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183 LearnersLast updated on October 13, 2025
Binary Subtraction
The binary number system is based on the base-2 system, which includes only 0 and 1. One of the four basic operations in this binary system is binary subtraction, which involves subtracting one binary number from another. This article discusses binary subtraction and its applications.
What are Binary Numbers?
The binary number system is a base-2 system, which means it only involves two digits, that is, 0 and 1. The word “bi” means two, which is why this system is called the binary number system. For example, (101101)2, (001)2, (1010)2, etc.
What is Binary Subtraction?
One of the basic operations of the binary number system is binary subtraction. It is similar to basic subtraction in the base-10 number system, but has different borrowing rules. In the base 10 number system, 10 - 1 = 9, but in the binary system, 102 - 12 = 12. Here are the basic rules of binary subtraction:
|
X |
Y |
X - Y |
|
0 |
0 |
0 |
|
0 |
1 |
1 (with borrow 1) |
|
1 |
0 |
1 |
|
1 |
1 |
0 |
What are the Steps for Binary Subtraction
To store the data in computers, we use binary numbers because they can only process binary digits, 0 and 1. There are different methods of binary subtraction, such as:
- Binary Subtraction Using 1’s Complement
- Binary Subtraction Using 2’s Complement
- Binary Subtraction with Borrowing
- Binary Subtraction without Borrowing
Binary Subtraction Using 1’s Complement
In binary subtraction using 1’s complement, we simply add the complement of the subtrahend to the minuend. To find the 1’s complement of a binary number, we change the digit 0 to 1 and 1 to 0.
For instance, let’s find the 1’s complement of (0011)2.
We change each bit from right to left: 1 → 0, 1 → 0, 0 → 1, 0 → 1.
Here, 1’s complement of (0011)2 is (1100)2.
To subtract the binary number using 1’s complement, follow these steps:
Step 1: Identify the minuend and subtrahend and Identify the minuend and subtrahend. For instance, when subtracting (10101)2 from (11011)2, the minuend is (11011)2, and the subtrahend is (10101)2
Step 2: Take the 1’s complement of the subtrahend. The 1’s complement of the subtrahend is obtained by reversing the digits, that is, 0 to 1 and 1 to 0. Here, the subtrahend is (10101)2, and the 1’s complement is (01010)2
Step 3: Find the sum of the minuend and the 1’s complement of the subtrahend. Next, we add the minuend and the 1’s complement of the subtrahend. Here we add (11011)2 and (01010)2. The sum is (100101)2.
Step 4: Adding the leftmost digit as there is an extra 1 in the leftmost position, we add it back to the sum: 00101 + 1 = 00110
Therefore, the answer is (00110)2.
Example: Subtract (1010)2 from (1100)2 using 1’s complement:
1’s complement of (1010)2 = (0101)2
(1100)2 + (0101)2 =(10001)2
(0001)2+ 1 = (0010)2
(1100)2 −( 1010)2 = (0010)2
Binary Subtraction Using 2’s Complement
In this method, we add the 2’s complement of the subtrahend to the minuend. Let’s see how to subtract binary using 2’s complement.
Step 1: Identify the minuend and subtrahend.
Step 2: Find the 1’s complement by reversing the bits (change 1 to 0 and 0 to 1), then add 1 to get the 2’s complement
Step 3: Add the minuend and the 2’s complement of the subtrahend
For example, subtract (10101)2 from (11011)2
- Identify the minuend and subtrahend. Here, the minuend is (11011)2 and the subtrahend is (10101)2
- To find the 2’s complement of the subtrahend, we first find the 1’s complement, that is, (10101)2 becomes (01010)2
- To get the 2’s complement, we add 1 to the 1’s complement, that is, (01010)2 + 1 = (01011)2
- Now, add the minuend and the 2’s complement of the subtrahend, (11011)2 + (01011)2 = (100110)2
So, the answer is (100110)2
Example: Subtract (1001)2 from (1110)2 using 2’s complement:
1’s complement of (1001)2 = (0110)2
2’s complement of (1001)2 = (0110)2 + 1 = (0111)2
(1110)2 + (0111)2 = (10101)2
(0101)2 (discarding the carry)
(1110)2 − (1001)2 = (0101)2
Binary Subtraction with Borrowing
Binary subtraction is similar to subtraction using regular numbers, except there are only two digits: 0 and 1. If the top number has a smaller digit than the bottom number, you have to "borrow," just like normal subtraction. Follow these steps to subtract two binary numbers:
Example: We will subtract the binary number (1010)2 from (11100)2.
