Binary Subtraction

Binary subtraction is one of the four binary operations, where we perform the subtraction method for two binary numbers (comprising only two digits, 0 and 1). This operation is similar to the basic arithmetic subtraction performed on decimal numbers in Maths. Hence, when we subtract 1 from 0, we need to borrow 1 from the next higher order digit, to reduce the digit by 1 and the remainder left here is also 1. Read other binary operations here.

Note: For fractional binary numbers, the same rule applies for subtraction, and the decimal should be appropriately placed.

What is Binary Subtraction?

Can you subtract binary numbers? The answer is yes. Subtraction of binary numbers is an arithmetic operation similar to the subtraction of decimal numbers or base 10 numbers. For example, 1 + 1 + 1 = 3 in base 10 and 1 + 1 + 1 = 11 in binary number system. When you add and subtract binary numbers, you will need to be careful when borrowing as these will take place more often.

When you subtract several columns of binary digits, you must take into account the borrowing. When 1 is to be subtracted from 0, the result is 1 where 1 is borrowed from the next highest order bit or digit.

Binary Subtraction Table

The subtraction of binary numbers is given by:

Note: 

The addition of two binary numbers 1 and 1 is 10, where we consider 0 and carry forward 1 to the next high order. But in the case of subtraction of 1 and 1, the answer is equal to 0, and nothing is carried forward.

In the case of decimal subtraction, when 1 is subtracted from 0, then we borrow 1 from the next preceding number and make it 10, and after subtraction, it results in 9, i.e. 10 – 1 = 9. But for binary subtraction, it results in 1 only.

Binary Subtraction Rules

Rules and tricks: Binary subtraction is much easier than the decimal subtraction when you remember the following rules:

• 0 – 0 = 0
• 0 – 1 = 1 ( with a borrow of 1)
• 1 – 0 = 1
• 1 – 1 = 0

Now, look at the example of the binary subtraction: 101 from 1010

How to Subtract Binary Numbers?

Learn how to do binary subtraction using the example: 1010 – 101

Procedure to do Binary Subtraction:

1010

(-) 101

Step 1: First consider the 1’s column, and subtract the one’s column,( 0 – 1 ) and it gives the result 1 as per the condition of binary subtraction with a borrow of 1 from the 10’s place.

Step 2: After borrowing 1 from the 10’s column, the value 1 in the 10’s column is changed into the value 0

1 Borrow

1 0 1 0

(-) 1 0 1

——————

1

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

1 Borrow

1 0 1 0

(-) 1 0 1

——————

0 1

Step 4: Now subtract the values in 100’s place. Borrow 1 from the 1000’s place ( 0 – 1 ) = 1.

1 1 Borrow

1 0 1 0

(-) 1 0 1

——————

0 1 0 1

So, the resultant of the subtraction operation is 0101.

When you cross-check the binary subtraction resultant value with the decimal value, the resultant value should be the same.

The binary value 1010 is equal to the decimal value 10, and 101 is equivalent to 5

So, 10 – 5 = 5

Therefore, the decimal number 5 is equal to the binary number 0101.

Binary Subtraction Using 1’s Complement

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

Procedures for Binary Subtraction by 1’s Complement

• Write the 1’s complement of the subtrahend
• Then add the 1’s complement subtrahend with the minuend
• If the result has a carryover, then add that carry over in the least significant bit
• If there is no carryover, then take the 1’s complement of the resultant, and it is negative.

Binary Subtraction Questions Using 1’s Complement

Question 1:

(110101)₂ – (100101)₂

Solution:

(1 1 0 1 0 1)₂ = 53₁₀

(1 0 0 1 0 1)₂ = 37₁₀ – subtrahend

Now take the 1’s complement of the subtrahend and add with minuend.

1 carry

1 1 0 1 0 1

(+) 0 1 1 0 1 0

——————

0 0 1 1 1 1

1 carry

——————

0 1 0 0 0 0

Therefore, the solution is 010000

(010000)₂ = 16₁₀

Question 2:

(101011)₂ – (111001)₂

Solution:

Take 1’s complement of the subtrahend

1 1 1

1 0 1 0 1 1

(+) 0 0 0 1 1 0 (1’s complement)

——————

1 1 0 0 0 1

Now take the 1’s complement of the resultant since it does not carry 1

The resultant becomes 0 0 1 1 1 0

Now, add the negative sign to the resultant value

Therefore the solution is – (001110)₂.

Binary Subtraction Examples

Consider other examples of binary subtractions are as follows:

Example 1: 0011010 – 001100

Solution:

1 1 Borrow

0 0 1 1 0 1 0

(-) 0 0 1 1 0 0

——————

0 0 0 1 1 1 0

Decimal Equivalent :

0 0 1 1 0 1 0 = 26

0 0 1 1 0 0 = 12

Therefore, 26 – 12 = 14

The binary resultant 0 0 0 1 1 1 0 is equivalent to 14.

Example 2: 0100010 – 0001010

Solution:

1 1 Borrow

0 1 0 0 0 1 0 = 34₁₀

(-) 0 0 0 1 0 1 0 = 10₁₀

——————

0 0 1 1 0 0 0 = 24₁₀

Practice Problems

1. Perform the binary subtraction: 111001 – 101011
2. Subtract 1000 from 1111.
3. What is the value of (1110110)₂ − (1010111)₂?

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