to

Octal or Hexa-decimal Numbers

Conversion of binary numbers to octal or hexa-decimal numbers and vice-versa may be accomplished very easily.

Since a string of 3 bits can have 8 different permutations, it follows that each 3-bit string is uniquely represented by one octal digit. Similarly, since a string of 4 bits has 16 different permutations each 4 bit string represents a hexa-decimal digit uniquely. The table below gives the decimal numbers 0 to 15 and their binary, octal and hexa-decimal equivalents and also the corresponding 3-bit and 4-bit strings.

**Conversion
of binary numbers to octal or hexa-decimal numbers and vice versa:**

## Conversion Table |
|||||||
---|---|---|---|---|---|---|---|

Decimal | Binary | Octal | 3-bit String | Hexa-decimal | 4-bit String | ||

0 | 0 | 0 | 000 | 0 | 0000 | ||

1 | 1 | 1 | 001 | 1 | 0001 | ||

2 | 10 | 2 | 010 | 2 | 0010 | ||

3 | 11 | 3 | 011 | 3 | 0011 | ||

4 | 100 | 4 | 100 | 4 | 0100 | ||

5 | 101 | 5 | 101 | 5 | 0101 | ||

6 | 110 | 6 | 110 | 6 | 0110 | ||

7 | 111 | 7 | 111 | 7 | 0111 | ||

8 | 1000 | 10 | - | 8 | 1000 | ||

9 | 1001 | 11 | - | 9 | 1001 | ||

10 | 1010 | 12 | - | A | 1010 | ||

11 | 1011 | 13 | - | B | 1011 | ||

12 | 1100 | 14 | - | C | 1100 | ||

13 | 1101 | 15 | - | D | 1101 | ||

14 | 1110 | 16 | - | E | 1110 | ||

15 | 1111 | 17 | - | F | 1111 |

Thus to convert a binary number to its octal equivalent we arrange the bits into groups of 3 starting at the binary point and move towards the MSB. We then replace each group by the corresponding octal digit. If the number of bits is not a multiple of 3, we add necessary number of zeros to the left of MSB. For binary fractions, we have to work towards the right of the binary point and follow the same procedure. Similarly, for conversion of octal numbers to binary numbers, we have to replace each octal digit by its 3-bit binary equivalent.

The same procedure is to be adopted in the case of hexa-decimal numbers and vice versa by converting the given numbers to binary numbers first with the help of above procedure and then converting these binary numbers to hexa-decimal numbers. Conversion to decimal may also be accomplished by the same procedure.

**Following
examples on conversion of binary numbers to octal or hexa-decimal numbers and
vice versa** **will elucidate the working method:**

(a) 1110101110

001

= 001 110 101 110

= 1656

(b) 111101.01101

111

= 75.32

(a) 1573

1573

= 001 101 111 011

= 1101111011

(b) 64.175

64.175

= 110 100 . 001 111 101

= 110100.001111101

(a) 1111101101

= 0011 1110 1101

= 3ED

(b) 11110.01011

1

= 0001 1110 . 0101 1000

= 1E.58

(a) A748

A748

= 1010 0111 0100 1000

= 1010011101001000

(b) BA2.23C

BA2.23C

= 1011 1010 0010 . 0010 0011 1100

= 101110100010.0010001111

1573

= 001101111011

= 0011 0111 1011 37B

A748

= 1010 0111 0100 1000

= 001 010 011 101 001 000

= 123510

(a) 725

725

= 256 + 128 + 64 + 16 + 4 + 1

= 469

(b) D9F

D9F

= 1101 1001 1111

= 110110011111

= 2048 + 1024 + 256 + 128 + 16 + 8 + 4 + 2 + 1

= 3487

- Why Binary Numbers are Used

- Binary to Decimal Conversion

- Conversion of Numbers

- Hexa-decimal Number System

- Conversion of Binary Numbers to Octal or Hexa-decimal Numbers

- Octal and Hexa-Decimal Numbers

- Signed-magnitude Representation

- Radix Complement

- Diminished Radix Complement

- Arithmetic Operations of Binary Numbers

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