## Binary Division Logic & Calculation

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The first article discusses binary addition ; the second article discusses binary subtraction ; the third article discusses binary multiplication ; this article discusses binary division. The pencil-and-paper method of binary division is the same as the pencil-and-paper method of decimal division, except that binary numerals are manipulated instead.

As it turns binary division calculator with remainder though, binary division is simpler. There is no need to guess and then check intermediate quotients; they are either 0 are 1, and are easy to determine by sight.

Pencil-and-paper division, also known as long division, is the hardest of the four arithmetic algorithms. Like the other algorithms, it requires you to solve smaller subproblems of the same type. Solving these division subproblems requires estimation, guessing, and checking.

In addition to these division subproblems, multiplication and subtraction are required as well. Here is an example:. Does 88 go into 8? Does 88 go into 83? Does 88 go into ? The first step of long division, as commonly practiced, combines several steps and their substeps into one. Technically, 88 goes into 8 zero times, so we should write down a 0, multiply 88 by 0, subtract 0 from 8, and then bring down the 3.

Next, we should write down a 0 because 88 goes into 83 zero times, multiply 88 by 0, subtract 0 from 83, and bring down the 1.

Binary division calculator with remainder tried to divide by 88 before — two steps ago. That means we have a two-digit cycle 45 from here on out. The answer is 9. The red digits are the carries that occur during the multiplication substeps the multiplication is done as if the divisor — the bigger number — is on top, by convention. Each red digit is crossed out before the next multiplication. To avoid clutter, I have chosen not to mark the borrows that occur during subtraction.

My example has a multi-digit divisor, and has an answer with a remainder that I wrote as a repeating decimal. I wanted one binary division calculator with remainder that showed long division to its fullest.

I could have picked a problem with a single-digit divisor which would require no guessing, assuming you know the multiplication factsor one that produced an integer quotient, or one that produced a quotient with a fractional part that terminated.

I could have expressed the fractional part as an integer remainder, or binary division calculator with remainder fraction form. Here it is broken down into steps, following the same algorithm I used for decimal numbers:. Does 11 go into 1? Does 11 go into 10? Does 11 go into binary division calculator with remainder We stop here, recognizing that we divided by 11 two steps ago. This means we have a two-digit cycle 10 from here on out. The quotient is When the answer has a repeating fractional part, checking it is not as binary division calculator with remainder as it is for the other arithmetic operations.

What we can do is approximate the quotient to a finite number of places and then check that it comes close to the expected answer. You can check the answer in a few ways. One way is by doing binary multiplication by hand: Another way to check is to convert the operands to decimaldo decimal division, and then convert the approximate decimal answer to binary. Estimating that as 3. That looks like it wants to be You can also check the answer using my binary calculator.

Again, that looks like It gives the decimal answer we expect: You can also use this tool to convert in the opposite direction, verifying that 3. There are also analytical ways to check the answer exactly: Like the other arithmetic algorithms, I described the division algorithm in a base-independent way.

I wanted to stress the mechanical procedure, not why it works in either decimal or binary. When you do binary long division, you might binary division calculator with remainder yourself doing some of the substeps in your head in decimal e. Be thankful my example only had a two-digit repeating cycle! Thank you for posting this series of article and emailing me to let me know it was up.

Can you share which tool is used to produce it? It is very clear. You gives so quick response. Recently I read several of your articles. Very well written and useful. I can post some testing I have done with some of your programs. One thing I find, on Ubuntu 64 v Gay caused dead loop compiled by gcc test.

However, it does work fine with gcc -m32 test. The issue seems due to integer size. BigFloat to convert binary to decimal, do the math operation and convert back to binary bits. Maybe you can email me with details see my contact page or continue this discussion on one of my David Gay articles. What are they for? Those are the carries during the multiplication see my article on binary multiplication.

Example of Binary Division The pencil-and-paper method of binary division calculator with remainder division is the same as the pencil-and-paper method of decimal division, except that binary numerals are manipulated instead. Get articles by e-mail. Tom, Those are the carries during the multiplication see my article on binary multiplication. Thank you so much!

I used it as model for a microcontroller routine of an electronics project. Please help me with these.

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Want to calculate with decimal operands? You must convert them first. This is an arbitrary-precision binary calculator. It can add , subtract , multiply , or divide two binary numbers. It can operate on very large integers and very small fractional values — and combinations of both. This calculator is, by design, very simple. You can use it to explore binary numbers in their most basic form. Similarly, you can change the operator and keep the operands as is.

Besides the result of the operation, the number of digits in the operands and the result is displayed. For example, when calculating 1. This means that operand 1 has one digit in its integer part and four digits in its fractional part, operand 2 has three digits in its integer part and six digits in its fractional part, and the result has four digits in its integer part and ten digits in its fractional part.

Addition, subtraction, and multiplication always produce a finite result, but division may in fact, in most cases produce an infinite repeating fractional value. Infinite results are truncated — not rounded — to the specified number of bits.

For divisions that represent dyadic fractions , the result will be finite , and displayed in full precision — regardless of the setting for the number of fractional bits. Although this calculator implements pure binary arithmetic, you can use it to explore floating-point arithmetic. For example, say you wanted to know why, using IEEE double-precision binary floating-point arithmetic, There are two sources of imprecision in such a calculation: Decimal to floating-point conversion introduces inexactness because a decimal operand may not have an exact floating-point equivalent; limited-precision binary arithmetic introduces inexactness because a binary calculation may produce more bits than can be stored.

In these cases, rounding occurs. My decimal to binary converter will tell you that, in pure binary, To work through this example, you had to act like a computer, as tedious as that was. First, you had to convert the operands to binary, rounding them if necessary; then, you had to multiply them, and round the result. For practical reasons, the size of the inputs — and the number of fractional bits in an infinite division result — is limited.

If you exceed these limits, you will get an error message. But within these limits, all results will be accurate in the case of division, results are accurate through the truncated bit position.

Skip to content Operand 1 Enter a binary number e.