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 0 1 ⇒

1 carry

**Hence the required sum is + 0001.**

**(ii) + 1101 and - 1011**

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

** **

**Solution: **

- 1 0 1 1 ⇒

1 carry

**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 1 0 0 ⇒

**Hence the required sum is – 0010.**

** **

**(ii) + 0011 and
- 1101.**

** **

** **

** **

**Solution:**

** **

- 1 1 0 1 ⇒

**Hence the required sum is – 1010.**

**B. When the two numbers are negative**

** **

For the addition of two negative numbers 1’s complements of both the numbers are to be taken and then added. In this case an end-around carry will always appear. This along with a carry from the MSB (i.e. the 4th bit in the case of sign-plus-magnitude 5-bit register) will generate a 1 in the sign bit. 1’s complement of the magnitude bits of the result of addition will give the final sum.

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

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

**(i) -1010 and
-0101**

**Solution:**

** **

- 0 1 0 1 ⇒

1 carry

1’s complement of the magnitude bits of sum is 1111 and the sign bit is 1.

**Hence the
required sum is -1111.**

**(ii) -0110 and
-0111.**

**Solution:**

** **

- 0 1 1 1 ⇒

1 carry

1’s complement of 0010 is 1101 and the sign bit is 1.

**Hence the required sum is - 1101.**

- Decimal Number System

- Why Binary Numbers are Used

- Binary to Decimal Conversion

- Conversion of Numbers

- Hexa-decimal Number System

- Conversion of Binary Numbers to Octal or Hexa-decimal Numbers

- Octal and Hexa-Decimal Numbers

- Signed-magnitude Representation

- Radix Complement

- Diminished Radix Complement

- Arithmetic Operations of Binary Numbers

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