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:
0 – 0 = 0
0 – 1 = 1
1 – 0 = 1 (Borrow 1 from the next high-order digit)
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:
Finding the 1’s complement of the subtrahend
Adding it with the minuend or the first number
If there is a carry, then we add it to the Step 2 result
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:
Finding the 2’s complement of the smaller number
Adding the result to the larger number
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
