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Last updated on October 13, 2025

Binary Division

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Binary division uses only two symbols, 0 and 1, with base 2. The prefix ‘bi’ means two. Division in this number system is a fundamental operation. Binary division is used in computer programming and data management. In this topic, we will explore the binary division method and its symbols in detail.

Binary Division for US Students
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What is Binary Division?

Binary division uses the long division method with only the digits ‘0’ and ‘1’ to divide one value by another. In this method, the dividend is divided by the divisor, and it results in a binary quotient and a remainder. In computers, this acts as a foundational system to store and represent information.

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What are the Rules for Binary Division?

Binary division follows the same method as decimal division, but only uses 0s and 1s. This form of division follows certain rules that must be understood. It focuses on just two symbols, 0 and 1, and 2 is the base value of this technique. The four basic rules of binary division are:

 

Binary Division Rules 

 

Explanation 

 

0 ÷ 0 = Undefined 

 

If zero is divided by zero, the result is undefined.  

0 ÷ 1 = 0 

The result of dividing 0 by 1 is zero.  

 

1 ÷ 1 = 1 

 

If 1 is divided by 1, the result is 1. 

1 ÷ 0 = Undefined 

 Division by zero is undefined; no number divided by zero gives a valid result.

 

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How to do Binary Division?

Using the long division method, we can easily divide binary numbers and find the result. We can perform binary division by following these steps:

 

 

Step 1: Before performing calculations, identify the dividend and the divisor. If the divisor is larger than the dividend, put 0 as the quotient and bring down the second bit of the dividend.


If the divisor is smaller than the dividend, multiply the divisor by 1, and the product becomes the subtrahend. After that, to get the remainder, subtract the subtrahend (the number we subtract) from the minuend (current part of the dividend we are working with). 

 

 

Step 2: After bringing down the next bit from the dividend, repeat step 1. 

 

 

Step 3: Continue the same steps until the whole dividend has been processed, or the remainder becomes zero.  
 

 

Let us take an example to understand the binary division in detail. For example, \(11010_2 \div 101_2 \)

 

Here, the given binary numbers are \(011010_2 \) and \(0101_2\). The leading zeros of the given numbers do not change the value, so we can simplify the numbers to \(11010_2 \) and \(101_2\)

Dividend = \(11010_2 \)

Divisor = \(101_2\)

 

 

Step 1: Since the divisor is smaller than the dividend, we must multiply the divisor by 1. Hence, the product is
\(101_2 \; (\,101_2 \times 1 = 101_2\,) \)).
 

The product (1012) becomes the subtrahend. Subtract the subtrahend from the current part of the dividend

\((110_2): \; 110_2 - 101_2 = 001_2 \).


Here, the quotient starts with 1. 

 

 

Step 2: Bring down the next bit (1) from the dividend, which makes the current portion of the dividend \(0011_2 \). \(101_2 \) is greater than \(0011_2\), we can put 0 in the quotient and bring down the next bit (0), making it \(0110_2\).

 

 

Step 3: Multiply the divisor by 1, and \(101_2 \times 1 = 101_2 \).

 

 

Step 4: Subtract \(101_2\) from \(110_2\)

\(110_2 - 101_2 = 001_2 \)
 

The quotient is, \(101_2\) and the remainder is \(1_2\) in binary (1 in decimal).

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Tips and Tricks to Master Binary Division

Here are some tips to keep in mind when performing binary division:
 

  • Understand the four basic rules of binary division. The four rules of binary division are: 

    \(0 ÷ 0 =\) Undefined 

    \(0 ÷ 1 = 0 \)

    \(1 ÷ 1 = 1 \)

    \(1 ÷ 0 =\) Undefined 

     
  • If the divisor is greater than the dividend, write 0 in the quotient. If the divisor is smaller than the dividend, multiply the divisor by 1.

     
  • Bring down the next bit from the dividend after each subtraction. 

     
  • Remove the leading zeros from the dividend and divisor, as they will not change the value of the binary numbers.

     
  • Use the long division method for binary division, as it is a simple and effective way to find the answer. 
     
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Common Mistakes and How to Avoid Them in Binary Division

The binary division uses only two digits 0 and 1, and 2 as a base. Sometimes, this method can be tricky for confusing students. Here are some common mistakes and helpful solutions to avoid these errors:

Mistake 1

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Forgetting the Rules of Binary Division 
 

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Four rule of Binary Divisions are: 

\(0 ÷ 0 =\) Undefined 

\(0 ÷ 1 = 0\) 

\(1 ÷ 1 = 1\) 

\(1 ÷ 0 =\) Undefined 

If students divide a binary number by 0, the result will be undefined or meaningless.
 

