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 si in the *second number* number, from right to left
1. For each digit dj in the first number, from right to left:
1. Multiply si × dj
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.

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Next step : internally memorise Taylor's expansion for 1/x and now you can do division, even getting TWO digits of precision for each iteration :-)

Now, at the time (and certainly still), i couldn't remember a ten digit number for beans. I thought this technique was so powerful that it must be that anyone could use it to perform mental arithmetic. This may be in error.

What is true is that the abacus is a very powerful way to teach arithmetic. Many have a fear of math. If you add 5 + 6 and get 12, it's off by one, but essentially wrong. Much fear of math is fear of failure. With the abacus, the procedure steps can be made extremely simple. In the procedure, one can learn that there is only one next step. One can learn to react mechanically to the instrument. Out pops the correct answer. With success, fear vanishes.

This website doesn't show how to multiply on the abacus! Please add something that will show how to multiply on an abacus! GRRRRRRRRRRR

By billybobjoe (not verified) on 06 Oct 2006 #permalink

It, um, does. Look up.

By Xanthir, FCD (not verified) on 06 Oct 2006 #permalink

this is quite complicated

By Anonymous (not verified) on 02 Jun 2010 #permalink