Exams
Test Series
Previous Year Papers
Syllabus
Books
Cut Off
Latest Updates
Eligibility
Binary Subtraction Definition, Rules, Steps & Solved Examples
IMPORTANT LINKS
Binary numbers are special numbers that use only two digits: 0 and 1. You won’t find digits like 2, 3, 4, or higher in binary. These numbers are used in computers and digital systems. Just like regular numbers, binary numbers can be added, subtracted, multiplied, and divided. Binary subtraction means taking one binary number away from another. The method is similar to normal subtraction, but since only 0 and 1 are used, there are special rules to follow, especially when borrowing. Binary subtraction is an important concept in computer and digital electronics.
In this mathematics article, we will learn what binary subtraction is, what the rules of binary subtraction are, steps to do binary subtraction, binary subtraction using 2’s and 1’s complement, and solve problems based on binary subtraction.
Maths Notes Free PDFs
Topic | PDF Link |
---|---|
Class 12 Maths Important Topics Free Notes PDF | Download PDF |
Class 10, 11 Mathematics Study Notes | Download PDF |
Most Asked Maths Questions in Exams | Download PDF |
Increasing and Decreasing Function in Maths | Download PDF |
Binary Subtraction
Binary subtraction is one of the basic arithmetic operations in the binary number system, which uses only the digits 0 and 1. It works much like decimal (base-10) subtraction but with simpler rules because there are fewer digits. For example, in base 10 we write 1 + 1 + 1 = 3, but in binary we write 1 + 1 + 1 = 11. When subtracting binary numbers, you must handle borrowing carefully, as it occurs more often than in decimal subtraction. If you subtract 1 from 0 in binary, you borrow 1 from the next higher bit and the result becomes 1. Borrowing always comes from a higher-order digit. When subtracting long binary numbers, work column by column and track each borrow to get the correct answer. Binary subtraction is widely used in digital electronics and computing.
Binary Subtraction Rules
Binary subtraction becomes much easier than decimal subtraction when the following rules are understood:
, ( with a borrow of 1)
Binary Subtraction Rules Table
In binary subtraction, we only deal with two digits: 0 and 1. Here are the basic rules you need to follow:
Binary Digits |
Result |
Explanation |
0 – 0 |
0 |
Subtracting 0 from 0 gives 0. |
1 – 0 |
1 |
Subtracting 0 from 1 gives 1. |
0 – 1 |
1 (with borrow) |
You can't subtract 1 from 0 directly. So, you borrow 1 from the next left digit, making the result 1. |
1 – 1 |
0 |
Subtracting 1 from 1 gives 0. |
- 3 Live Test
- 163 Class XI Chapter Tests
- 157 Class XII Chapter Tests
Binary Subtraction Examples
Example 1: Perform binary subtraction:
Solution: Using the rules of binary subtraction we have
Therefore, The subtraction of
Example 2: Perform binary subtraction:
Solution: Using the rules of binary subtraction we have
Therefore, the subtraction of
Steps To Do Binary Subtraction
Let us learn stepwise procedures that how to do binary subtraction with borrowing and without borrowing using the solved examples.
To learn about binary to decimal conversion, please click here.
Binary Subtraction With Borrowing:
Example: Subtract
Here the decimal equivalent of
Step 1: Arrange the numbers as shown below.
Step 2: Use the rules of binary subtraction to subtract
Here let us subtract the numbers starting from the right and move to the next higher order digit. The first step is to subtract
Step 3: Thus the difference of
To verify this, check the decimal equivalent of
To learn about addition and subtraction of decimals, please click here.
Binary Subtraction Without Borrowing:
Example: Subtract
Here the decimal equivalent of
Step 1: Arrange the numbers as shown below.
Step 2: Use the rules of binary subtraction to subtract
Step 3: Thus the difference of
To learn about the multiplication of binary numbers, please click here.
Binary Subtraction Using 1’s Complement
Binary subtraction using 1’s complement is a method to subtract two binary numbers. This method allows subtraction of two binary numbers by addition. The 1’s complement of a binary number can be obtained by replacing all
To perform binary subtraction using 1’s complement, please follow the steps given below:
Step 1: Determine the 1’s complement of the subtrahend (which means the second number of subtraction).
Step 2: Add it with the minuend or the first number.
Step 3: If there is a carryover left, then add it with the result obtained from step 2.
Step 4: If there are no carryovers, then the result obtained in step 2 is the difference of the two numbers using 1’s complement binary subtraction.
For example, Subtract
Here the decimal equivalent of
Step 1: 1’s complement of
Step 2: Add this with
Step 3: Arrange the numbers as given below and add them.
Step 4: So, the leftmost digit
Therefore, the answer is
To learn about binary addition using 1’s complement and 2’s complement, please click here.
