Conversion of binary numbers to octal or hexadecimal numbers and viceversa may be accomplished very easily.
Since a string of 3 bits can have 8 different permutations, it follows that each 3bit 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 hexadecimal digit uniquely. The table below gives the decimal numbers 0 to 15 and their binary, octal and hexadecimal equivalents and also the corresponding 3bit and 4bit strings.
Conversion of binary numbers to octal or hexadecimal numbers and vice versa:
Conversion Table 


Decimal  Binary  Octal  3bit String  Hexadecimal  4bit 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 3bit binary equivalent.
The same procedure is to be adopted in the case of hexadecimal 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 hexadecimal numbers. Conversion to decimal may also be accomplished by the same procedure.
Following examples on conversion of binary numbers to octal or hexadecimal numbers and vice versa will elucidate the working method:
1. Convert the following to octal numbers:
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