Mistake 2

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Ignoring the Values of Dividend and Divisor

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In binary division, students must remember to check the values of the divisor and dividend. If the divisor is greater than the dividend, they must write 0 in the quotient. If the divisor is smaller than the dividend, multiply the divisor by 1 and the result will be the subtrahend. If students forget these steps, they will get incorrect answers in binary division.

Mistake 3

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Finding the Difference Between Binary Numbers Incorrectly 
 

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When performing the subtraction of binary numbers, keep in mind to correctly borrow values from the next higher bit. By mistake, students find the incorrect difference between binary numbers and end up with wrong subtraction values.

For example, subtract \(100_2 - 011_2\)

The correct answer is: 


\(100_2 - 011_2 = 001_2 \).
 

Mistake 4

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Forgetting to Bring Down the Next Bit 
 

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Remember to bring down the next bit from the dividend at each step of the division. Sometimes, students forget to bring down the next bit and stop the subtraction too early. By bringing down the next bit, they can prevent errors and get the correct answer.

Mistake 5

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Not Removing Leading Zeros

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While performing binary division, students can remove the unnecessary zeros from the dividend and quotient. Since leading zeros do not change the value, they may be disregarded. It will help them to perform calculations easily and give accurate answers. For example, if the given binary numbers are \(011010_2 \) and \(0101_2\), removing the leading zeros simplifies them to \(11010_2\) and \(101_2\).

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Real-World Applications of Binary Division

Binary division has many real life applications and some of them are mentioned below. 

 

  • Computer arithmetic: Processors carry out division at the binary level when executing instructions like 'divide'. In CPUs and Arithmetic Logic Units(ALU), binary division circuits implement quotient and remainder operations.

     
  • Cryptography and encryption: Many cryptographic algorithms rely on arithmetic binary. For example, when implementing RSA or elliptic curve algorithms, binary division and modular inverses in binary form are essential for key generation, encryption and decryption.

     
  • Data storage and error-correcting codes: In RAID systems, ECC (Error Correcting Codes), and Storage Controllers, division in binary is used to compute parity bits, checksums and to correct data. For instance, dividing binary polynomials helps generate error-checking codes that detect or fix bit errors in hard discs, memory and network transmissions.

     
  •    IP address calculation: IP addresses and subnet masks are managed in binary formats. When dividing networks into subnets, routers compute divisions of binary address blocks to allocate addresses and define ranges. Binary division helps in dividing address space evenly and efficiently. 

      
  • Graphics, scaling and texture mapping: When computers scale images or textures, they often perform divisions in binary to adjust resolution or brightness. For instance, dividing pixel values by scaling factors. In rendering, shaders may divide binary color values so that things like blending or interpolation fast and precise.
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Solved Examples of Binary Division

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Problem 1

Evaluate 1010₂ ÷ 10₂ using the long-division method.

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Quotient = \(101_2\)


Remainder = \(0_2\)

 

Explanation

Here, we are dividing \(1010_2\) by \(10_2\).


Dividend = \(1010_2\)


Divisor = \(10_2\)


The divisor (\(10_2\)) is two digits, so start by looking at the first two digits of the dividend. 

 


Step 1: To begin with, we can start with the first two digits of the dividend.


The divisor is \(10_2\), and the first two digits of the dividend are also \(10_2\).


We can divide:


\(10_2  ÷ 10_2 = 1_2\)

 

 

Step 2: Now write the first digit of the quotient as 1.


Then subtract \(10_2 - 10_2 = 0_2\).

 

 

Step 3: Next, we can bring down the next digit from the dividend.


So, we have \(01_2\), which is smaller than \(10_2\).


 

Step 4: So, the next digit of the quotient is 0, and then bring down the next digit from the dividend


Now we have \(10_2\), which is equal to the divisor.


Hence, the next digit of the quotient is 1.


Subtract \(10_2 - 10_2 = 0_2\)


So, the quotient = \(101_2\)


Remainder = \(0_2\)

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Problem 2

Evaluate 1110₂ ÷ 10₂ using the long-division method.

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Quotient = \(111_2\)


Remainder = \(0_2\)

Explanation

Dividend = \(1110_2\)


Divisor = \(10_2\)

 


Step 1: We can start with the first two digits of the dividend.


\(11_2\) is the first two digits, which is greater than the divisor \(10_2\).


We can divide:


\(11_2 ÷ 10_2 = 1_2\)

 

 

Step 2: The first digit of the quotient is 1. 


Subtract \(11_2 - 10_2 = 01_2\)

 

 

Step 3: Bring down the next digit of the dividend. 


Now we have \(011_2\).


The first two digits \(11_2\) are greater than the divisor \(10_2\).


So, the second digit of the quotient is 1.


