Division in Binary Number System: Rules, Examples, and Methods
Learn how binary division works, the rules of division in the binary number system, and step-by-step examples using long division.
Division in Binary Number System
Division in the binary number system works much like decimal long division. The main difference is that binary uses only two digits: 0 and 1.
The basic binary division rules are:
| Division | Result |
|---|---|
| 0 ÷ 1 | 0 |
| 1 ÷ 1 | 1 |
| 0 ÷ 0 | Undefined |
| 1 ÷ 0 | Undefined |
For larger binary numbers, compare the divisor with the current part of the dividend, place either 0 or 1 in the quotient, subtract, bring down the next bit, and repeat.
For example:
1010₂ ÷ 10₂ = 101₂
In decimal, this is:
10 ÷ 2 = 5
So the binary result 101₂ is correct.
What Is Binary Division?
Binary division is the process of dividing one binary number by another.
Like decimal division, it uses three main terms:
Dividend ÷ Divisor = Quotient
For example:
1010₂ ÷ 10₂ = 101₂
Here:
- Dividend:
1010₂ - Divisor:
10₂ - Quotient:
101₂
Converting the numbers to decimal confirms the result:
1010₂ = 10₁₀
10₂ = 2₁₀
101₂ = 5₁₀
10 ÷ 2 = 5
The arithmetic is the same as decimal division; only the number system is different.
Rules of Division in the Binary Number System
There are four basic cases to understand.
0 ÷ 1 = 0
Zero divided by one is zero:
0 ÷ 1 = 0
1 ÷ 1 = 1
One divided by one is one:
1 ÷ 1 = 1
0 ÷ 0 Is Undefined
Division by zero is undefined, including:
0 ÷ 0
1 ÷ 0 Is Undefined
A nonzero value also cannot be divided by zero:
1 ÷ 0
These rules form the basis of binary division, but larger calculations also require binary subtraction.
How to Do Binary Division
The easiest manual method is binary long division.
The process follows the same pattern as decimal long division:
- Compare the divisor with the leftmost usable part of the dividend.
- If the divisor fits, write
1in the quotient. - If it does not fit, write
0. - Subtract the divisor when a
1is written. - Bring down the next bit of the dividend.
- Repeat until there are no bits left to bring down.
Any value left after the final subtraction is the remainder.
The important difference from decimal long division is that each quotient position can contain only 0 or 1.
Binary Long Division Example: 1010₂ ÷ 10₂
Divide:
1010₂ ÷ 10₂
The dividend is 1010₂ and the divisor is 10₂.
Step 1: Compare the First Bits
Start with the first two bits of the dividend:
10
The divisor is also:
10
So it fits exactly once.
Write 1 in the quotient:
1
----
10 ) 1010
Subtract:
10
10
--
00
Step 2: Bring Down the Next Bit
Bring down the next bit, which is 1.
Now the current value is:
1
The divisor 10₂ does not fit into 1₂, so write 0 in the quotient.
The quotient is now:
10
Bring down the final 0, producing:
10
Step 3: Divide Again
The divisor 10₂ fits into 10₂ once.
Write 1:
101
Subtract:
10
10
--
00
There are no more bits to bring down.
Therefore:
1010₂ ÷ 10₂ = 101₂
The remainder is 0.
Example: 1100₂ ÷ 10₂
Now divide:
1100₂ ÷ 10₂
Using binary long division:
110
-----
10 ) 1100
The result is:
1100₂ ÷ 10₂ = 110₂
We can verify this in decimal:
1100₂ = 12₁₀
10₂ = 2₁₀
110₂ = 6₁₀
12 ÷ 2 = 6
So 110₂ is the correct quotient.
Example: 1111₂ ÷ 11₂
Dividing by 11₂ is more useful for understanding the general method because the divisor is not simply a power of two.
Calculate:
1111₂ ÷ 11₂
Start with the first two bits:
11
The divisor 11₂ fits exactly once, so the first quotient bit is 1.
Subtract:
11
11
--
00
Bring down the next bit:
1
Since 11₂ does not fit into 1₂, write 0 in the quotient.
Bring down the final bit, giving:
11
The divisor fits once again, so write 1.
The quotient is:
101
Therefore:
1111₂ ÷ 11₂ = 101₂
Check in decimal:
1111₂ = 15₁₀
11₂ = 3₁₀
101₂ = 5₁₀
15 ÷ 3 = 5
Binary Division With a Remainder
Binary division does not always produce an exact quotient.
Consider:
101₂ ÷ 10₂
The decimal equivalents are:
101₂ = 5₁₀
10₂ = 2₁₀
Since:
5 ÷ 2 = 2 remainder 1
and decimal 2 is binary 10₂, the binary result is:
101₂ ÷ 10₂ = 10₂ R1
Here:
- Quotient =
10₂ - Remainder =
1₂
A remainder must always be smaller than the divisor.
Binary Division With Fractions
Instead of stopping with a remainder, binary division can continue after the binary point.
For example:
1₂ ÷ 10₂
Since:
1₁₀ ÷ 2₁₀ = 0.5₁₀
the binary result is:
0.1₂
This works because the first position after the binary point represents 1/2.
Another example is:
1₂ ÷ 100₂
Since 100₂ = 4₁₀:
1 ÷ 4 = 0.25
In binary:
1₂ ÷ 100₂ = 0.01₂
The second position after the binary point represents 1/4.
