How do computers subtract? They don't understand "minus" in the way humans do; instead, they use a clever set of rules based on the binary system of 0s and 1s. This fundamental operation is known as binary subtraction.
Mastering the subtraction of binary numbers is essential for anyone interested in computer science or digital electronics. It's the core process that enables everything from simple calculations to complex data manipulation.
This tutorial will break down the different methods for binary subtraction, including the complement method, to help you understand how computers really work. First start by understanding what Binary Subtraction is.
Enhance your programming skills with our Software Engineering courses and take your learning journey to the next level!
What is Binary Subtraction?
Binary subtraction is a basic arithmetic operation performed on binary numbers in the context of digital computation. It involves subtracting one binary number from another to find its difference. Starting from the rightmost bit, corresponding bits are subtracted. If the minuend bit is smaller than the subtrahend bit, a borrow operation is applied from the next higher bit.
This process continues until all bits are subtracted, yielding the binary difference. Binary subtraction is essential in computer arithmetic, data manipulation, and digital logic circuits, enabling computers to perform complex computations and data processing tasks.
Take your programming skills to the next level and gain expertise for a thriving tech career. Discover top upGrad programs to master data structures, algorithms, and advanced software development.
How to Subtract Binary Numbers?
The subtraction of binary numbers involves performing bit-wise subtraction and potential borrow operations for precise results. To perform binary subtraction, follow these step-by-step instructions and binary subtraction rules:
Step 1: Write down the binary numbers in columns with the minuend (the number from which you are subtracting) on top and the subtrahend (the number being subtracted) below it. Align the columns based on their place values (ones, twos, fours, eights, etc.).
Step 2: If the subtrahend has fewer bits than the minuend, add leading zeros to the subtrahend to match the number of bits in the minuend.
Step 3: Start subtracting the bits from right to left (from the ones place).
Step 4: If the minuend bit is larger or equal to the subtrahend bit, simply subtract the subtrahend bit from the minuend bit in the same column and write the result below.
Step 5: If the minuend bit is smaller than the subtrahend bit, you need to borrow. Borrow 1 from the next higher bit of the minuend.
Step 6: If the bit at the next higher position is also 0, continue borrowing until a 1 is found.
Step 7: After borrowing, subtract the adjusted minuend bit from the subtrahend bit and write the result below.
Step 8: Continue this process for all the bits, moving from right to left, until you have subtracted all the bits.
Step 9: If there is a borrow from the leftmost bit, it is necessary to perform additional borrow operations.
Step 10: The result is the difference between the two binary numbers.
The rules of subtracting binary numbers are straightforward and are based on the principles of binary arithmetic.
Here are some binary subtraction examples with answers:
Example 1: To subtract 1101 from 10010, reverse the order to 10010 - 01101. Complement the subtrahend to 1001101, then add it to the minuend, getting 10010 + 1001101 = 10100111. Remove the leading 1, and the result is 100111, equivalent to -39 in decimal.
Example 2: For 10101 - 1101, we reverse to 1101 - 10101. Fill with a leading zero, making 01101 - 10101. The complement of 10101 is 01011, so add it to 01101, resulting in 110. Remove the leading 1, and the answer is -10 in decimal.
Also Read: Comprehensive Guide to Binary Code: Basics, Uses, and Practical Examples
Binary Subtraction Table
Here is a binary subtraction table that shows the possible binary subtraction operations for 4-bit binary numbers:
Minuend (A) | Subtrahend (B) | Borrow-In | Difference (A - B) | Borrow-Out |
0000 | 0000 | 0 | 0000 | 0 |
0001 | 0000 | 0 | 0001 | 0 |
0010 | 0000 | 0 | 0010 | 0 |
0011 | 0000 | 0 | 0011 | 0 |
0100 | 0000 | 0 | 0100 | 0 |
0101 | 0000 | 0 | 0101 | 0 |
0110 | 0000 | 0 | 0110 | 0 |
0111 | 0000 | 0 | 0111 | 0 |
1000 | 0000 | 1 | 0111 | 0 |
1001 | 0000 | 1 | 1000 | 0 |
1010 | 0000 | 1 | 1001 | 0 |
1011 | 0000 | 1 | 1001 | 0 |
1100 | 0000 | 1 | 1011 | 0 |
1101 | 0000 | 1 | 1110 | 0 |
1110 | 0000 | 1 | 1111 | 0 |
1111 | 0000 | 1 | 1110 | 1 |
This table illustrates the subtraction of two 4-bit binary numbers (B) from another 4-bit binary number (A). It shows the borrow-in and borrow-out conditions, which are relevant when performing binary subtraction.