Subtract \(11_2 - 10_2 = 01_2\)

 

 

Step 4: Bring down the last digit of the dividend.


Now we have \(10_2\), which is equal to the divisor.


So, the quotient is 1.


Subtract \(10_2 - 10_2 = 0_2\)


Thus, \(1110_2 ÷ 10_2\)


Quotient = \(111_2\)


Remainder = \(0_2\)

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Problem 3

Evaluate 1100₂ ÷ 11₂ using the long-division method.

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Quotient = \(100_2\)


Remainder = \(0_2\)

Explanation

Dividend = \(1100_2\)


Divisor = \(11_2\)

 


Step 1: Start with the first two digits of the dividend. 


The first two digits are \(11_2\), which is equal to the divisor \(11_2 \).

 

 

Step 2: \(11_2 ÷ 11_2\)


Hence, the first digit of the quotient is 1.


Subtract \(11_2 - 11_2 = 0_2\)

 

 

Step 3: Bring down the next digit of the dividend.


Now we have \(0_2\), which is smaller than \(11_2\).


So, the quotient is 0.

 

 

Step 4: Bring down the next digit of the dividend.


Now we have \(00_2\), which is smaller than \(11_2\).


Hence, the quotient is 0.

 

Thus, \(1100_2 ÷ 11_2\)


Quotient = \(100_2\)


Remainder = \(0_2\)
 

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Problem 4

Evaluate 10100₂ ÷ 100₂ using the long-division method.

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Quotient = \(101_2\)


Remainder = \(0_2\)

Explanation

Dividend = \(10100_2\)


Divisor = \(100_2\)

 


Step 1: Start with the first three digits of the dividend. 


\(101_2\) is the first three digits, which is greater than the divisor \(100_2\).

 

 

Step 2: \(101_2 ÷ 100_2\) 


1 = Quotient and 


1 = Remainder.


Thus, the first digit of the quotient is 1.


 Subtract \(101_2 - 100_2 = 001_2\)

 

 

Step 3: We can bring down the next digit of the dividend.


Now we have \(010_2\), which is smaller than \(100_2\).


Hence, the quotient is 0.

 

 

Step 4: Now we can bring down the last digit of the dividend.


It is \(100_2\), which is equal to the divisor.


Hence, the quotient is 1.


Next, we can subtract \(100_2 - 100_2 = 0_2\)


Thus, \(10100_2 ÷ 100_2\) 


Quotient = \(101_2\)


Remainder = \(0_2\)
 

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Problem 5

Evaluate 1001₂ ÷ 1₂.

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Quotient = \(1001_2\)


Remainder = \(0_2\)

Explanation

Here, we divide the binary number \(1001_2\) by \(1_2\). The quotient is the same as the dividend.


Thus, \(1001_2 ÷ 1_2\)


Quotient = \(1001_2\)


Remainder = \(0_2\)

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FAQs of Binary Division

1.Differentiate decimal numbers and binary numbers.

In the decimal system, 10 digits are used to represent numbers and the base is 10. The 10 digits of a decimal number system range from 0 to 9 (0, 1, 2, 3, 4, 5, 6, 7, 8, 9). However, in the binary system, there are only 2 digits: 0 and 1. The base of this system is 2, and it is used to represent binary numbers.
 

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2.List the rules of binary division.

The four basic rules of binary division are: 
0 ÷ 0 = Undefined 
0 ÷ 1 = 0 
1 ÷ 1 = 1 
1 ÷ 0 = Undefined
 

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3.What is the significance of leading zeros in binary division?

In binary division, leading zeros do not have any significance over the result of the operation. They do not change the value of binary numbers, so we can remove them from the numbers to prevent confusion and errors. For example, if the given binary number is 011012 can be simplified as 11012.

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4.When will the quotient become 0?

If the divisor is larger than the given dividend, the value of the quotient becomes 0. Then the remainder becomes the value of the dividend. For example, 102 ÷ 1002 
Quotient = 02
Remainder = 102

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5.What will be the result of dividing a binary number by 0?

If we divide a binary number by 0, the result will be undefined. In mathematics, whatever number is divided by zero gives undefined answers. That is represented as: 
1 ÷ 0 = Undefined
 

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6.Why should my child learn binary division when we already use decimal division?

Binary division is essential for understanding how computers perform arithmetic operations. While we divide in base 10 in daily life, computers use base 2 (binary). Learning this helps children connect math with computer logic, coding, and digital systems.

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7.My child finds binary division difficult. How can I help at home?

Start by revising binary addition and subtraction, as division builds on them. Encourage using step-by-step column methods and visual aids like tables. Online binary calculators or interactive tools can help your child check answers and gain confidence.

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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.

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