# 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:

## Binary Subtraction with Borrowing

Let us subtract 1102 from 10102 using the rules.

Step 1: Aligning the numbers based on their place values

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.

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

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

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

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

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

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

_________

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

Step 3: Omitting this carry, we get

Thus, the required result is +0101