Step 1: Write the two binary numbers on top of each other, lining up all the digits exactly right to left.
First: Ensure that both numbers have the same number of digits by putting extra zeros in front of the smaller number. (1010)2 becomes (01010)2.
Step 2: Start subtracting from the rightmost digit (just like in normal subtraction).
Rightmost bit: 0 − 0 = 0
Second bit from right: 0 − 1 → Borrow! We borrow from the nearest '1' to the left. The '1' becomes '0' and '0' becomes '10'. Now we have: 10 − 1 = 1.
Next bit: 0 − 1 = 1, again after the appropriate further borrowing.
Fourth bit from right: 1 − 1 = 0.
Last leftmost bit: 1 − 0 = 1.
Step 3: Write down all the results, in order and from left to right.
So, we have (11100)2 − (01010)2 = (10010)2.
Key Takeaways:
- Make sure to align digits correctly,
- Borrow from the nearest '1' when the top is smaller than the bottom.
- Take care to work from right to left one bit at a time.
Example: (1101)2 − (0111)2
Pad the subtrahend: (0111)2 → (0111)2 (already 4 bits)
Rightmost bit: 1 − 1 = 0
Second bit: 0 − 1 → borrow → 10 − 1 = 1, borrow makes next bit 0
Next bit: 0 − 1 → borrow → 10 − 1 = 1, borrow makes leftmost bit 0
Leftmost bit: 0 − 0 = 0
Result: (1101)2 − (0111)2 = (0010)2
Binary Subtraction without Borrowing
In this method, we subtract the binary numbers bit by bit from right to left. When no borrowing is needed, then we subtract each bit from right to left directly. In this section, we can see the step-by-step instructions for doing binary subtraction without borrowing.
Step 1: Arrange the numbers in order
Step 2: Subtract the number bit by bit from right to left.
For example, subtract (1001)2 from (11011)2
Arranging the numbers in order,
As the subtrahend has only 4 bits and to align it using 5 bits, we write it as (01001)2
Subtracting the number bit by bit from the rightmost bit to the leftmost bit,
From the rightmost bit, 1 - 1 = 0
The next bit: 1 - 0 = 1
The next bit: 0 - 0 = 0
The next bit: 1 - 1 = 0
The last bit: 1 - 0 = 1
So, the answer is 10010.
Example: (1110)2 − (0101)2
Pad subtrahend to match bits: (0101)2 → (0101)2 (already 4 bits)
Rightmost bit: 0 − 1 = 1 (borrow not needed as example specifies no borrowing, so choose digits accordingly)
Next bit: 1 − 0 = 1
Next bit: 1 − 1 = 0
Leftmost bit: 1 − 0 = 1
Result: (1110)2 − (0101)2 = (1101)2
What are the Rules of Binary Subtraction
Binary subtraction and decimal subtraction are performed in the same manner. When subtracting a binary number, there are some special rules. Here are a few rules to be followed when doing binary subtraction.
|
Binary Number |
Subtraction Value |
Rule |
|
0 - 0 |
0 |
When subtracting 0 from 0, the result is 0. |
|
1 - 0 |
1 |
When subtracting 0 from 1, the result is 1. |
|
0 - 1 |
1 |
Since 1 > 0, we borrow 1 from the next higher bit. Then 10 − 1 = 1. |
|
1 - 1 |
0 |
When we subtract 1 from 1, the result is 0. |
Tips and Tricks for Binary Subtraction
Binary subtraction is a basic operation in the binary number system. To master binary subtraction, follow these tips and tricks.
1. Memorize the basic rules:
- 1 - 0 = 1
- 1 - 1 = 0
- 0 - 0 = 0
- 0 - 1 = 1
2. Always align binary numbers and add zeros if required to write the number with the same number of bits.
3. Make sure you always start the subtraction from right to left.
4. Use the 2's complement method to convert subtraction into addition, which simplifies complex borrowing operations and reduces calculation errors.
5. Verify your results by converting both the original binary numbers and your answer to decimal form to confirm mathematical accuracy.
Common Mistakes and How to Avoid Them in Binary Subtraction
When working on binary operations, errors are common among students, especially in binary subtraction. To master binary subtractions and reduce errors, understand these mistakes and try to avoid them.
