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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 out 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. We 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 example 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 facts , or 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 in 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 ? 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 straightforward 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 decimal , do 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 find 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 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.