Using the Abacus, part 2: Multiplication

Once you can add on an abacus, the next thing to learn is multiplication. Like addition, it follows pretty closely on the old pencil-and-paper method. But it's worth taking the time to look closely and see it step by step, because it's an important subroutine (to use a programming term) that will be useful in more complicated stuff.

Just for clarity, I'll write out the basic pencil and paper algorithm:

1. Write down a "0" for the initial value of the result.
1. For each digit s_i in the *second number* number, from right to left
1. For each digit d_j in the first number, from right to left:
1. Multiply s_i × d_j
2. Take the result of that multiplication, and add it to the result, starting at the *j+i-1*th column from the right.

When you finish, you'll have the product. Since this sounds a bit different from how you probably learned it in school, let's just step through it quickly, so that you can see that it really is the same thing. Let's multiply 219 × 163.

* Initial result=0.
* i=1,j=1: Multiply 3 × 9. That gives us 27. Add it starting in the (i+j-1)th column; i+j-1=1+1-1=1. So:

         0
        27

* i=1,j=2: 3×1=3. Add it to the 1+2-1=2nd column:

         0
        27
        3

* i=1,j=3: 3×2=6; add it to the 3rd column:

          0
         27
         3
        6
     --------
        657

* So the sum after adding the multiples from the first digit of "263" is 657. Now we start on the second digit. i=2,j=1: 6×9=54; add it to column 2+1-1=2.

       657
       54

* i=2,j=2: 6×1=6; add it to column 3:

        657
        54
        6

* i=2,j=3: 6×2=12; add it to column 4:

          657
          54
          6
        12
      ---------
        13797

* Now let's just rush through the last one, now that you have the idea.

          13797
            9
           1
          2
       ----------
          35697

On the abacus, we're basically going to do exactly the same thing that we just did. We start by zeroing the abacus out. Then we start right to left on the digits of the second number, doing the single digit multiplication in our head, and starting to add that to the result starting in the *(i+j-1)*th column from the right.

So let's do the same multiplication on the abacus. Start with it zeroed, and then just step through the algorithm - same steps as the "paper-and-pencil" example above.

The only thing that's really tricky about multiplication on the abacus is that you need to keep track of where you are by yourself. It's easy to lose your place, and when that happens, you'll end up with a wrong result. When you're learning to use the abacus, the goal is to get so accustomed to the abacus that it's just automatic, and so fast that you won't lose track and forget your position or the digits of either of the numbers you're multiplying.

One of the advantages of the Lee abacus is that it has sliding indicators to keep track of where you are, which are fast enough to use that they don't really slow down your pace when multiplying. The two upper mini abacuses can also be used to help you remember the two numbers you're multiplying.