Computer Science
Convert the following binary numbers to decimal, octal and hexadecimal numbers.
(i) 100101.101
(ii) 10101100.01011
(iii) 1010
(iv) 10101100.010111
Number System
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Answer
(i) 100101.101
Decimal Conversion of integral part:
| Binary No | Power | Value | Result |
|---|---|---|---|
| 1 | 20 | 1 | 1x1=1 |
| 0 | 21 | 2 | 0x2=0 |
| 1 | 22 | 4 | 1x4=4 |
| 0 | 23 | 8 | 0x8=0 |
| 0 | 24 | 16 | 0x16=0 |
| 1 | 25 | 32 | 1x32=32 |
Decimal Conversion of fractional part:
| Binary No | Power | Value | Result |
|---|---|---|---|
| 1 | 2-1 | 0.5 | 1x0.5=0.5 |
| 0 | 2-2 | 0.25 | 0x0.25=0 |
| 1 | 2-3 | 0.125 | 1x0.125=0.125 |
Equivalent decimal number = 1 + 4 + 32 + 0.5 + 0.125 = 37.625
Therefore, (100101.101)2 = (37.625)10
Octal Conversion
Grouping in bits of 3:
| Binary Number | Equivalent Octal |
|---|---|
| 101 | 5 |
| 100 | 4 |
| . | . |
| 101 | 5 |
Therefore, (100101.101)2 = (45.5)8
Hexadecimal Conversion
Grouping in bits of 4:
| Binary Number | Equivalent Hexadecimal |
|---|---|
| 0101 | 5 |
| 0010 | 2 |
| . | |
| 1010 | A (10) |
Therefore, (100101.101)2 = (25.A)16
(ii) 10101100.01011
Decimal Conversion of integral part:
| Binary No | Power | Value | Result |
|---|---|---|---|
| 0 | 20 | 1 | 0x1=0 |
| 0 | 21 | 2 | 0x2=0 |
| 1 | 22 | 4 | 1x4=4 |
| 1 | 23 | 8 | 1x8=8 |
| 0 | 24 | 16 | 0x16=0 |
| 1 | 25 | 32 | 1x32=32 |
| 0 | 26 | 64 | 0x64=0 |
| 1 | 27 | 128 | 1x128=128 |
Decimal Conversion of fractional part:
| Binary No | Power | Value | Result |
|---|---|---|---|
| 0 | 2-1 | 0.5 | 0x0.5=0 |
| 1 | 2-2 | 0.25 | 1x0.25=0.25 |
| 0 | 2-3 | 0.125 | 0x0.125=0 |
| 1 | 2-4 | 0.0625 | 1x0.0625=0.0625 |
| 1 | 2-5 | 0.03125 | 1x0.03125=0.03125 |
Equivalent decimal number = 4 + 8 + 32 + 128 + 0.25 + 0.0625 + 0.03125 = 172.34375
Therefore, (10101100.01011)2 = (172.34375)10
Octal Conversion
Grouping in bits of 3:
| Binary Number | Equivalent Octal |
|---|---|
| 100 | 4 |
| 101 | 5 |
| 010 | 2 |
| . | . |
| 010 | 2 |
| 110 | 6 |
Therefore, (10101100.01011)2 = (254.26)8
Hexadecimal Conversion
Grouping in bits of 4:
| Binary Number | Equivalent Hexadecimal |
|---|---|
| 1100 | C (12) |
| 1010 | A (10) |
| . | |
| 0101 | 5 |
| 1000 | 8 |
Therefore, (10101100.01011)2 = (AC.58)16
(iii) 1010
Decimal Conversion:
| Binary No | Power | Value | Result |
|---|---|---|---|
| 0 | 20 | 1 | 0x1=0 |
| 1 | 21 | 2 | 1x2=2 |
| 0 | 22 | 4 | 0x4=0 |
| 1 | 23 | 8 | 1x8=8 |
Equivalent decimal number = 2 + 8 = 10
Therefore, (1010)2 = (10)10
Octal Conversion
Grouping in bits of 3:
| Binary Number | Equivalent Octal |
|---|---|
| 010 | 2 |
| 001 | 1 |
Therefore, (1010)2 = (12)8
Hexadecimal Conversion
Grouping in bits of 4:
| Binary Number | Equivalent Hexadecimal |
|---|---|
| 1010 | A (10) |
Therefore, (1010)2 = (A)16
(iv) 10101100.010111
Decimal Conversion of integral part:
| Binary No | Power | Value | Result |
|---|---|---|---|
| 0 | 20 | 1 | 0x1=0 |
| 0 | 21 | 2 | 0x2=0 |
| 1 | 22 | 4 | 1x4=4 |
| 1 | 23 | 8 | 1x8=8 |
| 0 | 24 | 16 | 0x16=0 |
| 1 | 25 | 32 | 1x32=32 |
| 0 | 26 | 64 | 0x64=0 |
| 1 | 27 | 128 | 1x128=128 |
Decimal Conversion of fractional part:
| Binary No | Power | Value | Result |
|---|---|---|---|
| 0 | 2-1 | 0.5 | 0x0.5=0 |
| 1 | 2-2 | 0.25 | 1x0.25=0.25 |
| 0 | 2-3 | 0.125 | 0x0.125=0 |
| 1 | 2-4 | 0.0625 | 1x0.0625=0.0625 |
| 1 | 2-5 | 0.03125 | 1x0.03125=0.03125 |
| 1 | 2-6 | 0.015625 | 1x0.015625=0.015625 |
Equivalent decimal number = 4 + 8 + 32 + 128 + 0.25 + 0.0625 + 0.03125 + 0.015625 = 172.359375
Therefore, (10101100.010111)2 = (172.359375)10
Octal Conversion
Grouping in bits of 3:
| Binary Number | Equivalent Octal |
|---|---|
| 100 | 4 |
| 101 | 5 |
| 010 | 2 |
| . | . |
| 010 | 2 |
| 111 | 7 |
Therefore, (10101100.010111)2 = (254.27)8
Hexadecimal Conversion
Grouping in bits of 4:
| Binary Number | Equivalent Hexadecimal |
|---|---|
| 1100 | C (12) |
| 1010 | A (10) |
| . | |
| 0101 | 5 |
| 1100 | C (12) |
Therefore, (10101100.010111)2 = (AC.5C)16
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