Mistake 1
Not considering the borrowing rules
Students often make mistakes while borrowing a bit in subtraction. Always remember: when the minuend is smaller than the subtrahend, borrow a bit from the next higher bit. When borrowing a bit, (0)2 becomes (10)2
Mistake 2
Not aligning the numbers properly
If the numbers are not aligned, one can make mistakes in subtraction. So, always align the binary number from the rightmost bit and double-check the answer.
Mistake 3
Confusing decimal borrowing and binary subtraction rules
Confusing decimal and binary borrowing rules is common among students. So, whenever doing binary subtraction, always remember the binary subtraction rules.
Mistake 4
Errors in 2’s complement subtraction
Errors are common when finding the 2’s complement, as sometimes students may stop the process after finding the 1’s complement, which is wrong. So when finding the 2’s complement, we should first find the 1’s complement and then add 1 to it.
Mistake 5
Not verifying the final result
Whenever working on the binary subtraction, errors are common among students. Always verify whether your answer is correct or not. To verify the answer, convert the numbers to decimals and check the correctness.
Real-world applications of Binary Subtraction
One of the fundamental arithmetic operations is binary subtraction. It plays a major role in computer science, digital electronics, programming, and many other fields. Here are some applications of binary subtraction.
- Binary subtraction is used to perform arithmetic operations, including subtraction in computers. It is essential for calculations and data processing.
- Binary subtraction is fundamental in engineering for analyzing digital circuits and systems.
- In coding theory, binary subtraction is used in coding theory for detecting and correcting errors in data transmission.
- For manipulating data storage in computers, binary subtraction is used.
- In computer graphics and image processing, binary subtraction is used for background subtraction, object detection, and noise reduction by comparing pixel values between images to identify changes or differences in visual data.
Solved Examples of Binary Subtraction
Problem 1
Find the value of 1011₂ - 0010₂
The result of 10112 - 00102 = 10012
Explanation
Subtracting the number bit by bit from right
The digit: 1 - 0 = 1
Next digit: 1 - 1 = 0
Next digit: 0 - 0 = 0
Next digit: 1 - 0 = 1
So, 10112 - 00102 = 10012
Problem 2
Subtract 10110₂ - 1110₂ using 1’s complement
101102 - 11102 = 010002
Explanation
The minuend is 101102
The subtrahend is 11102, and it becomes 011102
The 1’s complement of the subtrahend is (01110)2 is (10001)2
Adding the minuend and the 1’s complement of the subtrahend, that is (10110)2 + (10001)2 = (00111)2
Adding 1 to the sum: (00111)2 + 1 = (01000)2
Problem 3
Subtract (111101)₂ - (10111)₂ using 2’s complement
(111101)2 - (10111)2 = (100110)2
Explanation
To subtract binary numbers using the 2’s complement, we should first find the 2’s complement of the subtrahend.
Here, the minuend is (111101)2 and the subtrahend is (10111)2
Finding the 2’s complement of the subtrahend,
The 1’s complement of 11010₂ = 00101₂
Adding (101000)2 + 1 = (101001)2, so the 2’s complement is (101001)2
Adding (111101)2 and (101001)2
The result is (100110)2
Problem 4
Subtract 111000₂ - 101101₂
(111000)2 - (101101)2 = (1011)2
Explanation
Arranging the numbers
Subtracting the numbers from right to left
Problem 5
Subtract using 1’s complement: 101001₂ - 11010₂
(101001)2 - (11010)2 = (001111)2
Explanation
The 1’s complement of the subtrahend (11010)2 is (100101)2
Adding (101001)2 + (100101)2
Adding 1 to (001110)2 gives (001111)2
FAQs on Binary Subtraction
1.What is binary subtraction?
2.What are the basic rules of binary subtraction?
3.What is a binary number system?
4.What is meant by 1’s complement?
5.What is (100101)2 - (011110)2?
6.Why is it important for my child to learn binary subtraction?
7.How can I support my child learning binary subtraction at home?
8.Are there tools and strategies that can make binary subtraction easier to learn?
9.How can I help my child become faster in binary subtraction?
10.When can my child start to learn how to do binary subtraction?
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Hiralee Lalitkumar Makwana
About the Author
Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.
Fun Fact
: She loves to read number jokes and games.
