Binary Subtraction

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.
 

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:
 

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 start from right to left, that is, 1 is changed to 0, then 1 is changed to 0, then 0 is changed to 1, and 0 is changed to 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
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.

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

Binary Subtraction with Borrowing

The binary subtraction is similar to the decimal subtraction. If necessary, borrowing is done, as we subtract bit by bit. Here is the step-by-step guide to binary subtraction with borrowing. 

Step 1: To subtract the binary numbers, we first align the numbers in order 

Step 2: Using the binary principle, subtract the number. The subtraction starts from right to left, and if needed, do the borrowing. 

For example, subtract (1010)2 from (11100)2
Arranging the numbers in proper order, that is 

As the subtrahend has 4 bits, we write it as (01010)2 to align, so it can be written as 
 

Subtracting from the rightmost bit, 
First digit from right: 0 - 0 = 0
The next digit is 0 - 1, so we need to borrow from the left. That is, the nearest 1 (the third bit) becomes 0, so 10 - 1 = 1
Next digit: 1 - 1 = 0
Last digit: 1 - 0 = 1 

So, (11100)2 - (01010)2 = (10010)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.
 

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 subtraction is a basic operation in the binary number system. To master binary subtraction, follow these tips and tricks. 

Memorize the basic rules: 1 - 0 = 1
1 - 1 = 0
0 - 0 = 0
0 - 1 = 1

Always align binary numbers and add zeros if required to write the number with the same number of bits.

Make sure you always start the subtraction from right to left.
 

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. 

• In engineering, binary subtraction is a fundamental part, as it is used to analyze 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