 − 3 = (mod 4)
(3 + 5) + 2 = (mod 7)
2 + (3 + 5) = (mod 8)
4 × (2 × 3) = (mod 6)
7 × (3 × 5) = (mod 9)
6 × (2 − 5) = (mod 8)
7 × (3 − 5) = (mod 10)
2 − (3 − 5) = (mod 6)
(4 − 7) − 3 = (mod 9)
8 − (2 − 5) = (mod 12)
2.
4.
6.
8.
10.
12.
14.
16.
18.
20.
22.
24.
26.
28.
30.
8+6=
5+6=
3×7=
(mod 9)
(mod 7)
(mod 8)
4 × 6 = (mod 7)
5 − 7 = (mod 10)
1 − 9 = (mod 11)
(4 + 4) + 4 = (mod 6)
2 + (3 + 4) = (mod 5)
(2 × 2) × 2 = (mod 3)
(2 × 6) + 4 = (mod 7)
5 × (8 + 3) = (mod 9)
4 × (1 − 7) = (mod 8)
3 − (1 − 4) = (mod 5)
(2 − 10) − 1 = (mod 11)
(1 − 1) − 1 = (mod 2)
For Exercises 31–40, find the values of each number in the given mod system.
31.
33.
35.
37.
39.
6-16
32 = (mod 6)
135 = (mod 7)
16 = (mod 9)
326 = (mod 3)
987 =
(mod 8)
32.
34.
36.
38.
40.
51 = (mod 4)
48 = (mod 5)
92 = (mod 10)
451 = (mod 5)
1656 =
(mod 11)
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255
For Exercises 41–50, find the value for y in each equation.
41.
43.
45.
47.
49.
3+y =1
1−y =6
7×y =6
y+4=1
4×y =6
(mod 6)
(mod 8)
(mod 8)
(mod 5)
(mod 7)
42.
44.
46.
48.
50.
y + 3 = 2 (mod 4)
3 − y = 5 (mod 9)
9 + y = 8 (mod 10)
y − (−1) = 7 (mod 8)
1 × y = 6 (mod 7)
Real World Applications
The days of the week can be thought of as a modular system using 0 = Sunday, 1 =
Monday, 2 = Tuesday, etc. Using this system, find the answer to each and give it as the
day of the week. (See the Critical Thinking Exercises.)
51. Sunday + 30 days
53. Tuesday + 45 days
55. Saturday + 360 days
52. Monday + 5 days
54. Friday + 120 days
56. Wednesday + 20 days
For Exercises 57–60, use the last digit in the identification number as the checking digit
and modulus 9 to see if the number is valid.
57. 76241382
59. 134804354
58. 5374193
60. 215805671
Writing Exercises
61.
62.
63.
64.
What is the identity for addition in a modular system?
Explain why rule 1 given in this section works.
Explain why rule 2 given in this section works.
Are all modular systems closed under addition and multiplication? Explain your answer.
Critical Thinking
65. Complete the addition and multiplication tables for mod 7.
+
0
1
2
3
4
5
6
0
1
2
3
4
5
6
×
0
1
2
3
4
5
6
0
1
2
3
4
5
6
6-17
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Chapter 6
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66. Find the additive inverse for each element in mod 7.
67. What properties (closure, commutative, etc.) are true for addition in mod 7?
68. What properties are true for multiplication in mod 7?
6-3 Mathematical Systems without Numbers
Mathematical systems do not need numbers or operations such as addition or multiplication. It is possible to create a mathematical system using any set of symbols and made-up
operations. For example, consider the elements w, x, y, and z, and the operation *. The
operation * is defined by
∗
w
x
y
z
w
x
y
z
w
x
y
z
w
x
y
z
w
x
y
z
w
x
y
z
Operations for Systems without Numbers
Operations are performed using this table in the same manner as they are performed in the
modular systems. For example, to find x ∗ y using the table shown, find x on the vertical
axis and y on the horizontal axis, and then draw a vertical line across from and a horizontal line down from y. The point of intersection is the answer, w.
w
x
y
z
→
∗
w
x
y
z
→w
Hence, x ∗ y = w.
Example 6-13
Perform the following operations using the system just described.
(a)
(b)
(c)
(d)
(e)
w∗y
z∗x
y∗y
z ∗ (w ∗ x)
(w ∗ w) ∗ y
Solution
(a)
(b)
(c)
(d)
(e)
6-18
w∗y = z
z∗x =x
y∗y=x
w ∗ x = y and z ∗ y = y; hence, z ∗ (w ∗ x) = y
w ∗ w = x and x ∗ y = w; hence, (w ∗ w) ∗ y = w
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Section 6-3 Mathematical Systems without Numbers
257
Try This One
6-E
Using the system just shown, perform these operations.
