2′s Complement

As we have gone through the basics of binary number systems, we now know that any number can be represented by the combination of 0 & 1. We can easily explain any number in binary numbers which are positive in magnitude, but what about the negative numbers. We can say that if we put negative(-) sign before the number it will represent the negative number, but the convention of negative sign before the numbers are not that popular in digital electronics system because they lack the logical representations. Rather there are methods like 1's complement and 2's complement which are used very much to represent the negative numbers in binary number system. We have already discussed about the 1's complement and 2's complement will be discussed in this article. It can be said that this is an extension of 1's complement as it can be gained after operating a binary number by 1's complement. We will come to the method at a later stage.


As we have already discussed 2's complement represents any negative binary number. The concept comes from subtracting the number from 2N, where N is the number of bits of that number. So if we want to represent a negative binary number we have to first consider its positive magnitude, then subtracting that value form 2N ( where N is the number of bits of that particular number as stated earlier) that number is the negative equivalent of that negative number in 2's complement method. Now we will go through the method of finding 2's complement of a number with an example.

Method and Example

In the easiest way to find out the 2's complement of a binary number we have to find out the 1's complement of that number first then add 1 with that. Now let us look at an example to understand the method more easily.
Suppose we have to represent ( - 5)10 in binary number.

Step 1
We have to consider binary equivalent of ( + 5)10 which is (0 0 0 01 0 0 1)2

Step 2
Now we have to find out 1's complement of 1 0 0 1 which is = 1 1 1 1 0 1 1 0

Step 3
And finally we have to add 1 with the result

This is the negative binary representation of (-5). If we convert the result in decimal number system we will get 1 1 1 1 0 1 1 1= 25110 which can be said as (24 - 1) which we already discussed is the actual definition of 2's complement.

Subtraction using 2’s complement

One of the most popular applications of 2's complement is the subtraction of binary numbers using 2's complement method. This method is preferred because here subtraction can be done by doing additions. With an example we will be able to grasp the method at once.

We want to do 7 - 12

Step 1
Taking 2's complement of minuend (12) which is = 1 1 1 1 0 1 0 0

Step 2

Adding the result with binary equivalent of 7

So the subtraction method using 2's complement method is hereby explained.