Binary Subtraction

Binary subtraction is one of the 4 binary operations performed using the 2 binary numbers 0 and 1; the other 3 are addition, multiplication, and division. 

It is similar to decimal addition in mathematics. However, we follow some rules while doing subtraction involving binary numbers. 

Rules

The subtraction involving 2 binary numbers follows the 4 rules:

1. 0 – 0 = 0
2. 0 – 1 = 1
3. 1 – 0 = 1 (Borrow 1 from the next high-order digit)
4. 1 – 1 = 0

We can summarize it as:

A + B	Subtraction	Borrow
0 – 0	0	0
0 – 1	1	0
1 – 0	1	1
1 – 1	0	0

Binary Subtraction with Borrowing

Let us subtract 110₂ from 1010₂ using the rules.

Step 1: Aligning the numbers based on their place values

	1	0	1	0
–		1	1	0

Step 2: We will consider the first column and subtract the first column, which gives the result 1 (0 – 1 = 1) with a borrow of 1 from the 10’s place. The value 1 in the 10’s column is converted to 0 after being borrowed from the 10’s column as 1.

	1	0	1	0
–		1	1	0
		1

Step 3: Subtracting the value in the 10’s place, (0 – 0) = 0

	1	0	1	0
–		1	1	0
	0	1

Step 4: Now, for subtracting the values in 100’s place, we will borrow 1 from the 1000’s place (0 – 1) = 1

		1	0	1	0
	–		1	1	0
0	1	0	1

Thus, subtracting 110₂ from 1010₂, we get 0101₂

Binary Subtraction without Borrowing

Let us subtract 10₂ from 111₂ to understand binary subtraction involving borrow.

Step 1: Aligning the numbers based on their place values

		1	1	1
	–		1	0

Step 2: There is no borrow in this case, as there is no instance of subtraction of 1 from 0. Thus, the result is:

		1	1	1
	–		1	0
	0	1	0	1

Thus, subtracting 10₂ from 111₂, we get 0101₂

Subtraction Using 1’s Complement

It is similar to binary addition using 1’s complement, where:

• 0 represents the positive sign
• 1 represents the negative sign

The basic steps to be followed while performing subtraction using 1’s complement are:

1. Finding the 1’s complement of the subtrahend
2. Adding it with the minuend or the first number
3. If there is a carry, then we add it to the Step 2 result
4. If there is no carry, the result from step 2 is the difference of the two numbers using 1’s complement

Let us subtract 100101₂ from 110101₂

Step 1: Aligning the numbers based on their place values

	1	1	0	1	0	1
–	1	0	0	1	0	1

Step 2: Finding the 1’s complement of the subtrahend and adding it to the minuend

	1	1	0	1	0	1
–	1	0	0	1	0	1
	0	0	1	1	1	1

Step 3:

	1	1	0	1	0	1
–	1	0	0	1	0	1
	0	0	1	1	1	1
+						1
	0	1	0	0	0	0

     1 1 0 1 0 1

   +1 0 0 1 0 1

_________

 0 0 1 1 1 1

+              1

_________

0 1 0 0 0 0

_________

Thus, the required result is + 010000

Subtraction Using 2’s Complement

Similar to 1’s complement, we can find 2’s complement of a number by adding a 1 to the 1’s complement of a number.

The method involves adding one integer to the 2’s complement of another number. It involves the following steps:

1. Finding the 2’s complement of the smaller number
2. Adding the result to the larger number
3. Omitting the carry forward

Using this method, let us subtract (1010)₂ from (1111)₂.

Step 1: Finding the 2’s complement of the smaller number, we get

(1010)₂ → (0110)₂

Step 2: Adding (0110)₂ to (1111)₂, we get

	1	1	1	1
+	0	1	1	0
1	0	1	0	1

Step 3: Omitting this carry, we get

	1	1	1	1
+	0	1	1	0
	0	1	0	1

Thus, the required result is +0101