Contemporary Mathematics

# Chapter 4

4.1
1.
416
2.
1,851
3.
17,488
4.2
1.
$9 \times {10^2} + 2 \times {10^1} + 4 \times {10^0}$
2.
$1 \times {10^3} + 2 \times {10^2} + 7 \times {10^1} + 9 \times {10^0}$
3.
$4 \times {10^6} + 1 \times {10^5} + 3 \times {10^4} + 0 \times {10^3} + 0 \times {10^2} + 4 \times {10^1} + 5 \times {10^0}$
4.3
1.
621
2.
3,203
3.
40,630,891
4.4
1.
1269
4.5
1.
42,136
4.6
1.
6,105,643
4.7
1.
257
4.8
1.
6,054
4.9
1.
1,248,073
4.10
1.
77
2.
240
3.
3,447
4.11
1.
XXVII
2.
XLIX
3.
DCCXXXIX
4.
MMMDCXLVII
4.12
1.
0, 1, 2, 3
4.13
1.
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B
4.14
1.
157
4.15
1.
2,014
4.16
1.
851
4.17
1.
27
4.18
1.
0, 1, 2, 3
10, 11, 12, 13
20, 21, 22, 23
30, 31, 32, 33
100
4.19
1.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B

10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1A, 1B

20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 2A, 2B

30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 3A, 3B

40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 4A, 4B

50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 5A, 5B

60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 6A, 6B

70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 7A, 7B

80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 8A, 8B

90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 9A, 9B

A0, A1, A2, A3, A4, A5, A6, A7, A8, A9, AA, AB

B0, B1, B2, B3, B4, B5, B6, B7, B8, B9, BA, BB

100

4.20
1.
0, 1, 2, 10, 11, 12, 20, 21, 22, 100
4.21
1.
20107
4.22
1.
554B12
4.23
1.
100010012
4.24
1.

2.

4.25
1.
The result has the digit 7 in it. In base 4, the 7 is an illegal symbol.
4.26
1.
The remainders include 10, which in base 6 is an illegal symbol.
4.27
1.
Since 12 is larger than 10, the base 10 number cannot have less digits than the base 12 number. Since it did, we know an error has been made.
4.28
1.
12426
4.29
1.
 + 0 1 2 3 0 0 1 2 3 1 1 2 3 10 2 2 3 10 11 3 3 10 11 12
4.30
1.
6337
4.31
1.
 + 0 1 2 3 4 5 6 7 8 9 A B C D 0 0 1 2 3 4 5 6 7 8 9 A B C D 1 1 2 3 4 5 6 7 8 9 A B C D 10 2 2 3 4 5 6 7 8 9 A B C D 10 11 3 3 4 5 6 7 8 9 A B C D 10 11 12 4 4 5 6 7 8 9 A B C D 10 11 12 13 5 5 6 7 8 9 A B C D 10 11 12 13 14 6 6 7 8 9 A B C D 10 11 12 13 14 15 7 7 8 9 A B C D 10 11 12 13 14 15 16 8 8 9 A B C D 10 11 12 13 14 15 16 17 9 9 A B C D 10 11 12 13 14 15 16 17 18 A A B C D 10 11 12 13 14 15 16 17 18 19 B B C D 10 11 12 13 14 15 16 17 18 19 1A C C D 10 11 12 13 14 15 16 17 18 19 1A 1B D D 10 11 12 13 14 15 16 17 18 19 1A 1B 1C
4.32
1.
13B912
4.33
1.
100100102
4.34
1.
326
4.35
1.
22712
4.36
1.
The symbols 4 and 5 are not legal symbols in base 4. Careful use of the base 4 addition table would correct this error.
 1 3 3 + 1 1 2 3 1 1
4.37
1.
This is correct if the numbers are base 10 numbers, but these numbers are base 14 numbers. In base 14, 9 + 9 is not 18, but instead is 13. Careful use of the base 14 addition table generates the correct answer, ${163_{14}}$.
4.38
1.
 * 0 1 2 3 0 0 0 0 0 1 0 1 2 3 2 0 2 10 12 3 0 3 12 21
Base 4 Multiplication Table
4.39
1.
 * 0 1 2 3 4 5 6 7 8 9 A B C D 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 2 3 4 5 6 7 8 9 A B C D 2 0 2 4 6 8 A C 10 12 14 16 18 1A 1C 3 0 3 6 9 C 11 14 17 1A 1D 22 25 28 2B 4 0 4 8 C 12 16 1A 20 24 28 2C 32 36 3A 5 0 5 A 11 16 1B 22 27 2C 33 38 3D 44 49 6 0 6 C 14 1A 22 28 30 36 3C 44 4A 52 58 7 0 7 10 17 20 27 30 37 40 47 50 57 60 67 8 0 8 12 1A 24 2C 36 40 48 52 5A 64 6C 76 9 0 9 14 1D 28 33 3C 47 52 5B 66 71 7A 85 A 0 A 16 22 2C 38 44 50 5A 66 72 7C 88 94 B 0 B 18 25 32 3D 4A 57 64 71 7C 89 96 A3 C 0 C 1A 28 36 44 52 60 6C 7A 88 96 A4 B2 D 0 D 1C 2B 3A 49 58 67 76 85 94 A3 B2 C1
4.40
1.
40006
2.
10101112
4.41
1.
436912
4.42
1.
${10_6} \div {3_6} = {2_6}$
2.
${50_{12}} \div {{\text{A}}_{12}} = {6_{12}}$
4.43
1.
The symbols 4 and 5 are not legal symbols in base 4. Careful use of the base 4 multiplication table would correct this error. The correct answer is 3334.
4.44
1.
This is correct if the numbers are base 10 numbers, but these numbers are base 14 numbers. In base 14, ${49_{14}} \times {9_{14}} = 2{\text{D}}{{\text{B}}_{14}}$ is not 81, but instead is 5B. Careful use of the base 14 addition table (Table 4.9) generates the correct answer, ${49_{14}} \times {9_{14}} = 2{\text{D}}{{\text{B}}_{14}}$.

1.
A system in which the position of a numeral determines the value associated with that numeral.
2.
279
3.
$4 \times {10^4} + 5 \times {10^3} + 2 \times {10^2} + 0 \times {10^1} + 9 \times {10^0}$
4.
10
5.
A numeral is a symbol representing a number. A number is a quantity or amount.
6.
601,947
7.
60
8.
20
9.
Roman numerals do not use place value.
10.
341
11.
209
12.
247
13.
CDLXXIX
14.
A base 25 system would require 25 symbols.
15.
Since the 4 is the second digit, its place value is 181 times 4, or 72.
16.

329

17.
409
18.
5118
19.
2,126
20.
In base 4, 5 is not a valid symbol. So, a mistake has been made.
21.
 + 0 1 2 3 4 5 6 7 0 0 1 2 3 4 5 6 7 1 1 2 3 4 5 6 7 10 2 2 3 4 5 6 7 10 11 3 3 4 5 6 7 10 11 12 4 4 5 6 7 10 11 12 13 5 5 6 7 10 11 12 13 14 6 6 7 10 11 12 13 14 15 7 7 10 11 12 13 14 15 16
22.
1216
23.
78
24.
8214
25.
512
26.
In base 8, 8 is not a valid symbol. So, a mistake has been made.
27.
A common base 14 error is performing the operation in base 10.
28.
29.
The process is the same, except the multiplication table for the base is used instead of the familiar base 10 rules.
30.
the multiplication table for the base
31.
22406
32.
B14
33.
The 8 is not a symbol used in base 5.
34.
A symbol that is not used in that base is present, or a base 10 rule is used.
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