One’s Complement Representation

One’s Complement Representation

Let A be an n-bit signed binary number in one’s complement form.  

The most significant bit represents the sign.  If it is a “0” the number is positive and if it is a “1” the number is negative.   The remaining (n-1) bits represent the magnitude, but not necessarily as a simple weighted number.  Consider the following one’s complement numbers and their decimal equivalents:

There are two representations of “0”, namely 000.....0 and 111.....1.  

From these illustrations we observe  

If the most significant bit (MSD) is zero the remaining (n-1) bits directly indicate the magnitude.  

If the MSD is 1, the magnitude of the number is obtained by taking the complement of all the remaining (n-1) bits.  

For example consider one’s complement representation of -6 as given above.  

Leaving the first bit ‘1’ for the sign, the remaining bits 111001 do not directly represent the magnitude of the number -6.  

Take the complement of 111001, which becomes 000110 to determine the magnitude. 

In the example shown above a 7-bit number can cover the range from +63 to -63.  In general an n-bit number has a range from +(2^n-1 - 1) to -(2^n-1 - 1) with two representations for zero.

The representation also suggests that if A is an integer in one’s complement form, then

         one’s complement of A  =  -A

One’s complement of a number is obtained by merely complementing all the digits.  

This relationship can be extended to fractions as well.  

For example if A = 0.101 (+0.625)₁₀, then the one’s complement of A is 1.010, which is one’s complement representation of (-0.625)₁₀.  Similarly consider the case of a mixed number.

                                      A = 010011.0101   (+19.3125)₁₀

         One’s complement of A = 101100.1010   (- 19.3125)₁₀

This relationship can be used to determine one’s complement representation of negative decimal numbers.

Example 1: What is one’s complement binary representation of decimal number -75?

Decimal number 75 requires 7 bits to represent its magnitude in the binary form.  One additional bit is needed to represent the sign.  Therefore,

         one’s complement representation of  75 =  01001011     one’s complement representation of -75 =  10110100