Division on the Abacus

Sep 25 2006 Published by under manual computing devices

Now we're going to try something challenging on the abacus: *division*. Like multiplication, abacus division is close to the way you'd do it on paper. But just like doing paper division is trickier than paper multiplication, abacus division is tricker than abacus multiplication. But the technique that is used to do division on the abacus is an important fundamental one: it's what makes it possible to use the abacus for more advanced operations, like roots.

Before going into the algorithm, there's one important new technique that we need, called *partitioning* on the abacus. The idea is that we're going to pick some column on the abacus, which we'll call the *reference column*; and *in our minds*, we're going to split the abacus so that the reference column and everything to its left is one abacus, and everything to its right is a second abacus.
The way that we're going to use this is that we're going to put the *dividend* into the section to the right of the reference column, and we're going to accumulate the *quotient* to the left.
So, let's start by reviewing the standard paper method.
1. Find the starting column for the quotient. This will be the *first* position *n* where the number formed in columns 1 through *n* of the dividend is greater than or equal to the divisor. Columns *1* through *n* are called the *working digits*; column *n* is called the *current quotient column*.
2. Using approximation, figure out the *largest* number *i* such that *i* times the divisor is *less than or equal* to the number formed by the working digits.
3. Write *i* in the current quotient column, and subtract *i* times the divisor from the working digits. The result *should be* a number *smaller* than the divisor. This is the *working remainder*.
4. Copy digits to the right of the working digits, and append them to the working remainder from step 3, until you get a number *greater than or equal to* the divisor. The working remainder + the copied digits become the new working digits. The last column that you copied is the new *current quotient column*. If there are any blank spaces between the old and new current quotient columns, fill them with zeros.
5. Go back to step 2, using the new working digits and current quotient column, until either the working remainder is zero, or you're bored and don't want to keep going.
As usual, it's hard to follow something like that without an example. Let's divide 4582 by 17.
* Find the starting column. It will be column 2, because 4<17, but 45>17.
* Find the largest multiple of 17 that's smaller than 45. That will be 2, and it will be the first digit of our answer. Subtract 2*17=34 from 45, leaving a working remainder of 11. We can pull down "8" from the dividend and append it, giving us new working digits 118; and the new current quotient column will be just one digit to the right of the old.

17 | 4582

* Find the largest multiple of 17 ≤ 118. That would be 6. 6*17=102, 118-102=16. The working remainder is 16. So we pull down a digit; 2. That gives us new working digits 162,

17 | 4582.0

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  • Doub says:

    Based on your description of the algorithm there seems to be some natural extension that you didn't mentionned (I have absolutely no knowledge of abacuses, so that's just guesses).
    First you don't have to explicitly define the number of columns of the quotient at the beginning, just keep adding columns until you have your remainder. It just save you some setup, but you still have to keep in mind the current column of the quotient and a working digit in the dividend.
    The other thing is that what I usually found most difficult on paper division is easy on an abacus. If the divisor is a big number, that is not necessarily easy to multiply in mind to do a single substraction. On an abacus you can simply iteratively substract it from your working digits, adding one to the current column of your quotient at each iteration. That way you only do in-place subtractions and increments, which can be pretty fast (and I think we don't do it on paper because it would waste enormous quantities of paper).
    Anyway it's a very interesting tool, and I no longer wonder how mathematicians did so complex jobs thousands of years before we mastered electricity. I'm looking forward for the algorithms on more complex operations 🙂

  • Stephen says:

    It appears you're ignoring a top and bottom bead. May as well use a soroban.

  • Corrections in the abacus instructions:
    Step 2, second last sentence: "So 17×2=34; we subtract 5 from the third column"; 5 should be 3.
    Step 4: second sentence: "and out working remainder"; "out" should be "our".

  • Stephen: MarkCC covers his choice of abacus type in his first abacus article:

    I'm going to talk about the Chinese abacus, the suan-pan. The main reason that I prefer the suan-pan is that the way that it's beads are set 5/2 lets you simplify some things; you can do things like delay a carry until you're ready; and it makes some 5's complement stuff easier to do.

  • Weiqi Gao says:

    I have written up a new blog entry about how division is done on the Chinese Suan Pan:
    The Chinese approach relies on a set of division rhymes that minimizes the amount of mental calculations, essentially turning computation into table lookup. The result is superimposed on to the divisend rather than put to the left side of the abacus to increase the locality of the memory location and reduce the hand's seek time. The quotient digits are obtained speculatively using just the first digit of the divisor and the first digit of the working remainder rather than precisely, again eliminating mental calculations.