2's Compliment

Computers cannot subtract so we use a mathematical "trick" so that we only add binary numbers together.

The trick involves taking the 2's compliment of the number we are subtracting. This can easily be done by "flipping" each of the binary digits (1 becomes a 0; 0 becomes 1) and then adding a 1 to the right-hand digit.

Consider the decimal number 33₁₀ which in binary is: 0010 0001₂ flipping the bits gives 1101 1110₂ and now adding 1 gives: 1101 1111₂ which is the 2's compliment representation.

Let's do the calculation: 39₁₀ - 33₁₀.

39₁₀ is 0010 0111₂ and to do this subtraction we add the 2's compliment value of the subtractor.

- 33₁₀ is 01101 1111₂

So 39₁₀ - 33₁₀ is 0010 0111₂ + 1101 1111₂

We now apply a simple algorithm:

IF the number has a leading 1

THEN removing this gives the final solution

ELSE

Evaluate the 2’s compliment of this number and

add a minus sign to get the final solution.

Since we have a 1 in the final column when this is removed we get our answer: 0000 0110₂ or 6₁₀.

There are several Web sites that can help with the evaluation of 2's compliment. This is one that I found most useful:

• Decimal/Two's Complement Converter