(a) y ∗ z
(b) z ∗ w
(c) x ∗ (y ∗ z)
For the answer, see page 266.
Properties for Systems without Numbers
The system shown in Figure 6-4(a) has these properties:
1.
2.
3.
4.
5.
Closure
Commutative
Associative
Identity property
Inverse property
Some of these properties are readily discernable by looking at the table. Recall that a
system is closed under an operation if all the elements in the body of the table appear in the
margins of the table. See Figure 6-4(a). The system in Figure 6-4(a) is closed for the operation while the system shown in Figure 6-4(b) is not closed for the operation.
The system shown in Figure 6-4(b) is not closed since it contains an element in the
body of the table that is not found in the margins.
A system can be checked for commutativity by looking at the body of the table. If the
elements are symmetrical with respect to the main diagonal, then the commutative property
for the operation is true. See Figure 6-4(c).
a
b
c
a
b
c
a
a
b
c
a
a
b
c
b
b
c
c
c
a
a
b
b
c
a
b
c
c
a
d
(a) Closed
(b) Not closed
a
b
c
a
b
c
a
a
b
c
a
a
c
b
b
b
c
a
b
b
c
b
c
c
a
b
c
c
a
b
(c) Commutative
(d) Not commutative
a
b
c
a
b
c
a
a
b
c
a
c
b
a
b
b
c
a
b
b
c
a
c
c
a
b
c
a
c
b
(e) Identity (a)
(f ) No identity
Figure 6-4
6-19
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The system shown in Figure 6-4(d) is not commutative. For example, we see that
c ∗ a = a ∗ c.
Finally, a system has an identity element if there is a row and a column of elements in
the table that are identical to the row and column outside. See Figure 6-4(e).
The system shown in Figure 6-4(f) does not have an identity element that can be used
for all the elements in the system.
Sidelight
Magic Squares
Have you ever heard of a magic square? These squares
have fascinated and intrigued people ever since the first
one was discovered more than 4000 years ago.
The sum of the numbers in each row is equal to 15:
4 + 9 + 2 = 15
3 + 5 + 7 = 15
8 + 1 + 6 = 15
The sum of the numbers in each column is also 15:
4 + 3 + 8 = 15
9 + 5 + 1 = 15
2 + 7 + 6 = 15
And the sum of the two diagonals is also 15:
4 + 5 + 6 = 15
2 + 5 + 8 = 15
A magic square!
4
9
2
3
5
7
8
1
6
The first magic square appears in the Chinese classic I-King and is called lo-shu. It is said that the Emperor
Yu saw the square engraved on the back of a divine
tortoise on the bank of the Yellow River in 2200 B.C.E. The
square was the same as the one shown in the above figure except that the numerals were indicated by black
and white dots, the even numbers in black and the odd
numbers in white.
Since some sort of magic was attributed to these
squares, they were frequently used to decorate the
abodes of gypsies and fortune tellers. They were very
popular in India, and they were eventually brought to
Europe by the Arabs.
Albrecht Dürer, a 16th-century artist from Germany,
used the magic square in his famous woodcut entitled
“Melancholia.” This square is indeed magic, for not only
are the sums of the rows, columns, and diagonals all
equal to 34, but also the sum of the four corners
(16 + 13 + 4 + 1) is 34. Furthermore, the sum of the four
center cells (10 + 11 + 6 + 7) is also 34, and the sum of
the slanting squares (2 + 8 + 9 + 15 and 3 + 5 + 12 + 14)
is 34. Finally, the year in which Dürer made the woodcut
appears in the bottom center squares (1514).
16
3
2
13
5
10
11
8
9
6
7
12
4
15
14
1
A great mathematician, Leonhard Euler
(1707–1783), constructed the magic square shown next.
The sum of the rows and columns is 260. Stopping
—Continued
6-20
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Section 6-3 Mathematical Systems without Numbers
259
Continued—
halfway on each row or column gives a sum of half of
260 or 130. Finally, a knight from a chess game can start
on square 1 and proceed in L-shaped moves and come
to rest on all squares in numerical order.
