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Last updated on October 13, 2025
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 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.
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. |
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).
Here are some tips to keep in mind when performing 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:
Binary division has many real life applications and some of them are mentioned below.
Evaluate 1010₂ ÷ 10₂ using the long-division method.
Quotient = \(101_2\)
Remainder = \(0_2\)
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 = \(0_2\)
Evaluate 1110₂ ÷ 10₂ using the long-division method.
Quotient = \(111_2\)
Remainder = \(0_2\)
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 = \(0_2\)
Evaluate 1100₂ ÷ 11₂ using the long-division method.
Quotient = \(100_2\)
Remainder = \(0_2\)
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 = \(0_2\)
Evaluate 10100₂ ÷ 100₂ using the long-division method.
Quotient = \(101_2\)
Remainder = \(0_2\)
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 = \(0_2\)
Evaluate 1001₂ ÷ 1₂.
Quotient = \(1001_2\)
Remainder = \(0_2\)
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 = \(0_2\)
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.
: She loves to read number jokes and games.