Binary Arithmetic

Binary arithmetic is a very important part of digital system. We have to perform arithmetic operations on binary numbers in many digital computers and other digital system. The arithmetic of binary number means addition, subtraction, multiplication and division. But binary arithmetic has its own rules to be followed. And most important of them is operations start from Least Significant Digit (LSD). We will undergo each binary operation with an example to learn in detail.

Binary addition

For binary addition, we have four simple rules to remember:

• 0 + 0 = 0
• 0 + 1 = 1
• 1 + 0 = 1
• 1 + 1 = 0 (with a carry to the adjacent left bit)

The first three cases are same as in ordinary addition. But, when we add 1+1, we get the value 2 in ordinary addition. And in binary numbers, 2 is represented as 10₂. Thus, we write a 0, and the 1 is overflown to the next bit i.e. it is taken as carry.

Let us see an example to understand the situation clearly.

53 + 41 = 94

 Now we will see this in binary numbers:

(53)₁₀ = (110101)₂

(41)₁₀ = (101001)₂

Here the operation starts from LSD where 1+1 = 0 and 1 is overflown to the next bit as a carry. Then the operation continues following the rules of binary addition. Again at the MSD 1+1 = 0 and carry taken to the next bit but as there is no other bit, 1 is added in the result as MSD. 

Binary Subtraction

We have four main rules to remember for the binary subtraction:

• 0 – 0 = 0
• 0 – 1 = 1  (borrow/take 1 from the adjacent bit to the left)
• 1 – 0 = 1
• 1 – 1 = 0

We consider the second case as a borrow case and borrow 1 from the immediate left bit. Thus, this becomes 10 (binary 2). Thus, 2-1 gives 1.

Let us see an example of binary subtraction.               

Now we will perform this using binary subtraction:

(26)₁₀ = (11010)₂

(12)₁₀ = (01100)₂

As we had discussed, according to the second rule in case of  0-1 we have to borrow 1 from the left adjacent bit. But when we borrowed 1 from that bit we again got the case 0-1. Hence, it again

formed the case 0-1 and again borrowed a bit from the left adjacent bit. And we got our result. 

Binary Multiplication

The binary multiplication is the simplest one when compared to the other operations as it is similar to the ordinary multiplication we perform. Let us consider the four rules under this operation :

• 0 x 0 = 0
• 0 x 1 = 0
• 1 x 0 = 0
• 1 x 1 = 1

Lets see an example of binary multiplication:

Binary Division

Binary division consists of two operations subtraction and multiplication. If you have understood binary subtraction properly then it will be easy for you to learn binary division.

Let us see an example to get a clear picture in our minds: