Binary Addition

Binary numbers, also known as base-2 number systems, are represented using two digits namely 0 and 1. The numbers in a binary number system look like this - 1100011010. Each digit in the binary number system is known as ‘Bit’.

All digital devices use a binary number system in their electronic circuit. The input 0 indicates OFF state and whereas input 1 indicates the On state. Because of these implementations, binary number systems are most widely used in modern computer technology. Read the article below to know how to perform Binary addition with and without regrouping.

These may include addition, multiplication, division, and subtraction. Each binary operation is represented by a different symbol. Besides being used in Mathematics, these operations play an important role in computer technology also. They help us make operating systems and circuits for various electrical devices like computers, laptops, smartphones, etc. 

Basic Binary Arithmetic Operations 

In this article, we will discuss binary addition in detail along with binary addition examples so students can perform calculations faster. 

What is Binary Addition?

Binary addition is the sum of two or more binary numbers. Binary addition is much similar to decimal addition, even a bit easier. In the decimal addition, if the sum of two numbers results in two digits, we carry the digit in the ten’s place to the next column to the left. Similarly in binary addition, if the sum of two numbers is greater than 1, we carry the 2’s digit over to the next column to the left  For example, 1+ 1 = 10₂. In this case,  we write 1’s digit (0) and carry the 2’s digit i.e. 1 of the result to the next column to the left. For this reason, the bit that is carried to the next column is known as the carry bit.

Binary Addition Rules

The addition of two binary numbers is as easy as the decimal number system. Just we have to take note of some rules while adding two binary numbers. There are four-five rules associated with binary addition. The binary addition rules are as follows.

• 0 + 0 = 0

• 0 + 1 = 1

• 1 + 0 = 1

• 1 + 1 =10 ( carry 1 to the next significant bit)

• 1 + 1 + 1 = 11( carry 1 to the next significant bit)

As binary numbers include only two digits i.e. 0 and 1, these four five rules are all the possible conditions for the addition of binary numbers.

Here is the stepwise procedure of how to add two binary numbers with regrouping and without regrouping.

Binary Addition without Regrouping

When the sum of two or more binary digits results in 0 or 1, then in such cases we don’t need any regrouping.  Let’s add binary numbers \[101_{2}\] and \[10_{2}\] to understand it in a better way.

Step 1: Write all digits of both the binary numbers in a separate column according to their place values as shown below

1 0 1

+ 1 0

………..

Step 2:  Starting from the rightmost column, add 1 and 0. Follow the binary addition rules which says 1 + 0 = 1.

1 0 1

+ 1 0

………..

       1

Step 3: Moving to the next column to the left, add 0 and 1. Follow the binary addition rules which says 0 + 1 = 1.

1 0 1

+ 1 0

 ………..

     1  1

  ………..

Step 4: Moving again to the next column to the left, we can see there is only one digit left i.e. 1.  Hence, we can apply the rule 1 + 0 = 1.

1 0 1

+ 1 0

………..

  1  1  1

 ………..

Therefore, \[101_{2} +  10_{2} =  111_{2}\].

Binary Addition with Regrouping

When the sum of two or more binary digits results in more than 0 or 1, then in such cases we need regrouping.  Let’s add binary numbers 1001₂ and 111₂ to understand it in a better way.

Step 1: Write all digits of both the binary numbers in a separate column according to their place values as shown below

1 0 0 1 

+ 1  1 1

………….

Step 2:  Starting from the rightmost column, add 1 and 1. Follow the binary addition rules which says 1 + 1 = 10. This is equivalent to 2₁₀. Hence, we will write 0 at the bottom and two take 1 as a carryover to the next place value. 

       1

1 0 0 1 

+ 1  1 1

………….

           0

Step 3: Move to the next column to the left. Follow the binary addition rules which says 1 + 0 + 1 = 10. This is again equivalent to 2₁₀. Hence, we will write 0 at the bottom and two take 1 as a carryover to the next place value. 

   1  1

1 0 0 1 

+ 1  1 1

………….

    0  0

Step 4:  Move again to the next column to the left. Follow the binary addition rules which says 1 + 1 + 0 = 10. This is again equivalent to 2₁₀.       

 1 1  1

1 0 0 1 

+ 1  1 1

………….

  0 0  0

Step 5: Move again to the next column to the left. Follow the binary addition rules which says 1 + 1 + 0 = 10. This is again equivalent to 2₁₀. As it is the last column left, we will not take 1 as carryover, instead, we will write 10 as the result at the bottom.

  1 1 1

   1 0  0  1 

 +    1  1  1

…………….

 1  0  0  0  0

…………….

Therefore, \[1001_{2} + 111_{2} =   10000_{2}\]

Binary Addition Examples with Solutions

Example 1:  

Add \[1010_{2} and 1111_{2}\]

Solution:     

1  1

1   0   1  0

+     1  1   1  1

---------------------------

1 1  0   0  1

---------------------------

Example 2:

Add: \[10011_{2} \, and \, 110001_{2}\]

Solution:

         1       1 1

            1 0 0 1 1 

    +   1 1 0 0 0 1

------------------------------

   1  0  0  0  1  0  0

------------------------------

Practice Problems

1. Add the binary numbers - 11001+10111

Ans: 0110000

2. What is the sum of 1111+0101? 

Ans: 010100

Summary 

Binary addition refers to adding more than one binary number. It is the same as the decimal system and covers binary numbers 0 and 1. For complex and fast calculations, we can use Binary addition converters. Binary numbers and their operations are used for various purposes, such as making electrical device circuits. Further, these operations are highly used in computer technology, where 0 indicates the OFF state of the circuit, and 1 indicates its ON state.