2s Complement Arithmetic
How to represent decimal number in 2’s complement?Let us take a number. The negative signed that number is such number sum of which with this number is zero. Take examples of our most familiar number system that is decimal system.
Where, In binary system there are only two digits 0 and 1. In binary system each and every number and its sign are represented by using only these two digits 0 and 1. As the negative and positive signs can not be written directly in binary system, we have to find some alternative method of representing them.
For that we take advantage of definition of negative numer. If we have to represent a negative binary number, we have to find a binary number sum of which with the original unsigned or positive binary number is zero. As the sum of these two numbers is zero, the second number is taken as negative equivalent of the first number.
Suppose we have to write - 5 in binary which is negative of 5 or + 5. Now 5 or + 5 can be represented in binary byte or 8 bits system as 0000 0101 and zero is represented in binary byte or 8 bits system as 0000 0000. Hence this result is representation of ( - 5)10.
Now let us examine 2's complement of 0000 0101 and this is Hence, it is proved zero minus any binary number is nothing but 2's complement of this number, hence 2's complement of any binary number can be represented as negative equivalent of that number. In 2's complement first we convert a decimal number into binary number system. Suppose we want to represent (5)10 into binary 2’s complement. Binary representation of (5)10 is (0000 0101)2.
Now the interesting fact is that the unsigned form (5)10 into 2’s complement is direct representation of binary form of (5)10 i.e. (0000 0101)2. Unsigned number is treated as positive number. Signed decimal number can be a positive number or negative. Now to calculate 2’s complement for a signed decimal number i.e. (-5)10, we have to find out the 1’s complement and then convert each digit into its opposite digit (i.e. 1 into 0 or 0 into 1 ) and add 1 at last. This result after addition is the 2’s complete representation of signed (negative) decimal number.
Now look at the chart below.
(+5)10 is same as (5)10. But for (-1)10 it is treated as unsigned (255)10. (-5)10 ⇔ (251)10
Always keep it in the mind that negative 2’s complement numbers are always represented as the binary number in such a way that when added to a positive number of the same magnitude must be zero.
The important thing is to be kept in the mind that the left most bit or significant bit indicates the sign of the integer. It is called the signed bit also.
- If the left most bit is zero then the number is positive.
- If the left most bit is one then the number is negative.
Arithmetic in 2’s Complement Method
2’s Complement AdditionAddition in the 2’s complement, it always follows the same rule as it is used in the normally binary addition. Suppose we want to add (8)10 and (-3)10.
First we have to convert them into 2’s complement and simply add them. As carry is 1 then number is positive. Here carry is 1 and it is out of 8 bits hence is not considered in byte and hence the answer is one byte long positive number. So it is clear that for signed and unsigned numbers addition can be performed easily in 2’s complement.