Chapter 4

416

1,851

17,488

$9\times <!--\; \times \; -->{10}^2+2\times <!--\; \times \; -->{10}^1+4\times <!--\; \times \; -->{10}^0$

$1\times <!--\; \times \; -->{10}^3+2\times <!--\; \times \; -->{10}^2+7\times <!--\; \times \; -->{10}^1+9\times <!--\; \times \; -->{10}^0$

$4\times <!--\; \times \; -->{10}^6+1\times <!--\; \times \; -->{10}^5+3\times <!--\; \times \; -->{10}^4+0\times <!--\; \times \; -->{10}^3+0\times <!--\; \times \; -->{10}^2+4\times <!--\; \times \; -->{10}^1+5\times <!--\; \times \; -->{10}^0$

621

3,203

40,630,891

1269

42,136

6,105,643

257

6,054

1,248,073

77

240

3,447

XXVII

XLIX

DCCXXXIX

MMMDCXLVII

0, 1, 2, 3

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

157

2,014

851

27

0, 1, 2, 3
10, 11, 12, 13
20, 21, 22, 23
30, 31, 32, 33
100

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

0, 1, 2, 10, 11, 12, 20, 21, 22, 100

2010₇

554B₁₂

10001001₂

The result has the digit 7 in it. In base 4, the 7 is an illegal symbol.

The remainders include 10, which in base 6 is an illegal symbol.

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.

1242₆

633₇

13B9₁₂

10010010₂

32₆

227₁₂

The symbols 4 and 5 are not legal symbols in base 4. Careful use of the base 4 addition table would correct this error.
The correct answer is 311₄.

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, ${163}_{14}$.

4000₆

1010111₂

4369₁₂

${10}_6÷<!--\; ÷\; -->3_6=2_6$

${50}_{12}÷<!--\; ÷\; -->{\text{A}}_{12}=6_{12}$

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.

This is correct if the numbers are base 10 numbers, but these numbers are base 14 numbers. In base 14, ${49}_{14}\times <!--\; \times \; -->9_{14}=2\text{D}{\text{B}}_{14}$ is not 81, but instead is 5B. Careful use of the base 14 addition table (Table 4.9) generates the correct answer, ${49}_{14}\times <!--\; \times \; -->9_{14}=2\text{D}{\text{B}}_{14}$.

A system in which the position of a numeral determines the value associated with that numeral.

279

$4\times <!--\; \times \; -->{10}^4+5\times <!--\; \times \; -->{10}^3+2\times <!--\; \times \; -->{10}^2+0\times <!--\; \times \; -->{10}^1+9\times <!--\; \times \; -->{10}^0$

10

A numeral is a symbol representing a number. A number is a quantity or amount.

601,947

60

20

Roman numerals do not use place value.

341

209

247

CDLXXIX

A base 25 system would require 25 symbols.

Since the 4 is the second digit, its place value is 18¹ times 4, or 72.

329

40₉

511₈

2,126

In base 4, 5 is not a valid symbol. So, a mistake has been made.

121₆

7₈

82₁₄

5₁₂

In base 8, 8 is not a valid symbol. So, a mistake has been made.

A common base 14 error is performing the operation in base 10.

the addition table

The process is the same, except the multiplication table for the base is used instead of the familiar base 10 rules.

the multiplication table for the base

2240₆

B₁₄

The 8 is not a symbol used in base 5.

A symbol that is not used in that base is present, or a base 10 rule is used.