# 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.

(ii) + 0011 and - 1101.

Solution:

+ 0 0 1 1      ⇒      0 0 0 1 1

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

1 0 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:

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

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

0 1 1 1 1

1      carry

1 0 0 0 0

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

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

1 0 0 0 1

1      carry

1 0 0 1 0

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
• Arithmetic Operations of Binary Numbers

Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.

## Recent Articles

1. ### Fraction in Lowest Terms |Reducing Fractions|Fraction in Simplest Form

Feb 28, 24 04:07 PM

There are two methods to reduce a given fraction to its simplest form, viz., H.C.F. Method and Prime Factorization Method. If numerator and denominator of a fraction have no common factor other than 1…

2. ### Equivalent Fractions | Fractions |Reduced to the Lowest Term |Examples

Feb 28, 24 01:43 PM

The fractions having the same value are called equivalent fractions. Their numerator and denominator can be different but, they represent the same part of a whole. We can see the shade portion with re…

3. ### Fraction as a Part of Collection | Pictures of Fraction | Fractional

Feb 27, 24 02:43 PM

How to find fraction as a part of collection? Let there be 14 rectangles forming a box or rectangle. Thus, it can be said that there is a collection of 14 rectangles, 2 rectangles in each row. If it i…

4. ### Fraction of a Whole Numbers | Fractional Number |Examples with Picture

Feb 24, 24 04:11 PM

Fraction of a whole numbers are explained here with 4 following examples. There are three shapes: (a) circle-shape (b) rectangle-shape and (c) square-shape. Each one is divided into 4 equal parts. One…