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Binary Addition and Subtraction

Here, we are going to learn about the Binary addition and subtraction: what are the binary addition and subtraction, and how to perform them?
Submitted by Saurabh Gupta, on October 20, 2019

1) Binary Addition

Since binary numbers consist of only two digits 0 and 1, so their addition is different from decimal addition. Addition of binary numbers can be done following certain rules:

ABSumCarry
0000
0110
1010
1101

The above table contains two bits a and b, their sum and carry.

On adding,

    0 + 0 = 0, 	
    0 + 1 = 1,	
    1 + 0 = 1,	
    1 + 1 = 10 (i.e., sum is 0 and carry is 1)

Let's do some exercise and solution some questions based on binary addition to get more of the topic.

Example 1: Perform (10)2 + (11)2

Solution:

Using the rules provided above, sum operation can be performed as:

Binary addition example 1

Therefore, (10)2 + (11)2 = (101)2

Verification:

We can verify our result by converting the above binary numbers into decimal numbers and then verifying the sum.

Here, (10)2 = (2)10, (11)2 = (3)2 and (101)2 = (5)10, thus when we will add 2 and 3 we get sum as 5.


Example 2: Perform (1)2 + (1)2 + (1)2 + (1)2

Solution:

Using the rules provided above, sum operation can be performed as:

Binary addition example 2

Example 3: Perform (110)2 + (111)2 + (101)2

Solution:

Using the rules provided above, sum operation can be performed as:

Binary addition example 3

Verification:

We can verify our result as (110)2=(6)10, (111)2=(7)10, (101)2= (5)10 and (10010)2= (18)10. So when we will add 6 + 7 + 5 =18, which we are getting as our answer.

2) Binary Subtraction

The binary subtraction is performed like decimal subtraction, the rules for binary subtraction are:

ABDifferenceBorrow
0000
0111
1010
1100

Example 1: Subtract (10)2 from (1001)2

Solution:

Binary subtraction example 1

In column C2, 1 can't be subtracted from 0 so, we have to borrow 1 from column C3, but C3 also has a 0, so 1 must be borrowed from column C4, the 1 borrowed from column C4 becomes 10 in column C3, now keeping 1 in column C3 bringing the remaining 1 to column C2 which becomes 10 in column C2 thus 10 – 1= 1 in column C2.

In column C3, 1 – 0 = 1

In column C4, 1 after providing borrow 1 is reduced to 0.

Therefore, (1001)2 – (10)2 = (111)2


Example 2: Subtract (111.111)2 from (1010.01)2

Solution:

Binary subtraction example 2

In Column C0, 1 can't be subtracted from 0, so we have to borrow 1 from column C1, which becomes 10 in column C0, thus 10 – 1 = 1,

In column C1, after providing borrow 1 to C0, C1 is reduced to 0. Now 1 can't be subtracted from so borrow 1 from C2, but it is also 0, so borrow 1 from C3 which is also 0, so borrow 1 from C4, reducing column C4 to 0. Now, this 1 borrowed from column C4 becomes 10 in column C3, keep 1 in the column C3 and bring other 1 to column C2, which makes column C2 as 10 now again bring 1 from C2 to C1, which reduces C2 to 1 and makes C1 as 10.

Thus, In Column C1, 10 – 1 = 1

In Column C2, 1 – 1 = 0

In Column C3, 1 – 1 = 0

In Column C4, we now have 1 to be subtracted from 0 which is not possible so we will borrow 1 from Column C5, but Column C5 has a 0 so borrow 1 from C6 making C6 to be 0 and bring it to C5 which makes it 10 in C5, keep 1 in C5 and bring the other 1 to C4 which makes C4 as 10 thus

In column C4, 10 – 1 = 1

In column C5, 1 – 1 = 0

In column C6, 0 – 0 = 0

Hence, the result is (1010.01)2 – (111.111)2 = (0010.011)2






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