Binary Addition using 1’s Complement

In binary addition using 1’s complement;

A. Addition of a positive and a negative binary number

We discuss the following cases under this.

Case I: When the positive number has greater magnitude.

In this case addition of numbers is performed after taking 1’s complement of the negative number and the end-around carry of the sum is added to the least significant bit.

The following examples will illustrate this method in binary addition using 1’s complement:

1. Find the sum of the following binary numbers:

(i) + 1110 and - 1101

Solution:

            + 1 1 1 0      ⇒      0 1 1 1 0

            - 1 1 0 1      ⇒      1 0 0 1 0      (taking 1’s complement)

                                      0 0 0 0 0

                                                 1      carry

                                      0 0 0 0 1

Hence the required sum is + 0001.

(ii) + 1101 and - 1011

(Assume that the representation is in a signed 5-bit register).

Solution:

           + 1 1 0 1      ⇒      0 1 1 0 1

            - 1 0 1 1      ⇒      1 0 1 0 0      (taking 1’s complement)

                                      0 0 0 0 1

                                                 1      carry

                                       0 0 0 1 0

Hence the required sum is + 0010.

Case II: When the negative number has greater magnitude.

In this case the addition is carried in the same way as in case 1 but there will be non end-around carry. The sum is obtained by taking 1’s complement of the magnitude bits of the result and it will be negative.

The following examples will illustrate this method in binary addition using 1’s complement:

Find the sum of the following binary numbers represented in a sign-plus-magnitude 5-bit register:

(i) + 1010 and - 1100

Solution:

           + 1 0 1 0      ⇒      0 1 0 1 0

            - 1 1 0 0      ⇒      1 0 0 1 1      (1’s complement)

                                      1 1 1 0 1

Hence the required sum is – 0010.