Binary Long Division Calculator
Divide two unsigned binary numbers using the traditional long division method. See quotient-bit decisions, partial dividends, divisor comparisons, subtraction steps, partial remainders and the final binary quotient and remainder.
| Step | Bit Brought Down | Partial Dividend | Compare With Divisor | Quotient Bit | Subtraction | New Remainder |
|---|---|---|---|---|---|---|
| Enter binary values and calculate to view the long division steps. | ||||||
What Is Binary Long Division?
Binary long division is the base-2 version of the familiar long division method used with decimal numbers. Instead of decimal digits 0 through 9, every decision is made using only binary digits 0 and 1.
Starting from the most significant bit of the dividend, each dividend bit is brought into the current partial remainder. The partial value is then compared with the divisor.
If the partial dividend is at least as large as the divisor, the current quotient bit is 1 and the divisor is subtracted. Otherwise, the quotient bit is 0 and no subtraction is performed.
How Binary Long Division Works
1. Start From the Left
Read the dividend from its most significant binary digit toward its least significant digit.
2. Bring Down a Bit
Append the next dividend bit to the current partial remainder.
3. Compare
Compare the resulting partial dividend with the binary divisor.
4. Write Quotient Bit
Write 1 when the divisor fits into the partial dividend; otherwise write 0.
5. Subtract
When the quotient bit is 1, subtract the divisor from the partial dividend.
6. Continue
Repeat the process until every dividend bit has been brought down.
Binary Long Division Algorithm
Quotient = empty
Read each dividend bit from left to right.
Bring down the next bit:
Partial = Remainder × 2 + Current Bit
If Partial is at least Divisor:
Quotient Bit = 1
Remainder = Partial – Divisor
Otherwise:
Quotient Bit = 0
Remainder = Partial
Repeat until all dividend bits are processed.
Binary Long Division Decision Table
| Partial Dividend | Comparison | Quotient Bit | Action |
|---|---|---|---|
| Less than divisor | Partial < Divisor | 0 | Do not subtract |
| Equal to divisor | Partial = Divisor | 1 | Subtract divisor |
| Greater than divisor | Partial > Divisor | 1 | Subtract divisor |
Example: 1101 ÷ 0011
The binary number 1101 represents decimal 13, while 0011 represents decimal 3.
0011 = 3
13 ÷ 3 = 4 remainder 1
Binary quotient:
0100
Binary remainder:
0001
Verification:
0011 × 0100 + 0001 = 1101
Decimal verification:
3 × 4 + 1 = 13
Example: 1111 ÷ 0011
1111 = 15
Divisor:
0011 = 3
Quotient:
0101 = 5
Remainder:
0000 = 0
Verification:
3 × 5 + 0 = 15
Binary Long Division With a Remainder
1011 = 11
0010 = 2
Quotient:
0101 = 5
Remainder:
0001 = 1
Therefore:
11 = 2 × 5 + 1
What If the Dividend Is Smaller Than the Divisor?
If the entire dividend is smaller than the divisor, every quotient-bit decision remains 0. The quotient is therefore zero and the original dividend becomes the remainder.
3 ÷ 5
Quotient = 0000
Remainder = 0011
Verification:
5 × 0 + 3 = 3
Binary Long Division Examples
| Dividend | Divisor | Quotient | Remainder | Decimal Check |
|---|---|---|---|---|
| 1000 | 0010 | 0100 | 0000 | 8 ÷ 2 = 4 |
| 1001 | 0010 | 0100 | 0001 | 9 = 2 × 4 + 1 |
| 1010 | 0011 | 0011 | 0001 | 10 = 3 × 3 + 1 |
| 1011 | 0010 | 0101 | 0001 | 11 = 2 × 5 + 1 |
| 1101 | 0011 | 0100 | 0001 | 13 = 3 × 4 + 1 |
| 1111 | 0011 | 0101 | 0000 | 15 = 3 × 5 |
Binary Long Division vs Decimal Long Division
| Feature | Binary Long Division | Decimal Long Division |
|---|---|---|
| Number base | Base 2 | Base 10 |
| Available digits | 0 and 1 | 0 through 9 |
| Quotient digit decision | Usually 0 or 1 | 0 through 9 |
| Core operations | Compare, subtract, bring down | Compare, multiply, subtract, bring down |
| Final outputs | Binary quotient and remainder | Decimal quotient and remainder |
Understanding the Binary Remainder
The value left after all dividend bits have been processed is the remainder. For valid unsigned integer division, the remainder must always be smaller than the divisor.
and
0 ≤ Remainder < Divisor
Why Learn Binary Long Division?
Binary Arithmetic
It helps explain division directly in base 2 instead of converting everything to decimal first.
Computer Science
Binary division is an important foundation for understanding arithmetic operations performed on binary data.
Digital Logic
The repeated compare-and-subtract process helps build intuition for more advanced hardware division algorithms.
Manual Verification
Step-by-step long division provides a transparent way to verify binary quotient and remainder calculations.
Important Binary Long Division Notes
Only binary digits 0 and 1 are accepted.
Leading zeros are allowed and do not change the numerical value.
The divisor cannot be zero.
This calculator returns an integer quotient and remainder rather than continuing into fractional binary digits.
The result is checked using:
Dividend = Divisor × Quotient + Remainder
For a valid result:
0 ≤ Remainder < Divisor