# Binary Addition Using 1s Complement – Definition, Examples | How to Add Binary Numbers in 1’s Complement?

Are you searching for a tool that computes the addition of two binary numbers using 1’s complement method? If yes, then you have reached the correct place. Here we are giving the detailed steps on how binary addition of two numbers with 1s complement. You can also get Binary Addition Using 1s Complement definition, how to get 1s complement of a binary number, and example questions to improve the levels of understanding.

A binary number is a number with the base 2. Binary addition is one of the basic arithmetic operations. The binary system has only two digits 0 and 1. The binary code uses the digits 0’s and 1’s to make certain processes turn on or off. The process of the binary addition operation is very familiar to the decimal system by adjusting to the base 2.

Before attempting the binary addition, you must have complete knowledge of how the place works in the binary number system. Almost all modern digital computers and electronic circuits perform the binary operation by representing each bit as a voltage signal. The binary bit 0 means OFF state, 1 means ON state.

### 1’s Complement of a Binary Number

We have a simple algorithm to convert a binary number into 1’s complement. To get 1’s complement of a binary number, invert the given binary number. You can also implement a logic circuit using only NOT gate for each bit of binary number input.

Example:

Calculate the 1’s complement of a binary number 10101110

Invert each bit of a given binary number

So, 1’s complement of 10101110 is 01010001.

### Binary Addition Using 1’s Complement

Binary addition means simply performing an addition operation between two binary numbers. You can add two binary numbers digit by digit just like decimal numbers. The rules for adding two binary digits is 0 + 0 = 0, 1 + 0 = 1, 0 + 1 = 1, 1 + 1 = 10(1 is carry). At first, convert the given numbers to the 1’s complement and add those numbers using thee rules. Also, have a look at the detailed process on how to add binary numbers using 1’s complement in the following section.

### How to do Binary Addition Using 1s Complement?

The process of adding binary numbers purely depends on their sign and magnitude. Here are the different cases.

Case I: Adding a positive and a negative number. If the positive number has a greater magnitude

Here, the addition of numbers is performed after taking 1’s complement of the negative number, and at the end round carry of the sum is added to the LSB (least significant bit).

Example: + 1101 – 1011

+ 1101 ⇒ 0 1 1 0 1

– 1011 ⇒ 1 0 1 0 0 (1’s complement)

⇒ 0 0 0 0 1 + 1 = 0 0 0 1 0

Hence the sum is +0010.

Case II: If the negative number has a greater magnitude

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

Example: 1011 – 1101

+ 1 0 1 1 ⇒ 0 1 0 1 1

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

⇒ 1 1 1 0 1

The sum is 1’s complement of 1101

Hence the sum is – 0010.

Case III: If two binary numbers are negative.

To add two negative binary numbers, 1’s complements of both the numbers are taken later addition is performed. In this case, an end-around carry will always appear. This along with a carry from the MSB will generate a 1 in the sign bit. 1’s complement of the magnitude bits of the result of an addition will give the final sum.

Example: -0110 and -0111

– 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 = 1 0 0 1 0

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

Hence the sum is -1101.

### Binary Addition Using 1s Complement Examples

Example 1:

Calculate the sum of 0100, -1000 using the 1’s complement.

Solution:

The given binary numbers are 0100, -1000

1’s complement of -1000 is 10111

0 0 1 0 0 + 1 0 1 1 1 = 1 1 0 1 1

1’s complement of 1011 is 0100

Hence the sum is -0100.

Example 2:

Find the sum of 10000, -00111 using the 1’s complement.

Solution:

The given binary numbers are 10000, -00111

Find the 1’s complement of the negative number i.e 00111

1’s complement of 000111 = 111000

010000 + 111000 = 1 0 0 1 0 0 0

0 0 1 0 0 0 + 1 = 0 0 1 0 0 1

Hence, the sum is 001001.

Example 3:

Find the sum of -0100, -0010 using the 1’s complement method.

Solution:

The given numbers are -0100, -0010

Find the 1’s complement of the negative numbers

1’s complement of 0100 is 1011

1’s complement of 0010 is 1101

1 0 1 1 + 1 1 0 1 = 1 1 0 0 0

1 1 0 0 0 + 1 = 1 1 0 0 1

1’s complement of 1 0 01 is 0 1 1 0 and 1 is the sign bit

Hence, the sum is -0110

Example 4:

Find the sum of 1000 and -0101 using the 1’s complement.

Solution:

The given binary numbers are 1000 and -0101.

Find the 1’s complement of the negative number

1’s complement of 0101 is 11010.

0 1 0 0 0 + 1 1 0 1 0 = 1 0 0 0 1 0

1 0 0 0 1 0 + 1 = 1 0 0 0 1 1

Hence the sum is 00011.

### FAQ’s on Binary Addition using 1’s Complement

1. How to find 1’s complement of a binary number?

To get the 1’s complement of binary numbers, just invert the number. Inversion means placing 1’s in place of 0’s and 0’s in place of 1’s. Otherwise, you can also use NOT logic gate to find the 1’s complement.

2. How to add two negative binary numbers using the 1’s complement?

For adding two negative binary numbers with the 1’s complement, just find the 1’s complement of both numbers. Add those 1’s complement numbers and add 1 to the obtained result. Then you will get 1 in place of the sign bit. Get the 1’s complement of the obtained sum to get the final result.

3. What’s the purpose of using complements in binary number operations?

Compared to other systems for representing the signed numbers 1’s complement, 2’s complement has the advantage that the fundamental arithmetic operations are identical to those for unsigned binary numbers.