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.
2 +--------------- 17 | 4582 34 ---- 118
* 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