Difference Between Binary Addition and Subtraction
Binary addition and binary subtraction are two fundamental arithmetic operations performed on binary numbers in digital computation.
Binary addition involves adding two binary numbers to find their sum. The addition process follows the same rules as decimal addition, where corresponding bits are added, carrying over any excess to the next higher bit when the result exceeds the binary base (2).
Example: 1011 (11 in decimal) + 0101 (5 in decimal) = 10000 (16 in decimal).
Binary subtraction involves subtracting one binary number from another to find its difference. The subtraction process is similar to decimal subtraction, but it may involve borrowing when the minuend bit is smaller than the subtrahend bit.
3.Example: 1011 (11 in decimal) - 0101 (5 in decimal) = 010 (2 in decimal).
Binary Multiplication Rules
Binary multiplication is analogous to decimal multiplication, but it involves only 0s and 1s since binary numbers are base-2 numbers. The rules for binary multiplication are as follows:
Multiplicand | Multiplier | Product |
0 | 0 | 0 |
0 | 1 | 0 |
1 | 0 | 0 |
1 | 1 | 1 |
How Do You Subtract Binary Numbers?
Subtracting binary numbers involves manipulating binary digits (0s and 1s) in a similar way to decimal subtraction. There are a few methods for performing binary subtraction, and two of them are using the 1's complement and 2’s complement method.
The 2's complement method is generally considered a better and more efficient method for binary subtraction compared to the 1's complement method. It simplifies the process by eliminating the need to deal with carries separately.
Subtraction with 2’s Complement Method
The 2's complement method is the standard technique used in modern computer systems to perform binary subtraction. It simplifies the process and avoids the need to handle carries explicitly, which can lead to errors and complexity in the 1's complement method.
Here's is the detailed procedure of using the 2's complement method:
- Take the 2's Complement of B: To find the 2's complement of a binary number, flip all the bits (change 0s to 1s and vice versa) and then add 1 to the least significant bit (rightmost bit).
Example: If B = 10101, its 2's complement is 01011.
- Perform Binary Addition: Add the 2's complement of B to A using binary addition. If you encounter a carry bit from the most significant bit (leftmost bit), simply ignore it. This is one of the key advantages of the 2's complement method over the 1's complement method.
Example:
A: 110110
B: 01011
+--------------
S: 101011
- Check for Overflow: If the leftmost carry bit after addition is 1, it indicates overflow, which means the subtraction result is negative and cannot be represented accurately using the given number of bits.
- Determine the Sign of the Result: If there's no overflow, the result is positive. If there is overflow, the result is negative.
Binary Subtraction Using 1’s Complement
Binary subtraction using 1's complement is a method of subtracting binary numbers by first finding the 1's complement of the number being subtracted and then adding it to the other number.
This method involves flipping the bits of the subtrahend (number being subtracted) to create its 1's complement, and then performing binary addition between the minuend (number from which subtraction is being performed) and the 1's complement of the subtrahend.
Procedures for Binary Subtraction With 1’s Complement
Here is the procedure for performing binary subtraction using the 1's complement method:
Let's say you want to subtract binary number B from binary number A.
- Determine the Larger Number: Compare the two binary numbers, A and B. If B is larger than A, swap A and B to ensure you're subtracting the smaller number from the larger one.
- Take the 1's Complement of B: To find the 1's complement of a binary number, flip all the bits, changing 0s to 1s and vice versa.
Example: If B = 10101, its 1's complement is 01010.
Add 1 to the 1's Complement of B: This step is called "2's complement," and it involves adding 1 to the 1's complement of B.
Example: Adding 1 to 01010 results in 01011.
- Perform Binary Addition: Add the modified B (1's complement + 1) to A using binary addition. If you encounter a carry bit from the most significant bit (leftmost bit), add it to the next bit.
Example:
A: 110110
B: 01011
+--------------
C: 001101 (carry)
S: 101011
- Check for Overflow: If the leftmost carry bit after addition is 1, it indicates overflow, which means the subtraction result is negative and cannot be represented accurately using the given number of bits.
- Determine the Sign of the Result: If there's no overflow, the result is positive. If there is overflow, the result is negative.
Example of Binary Subtraction
Here is an example of binary subtraction:
Subtract: 101110 (minuend) - 010101 (subtrahend)
1's Complement of the subtrahend: 101010
Binary addition:
101110 (minuend)
+ 101010 (1's complement of subtrahend)
----------------
101000
Check for overflow: No overflow occurred.
Determine the sign of the result: Since there's no overflow, the result is positive.
So, 101110 - 010101 = 101000.
