Binary Division Calculator

What the binary division calculator does

This divides one binary number by another and shows the result in both binary and decimal. Enter the two numbers, press the button, and read off the answer.

Binary division uses the same long division layout as ordinary numbers, and in one respect it is simpler, because at each step the divisor can only fit in either zero or one time. There is never a moment of guessing how many times it goes.

How to use it

1. Choose a bit size. This sets how wide your numbers are, with 8 as the default.
2. Enter the two binary numbers. The first is divided by the second. Use only 0s and 1s, and the second number cannot be zero.
3. Press Calculate. The result appears as both a binary result and a decimal result.

Press Reset to clear the inputs.

How binary long division works

You work across the dividend from the left, the same shape as long division in decimal: compare, subtract, and bring down the next digit. The part that makes binary easier is the comparing. The divisor either fits into the part you are looking at or it does not, so the quotient digit you write is always just 1 if it fits or 0 if it does not. When it fits, you subtract the divisor once and bring down the next bit. When it does not, you write 0 and bring down the next bit anyway. There is no working out whether it goes three times or seven, which is the slow part of decimal division.

A worked example with real numbers

Let us divide 1011 by 10. Work across 1011 from the left.

• Does 10 fit into 1? No, so write 0.
• Bring down the next bit to make 10. Does 10 fit into 10? Yes, once, so write 1 and subtract to leave 0.
• Bring down the next bit to make 1. Does 10 fit into 1? No, so write 0.
• Bring down the last bit to make 11. Does 10 fit into 11? Yes, once, so write 1 and subtract to leave 1.

The quotient reads 101 with 1 left over. To check it, 1011 is 11 and 10 is 2, and 11 divided by 2 is 5 with 1 remaining, and 101 is 5. The leftover is where the next part comes in.

Past the point: binary fractions

Rather than stopping at a leftover, this calculator keeps dividing past a binary point, the binary version of a decimal point. Carrying on with the 1 left over from the example, it brings down a 0 after the point and finds that 10 fits once more, giving a final result of 101.1. That is the binary way of writing 5.5, which is what 11 divided by 2 actually equals. So instead of a remainder, you get a complete number with a fractional part.

One thing worth knowing is that some divisions never come out exactly in binary. Dividing 1 by 11, which is one third, gives 0.010101 repeating forever, just as one third is 0.3333 repeating in decimal. When that happens, the calculator shows the fraction to a set number of bits rather than an endless string. This is the same reason computers can only hold many fractions approximately, a fact worth keeping in mind whenever exact fractional values matter.

Questions people ask

How does binary division work?

You use long division across the dividend, comparing, subtracting, and bringing down the next digit. Each quotient digit is simply 1 if the divisor fits or 0 if it does not, since it can only go zero or one times.

Does it give a remainder or a fraction?

A fraction. Instead of stopping at a leftover, the calculator continues past a binary point, so 1011 divided by 10 comes out as 101.1, which is 5.5, rather than 5 remainder 1.

Can I divide by zero?

No. Division by zero is not defined, so the calculator will not accept a second number of zero and asks you to change it.

Why do some results go on and on?

Because some fractions never end exactly in binary, like one third, which is 0.010101 repeating. The calculator shows such results to a set number of fractional bits rather than forever.

References

1. Weisstein, Eric W. "Binary." From MathWorld, A Wolfram Resource. https://mathworld.wolfram.com/Binary.html
2. Goldberg, David. What Every Computer Scientist Should Know About Floating-Point Arithmetic. ACM Computing Surveys. https://docs.oracle.com/cd/E19957-01/806-3568/ncg_goldberg.html