Dividing Binary Numbers by Powers of Two
Division becomes especially simple when the divisor is a power of two.
For unsigned integer values, dividing by 10₂ is equivalent to dividing by decimal 2:
1100₂ ÷ 10₂ = 110₂
Dividing by 100₂ means dividing by decimal 4:
1100₂ ÷ 100₂ = 11₂
When working with whole unsigned binary numbers, this can be viewed as shifting the bits to the right.
For example:
1100₂ → 110₂
is division by 2.
And:
1100₂ → 11₂
is division by 4.
Bits shifted out of an integer representation affect the remainder, so right shifting should not be treated as a universal replacement for division in every signed or fractional arithmetic system.
Binary Division vs Decimal Division
Binary and decimal long division use the same basic algorithm.
| Decimal Division | Binary Division |
|---|---|
| Uses digits 0–9 | Uses digits 0 and 1 |
| Quotient digits can range from 0–9 | Quotient digits are only 0 or 1 |
| Uses decimal subtraction | Uses binary subtraction |
| Based on powers of 10 | Based on powers of 2 |
The main advantage when learning binary division is that deciding the next quotient digit is simple: it can only be 0 or 1.
Common Binary Division Mistakes
Treating Binary Numbers as Decimal Numbers
The binary number:
1010₂
does not mean decimal one thousand and ten.
Its place values are:
1×8 + 0×4 + 1×2 + 0×1 = 10
Therefore:
1010₂ = 10₁₀
Forgetting to Write Zero in the Quotient
Suppose the current part of the dividend is smaller than the divisor.
You still need to place a 0 in the appropriate quotient position before bringing down the next bit.
Skipping that zero changes the place value of the remaining quotient digits and can produce the wrong answer.
Dividing by Zero
Both of these expressions are undefined:
0 ÷ 0
1 ÷ 0
The divisor must be nonzero.
Mixing Binary and Decimal During the Calculation
Decimal conversion is useful for checking an answer, but the long-division calculation itself should remain consistent.
If you are performing binary long division, carry out the comparison and subtraction in binary.
How to Check a Binary Division Answer
A division result can be checked using the relationship:
Divisor × Quotient + Remainder = Dividend
For example:
101₂ ÷ 10₂ = 10₂ R1
Check it:
10₂ × 10₂ = 100₂
100₂ + 1₂ = 101₂
The result matches the original dividend, so the quotient and remainder are correct.
Converting to decimal provides another useful check:
2 × 2 + 1 = 5
Why Binary Division Matters
Binary division is part of binary arithmetic, which is fundamental to digital computing.
At the hardware and software level, division may be implemented using different algorithms or optimized operations depending on the processor, data type, and divisor. For example, division by powers of two can often be handled efficiently with bit-shift operations when the numeric representation allows it.
Understanding the manual method is still useful because it demonstrates the relationship between:
- binary place values
- comparison
- binary subtraction
- quotients
- remainders
- bit shifting
These concepts appear throughout computer science, digital electronics, computer architecture, and low-level programming.
Binary Division Practice Problems
Try solving these before checking the answers.
Question 1
1000₂ ÷ 10₂
Answer:
100₂
Question 2
110₂ ÷ 10₂
Answer:
11₂
Question 3
1111₂ ÷ 1₂
Answer:
1111₂
Question 4
101₂ ÷ 10₂
Answer:
10₂ R1
Question 5
1100₂ ÷ 100₂
Answer:
11₂
Question 6
1001₂ ÷ 11₂
Answer:
11₂
Check in decimal:
9 ÷ 3 = 3
Frequently Asked Questions
How do you divide binary numbers?
Use the same basic process as decimal long division. Compare the divisor with the current part of the dividend, write 1 if it fits or 0 if it does not, subtract when necessary, bring down the next bit, and repeat.
What are the four rules of binary division?
The basic rules are:
0 ÷ 1 = 0
1 ÷ 1 = 1
0 ÷ 0 = Undefined
1 ÷ 0 = Undefined
Division by zero is undefined.
Is binary division the same as decimal division?
The underlying long-division algorithm is essentially the same. Binary uses base 2 rather than base 10, so each quotient digit can only be 0 or 1.
Can binary division have a remainder?
Yes. For example:
101₂ ÷ 10₂ = 10₂ R1
Can binary division produce fractions?
Yes. For example:
1₂ ÷ 10₂ = 0.1₂
The value 0.1₂ is equivalent to decimal 0.5.
Why does dividing by 10₂ shift bits to the right?
10₂ represents decimal 2. Each binary position has twice the value of the position to its right, so moving an unsigned whole-number bit pattern one position to the right corresponds to integer division by two, subject to how any shifted-out bits are handled.
Related Tool and Source
For practice with larger binary values, use the Binary Division Calculator. For more background on how binary represents numbers by powers of two, Khan Academy’s binary numbers guide is a useful companion.
Final Thoughts
Division in the binary number system follows the same basic logic as decimal long division, but every quotient digit is either 0 or 1.
Start by learning the four basic division cases, then practice comparing the divisor, subtracting in binary, and bringing down each successive bit. Once that process is familiar, remainders, binary fractions, and division by powers of two become much easier to understand.
A useful way to verify any result is:
Divisor × Quotient + Remainder = Dividend
That check works whether the numbers are written in binary or decimal.