More Binary Subtraction Questions Using 1’s Complement
Question 1
Perform the following binary subtraction using the 1's complement method: 110101 - 100011
Solution:
Find the 1's complement of the subtrahend (100011): 1's complement of 100011 = 011100
Perform binary addition:
110101 (minuend)
+ 011100 (1's complement of subtrahend)
----------------
010001
Check for overflow: No overflow occurred.
Determine the sign of the result: Since there's no overflow, the result is positive.
So, 110101 - 100011 = 010001.
Question 2
Perform the following binary subtraction using the 1's complement method: 101010 - 110101
Solution:
Find the 1's complement of the subtrahend (110101): 1's complement of 110101 = 001010
Perform binary addition:
101010 (minuend)
+ 001010 (1's complement of subtrahend)
----------------
110100
Check for overflow: No overflow occurred.
Determine the sign of the result: Since there's no overflow, the result is positive.
So, 101010 - 110101 = 110100.
Question 3
Perform the following binary subtraction using the 1's complement method: 111000 - 100011
Solution:
Find the 1's complement of the subtrahend (100011): 1's complement of 100011 = 011100
Perform binary addition:
111000 (minuend)
+ 011100 (1's complement of subtrahend)
----------------
010100
Check for overflow: No overflow occurred.
Determine the sign of the result: Since there's no overflow, the result is positive.
So, 111000 - 100011 = 010100.
Conclusion
Binary subtraction is a cornerstone of digital computation. By mastering the rules and the complement method, you have learned the exact process that computers use to handle subtraction. This skill is essential for anyone in computer science or digital electronics. Understanding the subtraction of binary numbers is not just an academic exercise; it's a window into how modern technology processes data at its most fundamental level. Keep practicing these techniques, and you'll have a solid foundation for more advanced topics.
FAQs
1. What are the four basic rules for bit-wise binary subtraction?
The four fundamental rules for the subtraction of binary numbers at a bit-by-bit level are very simple and form the basis of the direct subtraction method:
- 0 - 1 = 1 (with a borrow of 1 from the next most significant bit)
The concept of "borrowing" is the most crucial part of this process, similar to borrowing in decimal subtraction.
2. How does borrowing work in binary subtraction?
Borrowing in binary subtraction works similarly to decimal subtraction, but instead of borrowing a group of 10, you borrow a group of 2. When you need to subtract 1 from 0, you must borrow from the next column to the left. When you borrow from a 1, it becomes a 0, and the bit you are working on becomes "10" (which is 2 in decimal). You can then perform the subtraction 10 - 1 = 1. This process can cascade if the next bit is also a 0.
3. Why are complement methods used for binary subtraction?
While direct subtraction with borrowing is easy for humans to understand, it is complex for digital logic circuits to implement. Complement methods, specifically 2's complement, are used because they allow a computer's processor to perform subtraction using only addition. The processor can take the 2's complement of the subtrahend (the number being subtracted) and then add it to the minuend. This simplifies the hardware design of the Arithmetic Logic Unit (ALU) as it only needs circuits for addition.
4. What is the 1's complement of a binary number?
The 1's complement of a binary number is found by simply inverting all of its bits. This means you change every 0 to a 1 and every 1 to a 0. For example, the 1's complement of the binary number 10110 is 01001. This is the first step in the 1's complement method for the subtraction of binary numbers.
5. How do you perform binary subtraction using the 1's complement method?
To perform binary subtraction using this method, you first find the 1's complement of the subtrahend. Then, you add this complemented number to the minuend. If there is a carry-over bit (an "end-around carry") from the most significant bit, you add it back to the least significant bit of the result. If there is no carry-over, the result is negative, and you find its 1's complement to get the final answer.
6. What is the 2's complement of a binary number?
The 2's complement of a binary number is the standard way that computers represent negative integers. It is calculated in a two-step process: first, you find the 1's complement by inverting all the bits, and then you add 1 to the result. For example, for the binary number 0101 (5), the 1's complement is 1010, and adding 1 gives the 2's complement 1011 (-5).
7. How do you perform binary subtraction using the 2's complement method?
This is the method most commonly used by computers. To perform the subtraction of binary numbers using 2's complement, you first find the 2's complement of the subtrahend. Then, you add this result to the minuend. If there is a final carry-over bit from the most significant bit, you simply discard it. The remaining result is your final answer. If there is no carry-over, the result is negative and is already in its 2's complement form.
8. What happens if I subtract a larger binary number from a smaller one?
When you subtract a larger number from a smaller one, the result will be negative. When using the 2's complement method for this binary subtraction, the final result will naturally be a negative number represented in its 2's complement form. There will be no final carry-over bit to discard. To find the magnitude of the negative result, you would need to find the 2's complement of the answer and treat it as a positive number.
