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# Binary Subtraction

Updated on 19/07/20242,936 Views

Binary subtraction is a fundamental operation in digital computation, playing a crucial role in various aspects of computer science and electronics.

Understanding how binary subtraction works is essential for performing complex arithmetic calculations and data manipulation in modern computing systems. In this tutorial, we will delve into the principles of binary subtraction, exploring the methods used to find the difference between binary numbers.

By the end of this guide, you will gain a clear understanding of the techniques involved in this process and its significance in computer arithmetic.

## Overview

Binary subtraction is the process of finding the difference between two binary numbers. This essential arithmetic operation is performed in digital systems and logic circuits to perform various tasks, including data processing and numerical calculations. To subtract one binary number from another, a systematic approach is followed, involving bit-wise subtraction and possible borrow operations

In this tutorial, we will explore the step-by-step procedure for binary subtraction and highlight its practical applications in computer science. Whether you're a computer enthusiast or a student of electronics, this guide will equip you with the knowledge to perform binary subtraction efficiently and apply it in real-world scenarios.

## 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.

## 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.

## 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:

1. 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.

1. 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`
1. 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.
1. 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.

1. 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.
1. 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.

1. 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`
1. 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.
1. 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

`   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

`  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

`  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

`  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 fundamental arithmetic operation in digital computation and plays a crucial role in various fields such as computer science, digital electronics, and programming. By understanding the rules of binary subtraction and mastering the complement method, one can efficiently perform binary subtraction, even when the subtrahend has more digits than the minuend.

Binary subtraction enables us to find the difference between binary numbers and handle negative results using two's complement representation. It is an essential skill for working with binary data, performing complex computations, and designing digital systems.

## FAQs

1. How to subtract in binary?

To subtract in binary, follow these steps: Start from the rightmost bit and subtract corresponding bits. If the minuend bit is smaller, borrow from the next higher bit. Continue this process until all bits are subtracted.

1. What happens if I subtract a binary number from itself?

Subtracting a binary number from itself results in 0 in all bits, representing that the difference between the two numbers is zero.

1. What is the significance of binary subtraction in computer science?

Subtraction in binary is essential for computer arithmetic, data manipulation, and digital logic circuits. It enables computers to perform complex computations and process data accurately.

1. Can I use binary subtraction in programming?

Yes, binary subtraction is commonly used in programming when dealing with binary data, bitwise operations, and numerical calculations.

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