1
48
31
50
33
16
63
18
30
51
46
3
62
19
14
35
47
2
49
32
15
34
17
64
52
29
4
45
20
61
36
13
5
44
25
56
9
40
21
60
28
53
8
41
24
57
12
37
43
6
55
26
39
10
59
22
54
27
42
7
58
23
38
11
Magic squares have been a source of amusement
to people through the ages. They are easy to construct
and the method of constructing them can be found in
many mathematics books.
The other properties need to be checked using the elements of the system and the
operation.
Example 6-14
Find the inverse of the elements in the system defined as
∗ w x y z
w x y z w
x y z w x
y z w x y
z w x y z
Solution
Since the identity is z, the inverse of each element can be found by solving these
equations.
w∗
=z
x ∗
=z
y ∗
=z
z ∗
=z
The game scissors, paper, rock
forms a system, as shown here. 0
means a tie, and the operation
is “game,” called g here. What
properties hold for this system?
w ∗ y = z, so y is the inverse of w.
x ∗ x = z, so x is the inverse of x.
y ∗ w = z, so w is the inverse of y.
z ∗ z = z, so z is the inverse of z.
6-21
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Chapter 6
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Try This One
6-F
Answer these questions using the system shown here.
a
c
a
b
a
b
c
b
a
b
c
c
b
c
a
(a) Is the system closed for ?
(b) Is the system commutative?
(c) What is the identity element?
(d) What is the inverse of a?
For the answer, see page 266.
Mixing paint using the three primary colors, red (R), blue (B), and yellow (Y), can be
thought of as a mathematical system with a binary operation. When red and blue are mixed,
the color purple (P) is obtained. When blue and yellow are mixed, green (G) is obtained.
When red and yellow are mixed, orange (O) is obtained. Mixing paint can be set up in table
form, as shown.
Let M be the operation of mixing two colors of paint.
M R
B
Y
R
R
P
O
B
P
B
G
Y
O G Y
What properties does the system have?
As shown in this section, mathematical systems need not be a set of numbers with
operations such as addition or multiplication. They can use any set of elements and any
clearly-defined operations.
Exercise Set 6-3
Computational Exercises
For Exercises 1–15, use the elements C, D, E, and F, and the operation ? as defined by
? C D
C D F
D F E
E C D
F E C
6-22
E F
C E
D C
E F
F D
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Section 6-3 Mathematical Systems without Numbers
1.
3.
5.
7.
9.
11.
12.
13.
14.
15.
C?E
E?E
C?F
E ? (C ? C)
(E ? D) ? E
261
F?D
F?F
(D ? E) ? D
F ? (D ? C)
C ? (D ? E)
2.
4.
6.
8.
10.
Is the system closed under the operation?
Is there an identity for the operation?
Is the operation commutative?
What are the inverses for each element?
Find the value for x when E ? x = D.
For Exercises 16–30, use the elements and the operation ∗ as defined by
16. 17. 18. 19. 20. 21. ( )
22. ( )
23. ( ) 24.
26.
27.
28.
29.
30.
31.
(
)
25. ( ) Is the system closed under ∗?
Is ∗ commutative?
Is there an identity for the system?
Is ∗ associative?
What is the inverse of ?
Given a universal set U, a specified set A, and the null set , construct a table for the
operation of the union of sets. For example, U ∪ A = U, A ∪ = A, etc.
∪
U
A
U
U
A
A
For Exercises 32–34, use the elements and operation shown in Exercise 31.
32.
33.
34.
35.
Is the operation commutative?
Is the system closed under set union?
Is there an identity for the operation?
For the universal set U, a specified set A, and the null set, construct a table for the
operation of intersection of two sets.
6-23
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Chapter 6
Other Mathematical Systems
∩
U A U
A
For Exercises 36–38, use the elements and operations shown for ∩.
36. Is the operation commutative?
37. Is the system closed under intersection?
38. Is there an identity for the operation?
Writing Exercises
39. What four things are necessary to create a mathematical system?
40. Using the operation table, describe how you can tell if a system is closed for the
operation.
41. Using the operation table, describe how you can tell if a system is commutative for
the operation.
42. Using the operation table, describe how you can find the inverse of an element if it
exists.
Critical Thinking
A truth table similar to one shown in Chapter 3 for the conjunction is shown in Figure 6-5(a).
This can be converted to a mathematical system using T and F as the elements and ∧ as
the operation. This is shown in Figure 6-5(b).
p
T
T
F
F
q
T
F
T
F
(a)
p⵩q
⵩
T
F
T
F
F
F
T
F
T
F
F
F
(b)
Figure 6-5
43. Construct a table for a mathematical system for p ∨ q using ∨ as the operation.
44. Construct a table for a mathematical system for p → q. What properties are valid for
this system?