9. What happens if I subtract a binary number from itself?
When you perform a binary subtraction where a number is subtracted from itself, the result is always zero. In binary, this will be represented by a string of zeros (e.g., 101 - 101 = 000). This is a fundamental property of arithmetic and is a useful check to ensure your subtraction logic, whether direct or complement-based, is working correctly.
10. How do you subtract in binary?
To subtract in binary using the direct method, you follow a process similar to decimal subtraction. You start from the rightmost bit and subtract the corresponding bits. If the top bit (minuend) is smaller than the bottom bit (subtrahend), you must "borrow" from the next higher bit to the left. This process is continued column by column from right to left until all bits have been subtracted. For computers, binary subtraction is typically performed using the more efficient 2's complement addition method.
11. Can I use binary subtraction in programming?
Yes, while high-level programming languages handle subtraction for you with the - operator, understanding binary subtraction is crucial for low-level programming, bitwise operations, and embedded systems. Many programming languages provide bitwise operators that allow you to directly manipulate the binary representation of numbers, and knowledge of 2's complement is essential for understanding how negative numbers are stored and processed.
12. What is the significance of binary subtraction in computer science?
The subtraction of binary numbers is a fundamental operation in computer science and the foundation of all computer arithmetic. It is essential for everything from basic calculations in a spreadsheet to complex data manipulation in scientific computing. The design of digital logic circuits within a computer's processor (the ALU) is heavily based on the principles of binary arithmetic, particularly the efficiency of performing subtraction via 2's complement addition.
13. How do you check if your binary subtraction result is correct?
The easiest way to verify your result is to convert all the numbers—the minuend, the subtrahend, and the result—from binary to their decimal equivalents. You can then perform the subtraction in the familiar decimal system and check if your decimal result matches the decimal equivalent of your binary answer. This is a reliable way to check your work when you are first learning binary subtraction.
14. Can you subtract binary numbers with fractional parts?
Yes, you can. The process for subtracting binary numbers with a fractional part (separated by a binary point) is very similar to the process for integers. You align the binary points and then perform the subtraction column by column, just as you would with decimal numbers. The same rules of borrowing apply across the binary point. The complement methods can also be adapted for fixed-point binary numbers.
15. What is an "overflow" in the context of binary subtraction?
An overflow is an error condition that can occur when the result of an arithmetic operation is too large to be represented by the number of bits available. In 2's complement binary subtraction, an overflow can happen if you subtract a negative number from a positive number and the result is too large, or if you subtract a positive number from a negative number and the result is too small (a negative overflow).
16. Why is 2's complement preferred over 1's complement?
The 2's complement representation is preferred in modern computers for a few key reasons. First, it has only one representation for zero (0000...), whereas 1's complement has two (0000... and 1111...), which simplifies the hardware logic. Second, the arithmetic for 2's complement is more straightforward; for subtraction, you simply add the two numbers and discard the carry, whereas 1's complement requires an extra "end-around carry" step if there is a carry-out.
17. How does a computer's ALU (Arithmetic Logic Unit) handle subtraction?
An ALU is the part of a CPU that performs arithmetic and logic operations. To perform subtraction, the ALU is designed to use the 2's complement method because it simplifies the hardware. Instead of having a dedicated and complex "subtractor" circuit, the ALU has a circuit to calculate the 2's complement of a number and then reuses its "adder" circuit to perform the addition. This is a more efficient design.
18. What are some common mistakes to avoid when learning binary subtraction?
A common mistake in the direct method is forgetting how to handle a cascaded borrow, where you need to borrow across multiple columns of zeros. When using the complement methods, a frequent error is forgetting to add 1 when calculating the 2's complement, or forgetting the "end-around carry" step in the 1's complement method. Careful, step-by-step work is the key to avoiding these issues.
19. How can I learn more about binary arithmetic?
The best way to learn is through a combination of structured education and hands-on practice. A comprehensive program, like the software engineering courses offered by upGrad, can provide a strong foundation in digital logic and computer architecture. You should also regularly practice solving problems by hand to solidify your understanding of concepts like binary subtraction and the complement methods.
20. What is the main takeaway about the subtraction of binary numbers?
The main takeaway is that while the direct "borrowing" method is intuitive for humans, the real power and efficiency in computer systems come from the complement methods. Understanding how to perform binary subtraction using 2's complement is crucial because it reveals the clever way that computers simplify complex operations, turning every subtraction problem into an addition problem. It is a fundamental concept that underpins all digital computation.
