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

● Binary Numbers