Converting Hex Representation into Binary Representation

It is easy to convert between base sixteen and base two representation:

• (In base two) Shifting left by four bits is equivalent to multiplication by sixteen.
• (In base hex) Shifting left by one digit is equivalent to multiplication by sixteen.

To see how this works, look at this integer represented in base two and in base sixteen:

base two           base sixteen

1010        =      A

Now multiply each by sixteen:

base two           base sixteen

1010 0000   =      A0

Groups of four bits (starting from the right) match powers of sixteen, so each group of four bits matchs a digit of the hexadecimal representation. Let us rewrite the integer C6D in binary:

C6D  =    C × sixteen2 +    6 × sixteen1   +    D × sixteen0  

     =    C × (24)2    +    6 × (24)1      +    D × (24)0  

     = 1100 × (24)2    + 0110 × (24)1      + 1101 × (24)0   

     = 1100 ×  28      + 0110 ×  24        + 1101 × 1  

Using the idea that each multiplication by two is equivalent to appending a zero to the right, this is:

     = 1100 0000 0000  + 0110 0000         + 1101  

C6D  = 1100 0110 1101

Each digit of hex can be converted into a 4-bit binary number, each place of a hex number stands for a power of 2⁴. It stands for a number of 4-bit left shifts. So you can convert a hexadecimal representation of a number into an unsigned binary represenation directly by replacing each hex digit with its 4-bit binary equivalent. For example:

   1    A    4    4    D      (Hex    Representation) 

0001 1010 0100 0100 1101      (Binary Representation)