Procedure for Binary Subtraction using 1’s Complement
- Identify the minuend and the subtrahend
- Minuend is the number from which you subtract.
- Subtrahend is the number you are subtracting.
- Find the 1’s complement of the subtrahend
- Flip all the bits: change all 0s to 1s and all 1s to 0s.
- Add the 1’s complement of the subtrahend to the minuend
- Check for a carry bit:
- If there is a carry, add that carry (1) to the least significant bit (rightmost digit) of the result.
- If there is no carry, take the 1’s complement of the result and add a negative sign to the answer.
- Minuend is the number from which you subtract.
- Subtrahend is the number you are subtracting.
- Flip all the bits: change all 0s to 1s and all 1s to 0s.
- If there is a carry, add that carry (1) to the least significant bit (rightmost digit) of the result.
- If there is no carry, take the 1’s complement of the result and add a negative sign to the answer.
Binary subtraction Using 2’s Complement
Binary subtraction using 2’s complement is a method to subtract two binary numbers. For finding 2’s complement of the binary number, we will first find the 1’s complement of the binary number and then add
To perform binary subtraction using 2’s complement, please follow the steps given below:
Step 1: In the first step, find the 2’s complement of the subtrahend.
Step 2: Add the complement number with the minuend.
Step 3: If we get the carry by adding both the numbers, then we discard this carry and the result is positive, else take 2’s complement of the result which will be negative.
For example, Subtract
Here the decimal equivalent of
Step 1: 2’s complement of
Step 2: Add this with
Step 3: We get the carry bit 1. So we discard this carry bit and remaining is the final result and a positive number. Therefore the answer is
Since the difference of
To learn about different types of numbers with definitions and examples, please click here.
Procedure for Binary Subtraction using 2’s Complement
- Identify the minuend and subtrahend
- Find the 2’s complement of the subtrahend:
- First, take the 1’s complement (flip all bits)
- Then add 1 to the result
- Add this 2’s complement to the minuend
- Check for a carry bit:
- If there is a carry, ignore it. The result is positive and correct.
- If there is no carry, the result is in 2’s complement form, so take the 2’s complement of the result and add a negative sign.
- First, take the 1’s complement (flip all bits)
- Then add 1 to the result
- If there is a carry, ignore it. The result is positive and correct.
- If there is no carry, the result is in 2’s complement form, so take the 2’s complement of the result and add a negative sign.
Binary Subtraction Solved Example
- Subtract the following:
(i).
(ii).
Solution: Using the rules for subtraction of binary numbers, solve the following.
(i). Subtract
Thus the answer is
(ii). Subtract
Thus the answer is
- Subtract
using 1’s complement.
Solution: Here the decimal equivalent of
Now 1’s complement of subtrahend,
Add this with minuend,
So, the leftmost digit
Therefore, the answer is
3. Subtract 1001₂ – 0110₂ using 2’s Complement Method
Step 1: Identify the Minuend and Subtrahend
- Minuend = 1001₂ (this is the number we subtract from)
- Subtrahend = 0110₂ (this is the number we subtract)
Step 2: Find the 2’s Complement of the Subtrahend
- Write the subtrahend: 0110₂
- Find the 1’s complement (flip the bits): 1001₂
- Add 1 to the 1’s complement:
1001
- 1
———
1010 → This is the 2’s complement of 0110₂
Step 3: Add the Minuend and the 2’s Complement
1001 (Minuend)
+1010 (2’s complement of Subtrahend)
------
10011
Step 4: Ignore the carry (the leftmost 1)
The result is: 0011₂
1001₂ – 0110₂ = 0011₂
We hope that the above article is helpful for your understanding and exam preparations. Stay tuned to the Testbook App for more updates on related topics from Mathematics, and various such subjects. Also, reach out to the test series available to examine your knowledge regarding several exams.
FAQs For Subtraction of Binary Numbers
How to subtract binary numbers?
Binary subtraction can be performed by the normal borrow method of arithmetic subtraction or by finding the 1's complement of the subtrahend and adding it with the minuend and add carryovers if any with the sum.
What are the rules of binary subtraction?
Rules of binary subtraction are as follows:
What is the difference between binary addition and binary subtraction?
In case of binary addition, when
What signs do the binary digits 0 and 1 represent?
represents a positive sign
What is a binary number system?
A binary number system represents a number in terms of only two digits, i.e.,
Can you subtract a larger binary number from a smaller one?
Yes, but the result will be negative, and you may need to use two's complement method to represent the result.
What is the two's complement method?
It's a way to perform binary subtraction by: Taking the two’s complement of the subtrahend. Adding it to the minuend. Ignoring overflow (if any) to get the final answer.