# Conversion of Binary Numbers 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:

1. Convert the following to octal numbers:

(a) 11101011102

Solution:

001110101110

= 001 110 101 110

= 16568

Hence the required octal equivalent is 1656.

(b) 111101.011012

Solution:

111101.0110102

= 75.328

Hence the required octal equivalent is 75.32.

2. Convert the following to their binary equivalents:

(a) 15738

Solution:

15738

= 001 101 111 011

= 11011110112

Hence the required binary number is 1101111011.

(b) 64.1758

Solution:

64.1758

= 110 100 . 001 111 101

= 110100.0011111012

Hence the required binary number is 110100.001111101.

3. Convert the following to hexa-decimal numbers:

(a) 11111011012

Solution:

001111101101

= 0011 1110 1101

= 3ED16

Therefore, 11 1110 11012 = 3ED16

(b) 11110.010112

Solution:

11110.010112

= 0001 1110 . 0101 1000

= 1E.5816

Therefore, 11110.010112 = 1E.5816

4. Convert the following to binary equivalents:

(a) A74816

Solution:

A74816

= 1010 0111 0100 1000

= 10100111010010002

Hence the required binary equivalent is 1010011101001000.

(b) BA2.23C16

Solution:

BA2.23C16

= 1011 1010 0010 . 0010 0011 11002

= 101110100010.0010001111

Hence the required binary equivalent is 101110100010 . 0010001111.

5. Convert 15738 to hexa-decimal

Solution:

15738

= 001101111011

= 0011 0111 1011 37B16

Hence 15738 = 37B16

6. Convert A74816 to octal equivalents.

Solution:

A74816

= 1010 0111 0100 1000

= 001 010 011 101 001 000

= 1235108

Therefore, A74816 = 1235108

7. Convert the following to decimal numbers:

(a) 7258

Solution:

7258 = 111010101

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

= 46910

Therefore, 7258 = 46910

(b) D9F16

Solution:

D9F16

= 1101 1001 1111

= 110110011111

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

= 348710

Therefore, D9F16 = 348710

• 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
• Arithmetic Operations of Binary Numbers

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