Chapter 2 - Encoding Schemes and Number System
Exercise Solutions ( continued .. )
6. Express the following decimal numbers to hexadecimal numbers:
(i) 548
Steps:
548 ÷ 16 =
34, remainder 4
34 ÷ 16 =
2, remainder 2
2 ÷ 16 =
0, remainder 2
Thus 548 is 224 in Hexadecimal
(ii) 4052
Steps:
4052 ÷ 16 = 253, remainder 4
253 ÷ 16 = 15, remainder 13 (which is represented as D in hexadecimal)
15 ÷ 16 = 0, remainder 15 (which is represented as F in hexadecimal)
Thus 4052 is FD4 in Hexadecimal
(iii) 58
Steps:
58 ÷ 16 = 3, remainder 10 (which is represented as A in hexadecimal)
3 ÷ 16 = 0, remainder 3
Thus 58 is 3A in Hexadecimal
(iv) 100.25
Step 1: Convert the Integer Part (100)
100 ÷ 16 = 6, remainder 4
6 ÷ 16 = 0, remainder 6
Step 2: Convert the Fractional
Part (0.25)
Multiply by
16:
0.25 × 16 =
4.0
Thus 100.25 in Hexadecimal is 64.4
7. Express the following hexadecimal numbers into equivalent decimal
numbers.
(i) 4A2
Identify the positions and their values:
The hexadecimal number is read
from right to left, where each digit's position corresponds to a power of 16.
The digits in 4A2 are:
- 22 in the 160 (ones)
place
- AA (which is 10 in decimal) in
the 161 (sixteens) place
- 4 in the 162 (two
hundred fifty-sixes) place
Calculate the decimal value
for each digit:
10 × 161 = 10 × 16 = 160
Sum all the values:
1024 + 160 + 2 = 1186
Thus 4A2 in Decimal is 1186
(ii) 9E1A
Calculate decimal value for each digit:
9 × 163 = 9 × 4096 = 36864
14 × 162 = 14 × 256 = 3584
1 × 161 = 1 × 16 = 16
10 × 160 = 10 × 1 = 10
Sum all the values: 40474
(iii) 6BD
Calculate decimal value for each digit:
6 × 162 = 6 × 256 = 1536
11 × 161 = 11 × 16 = 176
13 × 160 = 13 × 1 = 13
Sum all the values: 1725
(iv) 6C.34
Calculate decimal value for integer part (6C)
6 × 161 = 6 × 16 = 96
12 × 160 = 12 × 1 = 12
Convert the fractional part (.34)
3 × 16−1 = 0.1875
4 × 16−2 = 0.015625
Sum all values and combine both parts to get the answer: 108.203125.
8. Convert the following binary numbers into octal and hexadecimal numbers.
(i) 1110001000
Convert to Decimal first: 904
Now, convert 904 to Octal:
904 ÷ 8 =
113 with a remainder of 0
113 ÷ 8 =
14 with a remainder of 1
14 ÷ 8 =
1 with a remainder of 6
1 ÷ 8 =
0 with a remainder of 1
Octal value: 1610
Now, convert 904 to Hexadecimal:
904 ÷ 16 =
56 with a remainder of 8
56 ÷ 16 =
3 with a remainder of 8
3 ÷ 16 =
0 with a remainder of 3
Hexadecimal value: 388
(ii)
110110101
Convert to Decimal first: 437
Now convert 437 to Octal:
437 ÷ 8 =
54 with a remainder of 5
54 ÷ 8 =
6 with a remainder of 6
6 ÷ 8 =
0 with a remainder of 6
Octal value: 665
Now convert 437 to Hexadecimal:
437 ÷ 16 =
27 with a remainder of 5
27 ÷ 16 =
1 with a remainder of 11 (which is represented as B in
hexadecimal)
1 ÷ 16 =
0 with a remainder of 1
Hexadecimal value: 1B5
(iii) 1010100
Convert to Decimal first: 84
Now convert 84 to Octal:
84 ÷ 8 =
10 with a remainder of 4
10 ÷ 8 =
1 with a remainder of 2
1 ÷ 8 =
0 with a remainder of 1
Octal value: 124
Now convert 84 to Hexadecimal:
84 ÷ 16 =
5 with a remainder of 4
5 ÷ 16 =
0 with a remainder of 5
Hexadecimal value: 54
(iv) 1010.1001
Convert to Decimal first: 10.5625
Now convert 10.5625 to Octal:
Integer Part:
10 ÷ 8 =
1 with a remainder of 2
1 ÷ 8 =
0 with a remainder of 1
Fractional Part:
0.5625 × 8
= 4.5 (Take the integer part, which is 4)
0.5 × 8 =
4.0 (Take the integer part, which is 4)
Octal value: 12.44
Now convert 10.5625 to Hexadecimal
Integer Part:
10 ÷ 16 =
0 with a remainder of 10 (which is represented as A in
hexadecimal)
Fractional Part:
0.5625 × 16
= 9.0 (Take the integer part, which is 9)
Hexadecimal value: A.9
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