45. Construct a table for a mathematical system for p ↔ q. What properties are valid for
this system?
6-24
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Chapter
Six
Summary
Section
Important Terms
Important Ideas
6-1
mathematical system
infinite system
finite system
clock arithmetic
binary operation
group
abelian group
A mathematical system consists of a nonempty set of elements, operations
on the elements, definitions, and properties of the operations. A finite
mathematical system has a specific number of elements, whereas an infinite
mathematical system has an unlimited number of elements. A finite
mathematical system using the 12-hour clock is explained in this section.
When an operation is performed using two elements of a system, it is called
a binary operation. A mathematical system is called a group if it is closed
under the operation, has an identity element, satisfies the associative
property for the operation, and every element has an inverse. If the
operation is commutative, then the group is called an abelian group.
6-2
6-3
modular system
A modular system uses principals similar to those shown for the 12-hour
clock. It can have as few as two elements.
It is possible to create mathematical systems using elements other than
numbers. Some of these systems are shown in this section.
Review Exercises
For Exercises 1–20, find the equivalent number for the
given mod system.
1. 67 =
(mod 5)
2. 41 =
(mod 3)
3. 532 =
(mod 8)
4. 861 =
(mod 6)
14. 64 =
(mod 3)
15. 18 =
(mod 3)
16. 1235 =
(mod 6)
17. 4721 =
(mod 8)
18. 856 =
(mod 11)
19. 1000 =
20. 25 =
(mod 12)
(mod 4)
5. 22 =
(mod 4)
6. 10 =
(mod 2)
For Exercises 21–40, perform the indicated operation for the
given mod system.
7. 37 =
(mod 10)
21. 5 + 9 =
8. 999 =
(mod 7)
22. 2 − 10 =
(mod 11)
(mod 12)
9. 56 =
(mod 9)
23. 6 × 6 =
(mod 7)
10. 80 =
(mod 5)
24. 7 + 8 =
(mod 9)
25. 3 − 7 =
(mod 8)
26. 4 × 5 =
(mod 6)
27. 3 + 2 =
(mod 4)
11. 173 =
12. 45 =
13. 250 =
6-25
(mod 9)
(mod 7)
(mod 10)
263
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Chapter 6
28. 5 − 12 =
29. 6 × 7 =
Other Mathematical Systems
(mod 13)
(mod 10)
30. 10 × 10 =
(mod 12)
54. i · (i · i)
55. (−i · i) · (−1)
56. (1 · 1) · (−i)
31. 3 − 4 =
(mod 5)
57. i 2
32. 5 × 5 =
(mod 6)
58. i 3
33. 5 × (3 + 7) =
(mod 8)
59. i 10
34. 2 × (2 + 9) =
(mod 12)
60. Find the value of y when i · y = 1.
35. 3 − (3 − 5) =
(mod 6)
61. Find the value of y when y · (−1) = i .
36. (10 − 6) − 9 =
37. 5 × (7 − 9) =
(mod 11)
(mod 12)
62. Find the value of y when i · (y · i) = 1.
63. Is (−i · 1) · i = −i · (1 · i)?
38. 8 + 8 + 8 =
(mod 10)
64. Is i 3 = i 7 ?
39. 4 × 3 × 5 =
(mod 9)
65. Is the system closed under · ?
40. 3 × (4 + 5) =
(mod 7)
For Exercises 41–50, find the value of y in each equation.
41. 6 + y = 2 (mod 8)
42. y + 7 = 1 (mod 10)
43. y + 7 = 1 (mod 9)
66. Is the system commutative?
67. What is the identity for the system?
68. What is the inverse of i ?
69. What is the inverse of −1?
70. What is the inverse of −i ?
44. 3 − y = 6 (mod 8)
45. y − 2 = 5 (mod 6)
46. 3 × y = 6 (mod 8)
47. y + 2 = 1 (mod 12)
Chapter Test
For Exercises 1–6, find the equivalent number for the given
mod system.
48. 5 − y = 6 (mod 9)
1. 43 =
49. 3 × 5 = y (mod 7)
2. 518 =
50. 5 × (2 + y) = 1 (mod 12)
3. 15 =
(mod 2)
4. 56 =
(mod 12)
5. −6 =
(mod 4)
For Exercises 51–70, use this system:
·
i
−1
−i
1
i −1 −i
1
−1 −i
1
i
−i
1
i −1
1
i −1 −i
i −1 −i
1
(mod 6)
(mod 3)
6. −15 =
(mod 5)
For Exercises 6–12, perform the indicated operation for the
given mod system.
7. 8 + 6 =
(mod 10)
51. i · i
8. 3 + 7 =
(mod 9)
52. −1 · i
9. 4 − 6 =
(mod 7)
53. i · 1
6-26
10. 8 − 10 =
(mod 12)
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265
Chapter Test
11. 5 × 9 =
(mod 11)
25. Is the operation ∗ commutative?
12. 4 × 7 =
(mod 10)
26. Is the system closed?
For Exercises 13–18, find the value of y in each equation.
13. 2 + y = 4 (mod 6)
Projects
14. y + 8 = 6 (mod 10)
A mathematical system can be created by rotating a
five-pointed star about its center. See Figure 6-6(a). The
operation of rotating the star is designated by the symbol
R . The operation R is defined as follows. The first
element is the starting position, and the second element is
the number of degrees the star is rotated. For example, B R
C means to start at B and rotate the star 144◦ . B will then
be in the position of D; hence, B R C = D. See Figure 66(b). C R E means to start at C and rotate the star 288◦ ;
hence, C R E = B.
15. y − 3 = 7 (mod 8)
16. y − 2 = 4 (mod 5)
17. 3 × y = 0 (mod 12)
18. 3 × y = 0 (mod 5)
For Exercises 19–26, use the system shown.
* x y z
x x y z
y y x z
z z z s
A 0°
19. x ∗ z
E 288°
20. (y ∗ x) ∗ z
D
B 72°
C
E
Start at B
21. z ∗ z
D 216°
22. x ∗ (z ∗ y)
B
A
End at D
23. (z ∗ x) ∗ x
(a)
24. What is the inverse of y?
Figure 6-6
Rotate 144°
(b) B R C D
Revisited
▲
Mathematics in Our World
C 144°
Game or Mathematical System?
As stated previously, games can be thought of as mathematical systems. Recall that a mathematical system
consists of a set of elements, operations on elements,
definitions and properties for the operations and elements. In the case of “tic-tac-toe,” the elements are Xs
and Os. The operation is placing an X or an O in a space
on the grid. A win by either player is defined as getting
three Xs or three Os in a row, column, or diagonal. A draw
is defined as neither X nor O winning. There are many
properties that can be stated for this game. A few are
given here.
1. There are nine opening moves. (Actually, three if
symmetry is considered.)
2. If X begins and there is no winner, X has five moves.
3. If X begins and there is no winner, O has four
moves.
4. There are eight ways to win.
5. There are 126 possible distinct games.
There are two properties that might surprise you.
6. Out of the 126 possible distinct games, X wins 120
when X begins.
7. If X begins, O can (using the right strategy) end
every game in a draw.
Other games, such as tennis, chess, and some card
games can be considered as mathematical systems.
Adapted from An Introduction to the Elements of Mathematics by John N. Fujii, published by John Wiley and Sons, 1961.
6-27
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Chapter 6
Other Mathematical Systems
Answer these questions:
Answers to Try This One
1. Construct a table for the operation.
2. Perform each operation.
(a) B R B
(b) C R A
(c) D R E
(d) E R C
(e) A R B
(f) C R B
3. Is the system commutative?
4. Is there an identity? If so, what is it?
5. What is the inverse of B?
6. What is the inverse of C?
7. Is the system closed?
6-A. (a) 10; (b) 9; (c) 11; (d) 12; (e) 9
6-B. (a) 5; (b) 5; (c) 9
6-C. (a) 6 = 0 (mod 6); (b) 8 = 2 (mod 6);
(c) 20 = 2 (mod 6); (d) 8 = 2 (mod 6);
(e) 24 = 0 (mod 6)
6-D. +
0
1
0
0
1
1
1
0
(a) 0 + 0 = 0 = member of table;
0 + 1 = 1 = member of table
1 + 0 = 1 = member of table
1 + 1 = 0 = member of table
(b) 0 + 0 = 0 = 0 + 0;
0 + 1 = 1 = 1 + 0;
1 + 1 = 0 = 1 + 1.
(c) The identity is 0 since
0+0=0
1+0=1
(d) The inverse of 0 is 0 since
0+0=0
The inverse of 1 is 1 since
1+1=0
6-E. (a) y; (b) w; (c) w
6-F. (a) Yes; (b) Yes; (c) b; (d) c
6-28