Thursday, December 14, 2006

arithmetic unplugged - the abacus

Somewhere in the intersection of the sets "unnecessary activies", "math-related stuff", "exotic devices", "cheap equipment", and "skills requiring long practice" is "the use of an abacus". How could a frugal ComSci-major/Math-minor with way too much time on his hands resist?

I bought a surprisingly little but quite usable 13-rod Japanese abacus or soroban on eBay for cheap, and also an old but lightly used copy of The Japanese Abacus: Its Use and Theory by Takashi Kojima (ISBN 0-8048-0278-5, although I later stumbled on a page with links to a verrrrry similar pdf). If I had the requisite desire and/or discipline I would probably be pretty good at using my soroban--I'm not. Part of the lack of motivation stems from the fact that I have no practical reason for improvement; after all my cell phone (the same one that once made me feel like I had the power of the sun in the palm of my hand) has a calculator function. Nevertheless, I'm progressing slowly.

On the soroban, each rod has one 5-unit bead and four 1-unit beads, with a separator between the fiver and the rest. This means that the soroban is primarily for regular decimal-based applications, although I suppose one could use just the 5-beads and do binary math if one wanted to (setting the 5-bead would represent a 1, and each rod would represent a power of two). Note that more than 6 or so items in a group could be hard to accurately recognize and distinguish at a glance, so more beads might actually make calculations slower, because it would force the operator to laboriously count bead-by-bead.

One exceedingly simple drill for beginners, that I haven't seen mentioned anywhere, is just to add a single-digit number to itself a set number of times. It's easy to tell when you've made a mistake: merely check to see if the result is a multiple of the number. You can also start with a high multiple and perform a series of subtractions. The point of this drill, which clearly would never be performed with an abacus in a real situation, is to increase your speed. In my opinion, it makes sense to get really good at this drill before adding and subtracting two digit numbers, which seem to be the usual starting exercises for the abacus.

As the little book explained, the abacus procedures simplify a calculation by breaking it up into lots of rapid, little single-digit calculations. The usual pencil-and-paper method is almost exactly like the abacus method (in spirit anyway), but with some significant differences. The first, of course, being that the paper method involves writing a problem out, while an abacus operator just flicks his fingers to shift some beads--a considerably simpler if not faster motion. The second difference is that numbers in the paper method are distinguished by having differently shaped symbols, while numbers on an abacus are actual quantities of beads--numbers you can feel. Hence the usefulness of the "toy" abacus in teaching children.

The third and most confusing difference is that the paper method is free-form or malleable, while an abacus cannot be "rewritten" to have more or less beads in any given case! On the abacus, "carrying" and "borrowing", meaning the overflow or underflow of one power of ten to another, are achieved by making a one-bead adjustment in the higher power and then offsetting that adjustment by moving beads in an inverse operation in the lower power. You undo the excess. For instance, to add 4 to 8, you:
1. Check to see if you have enough beads on the rod under consideration. Since there is only one 1-bead left (8 is one 5-bead and three 1-beads), proceed to step 2.
2. In step 3, you will add a 1-bead on the tens rod to the left, which will be too much by 10 - 4 = 6. So reset six beads, that is, the 5-bead and one 1-bead, leaving two 1-beads.
3. Add the 1-bead in the tens rod. By not doing this until last, you can keep your attention on the unit rod for both steps 1 and 2. Otherwise, you would focus on the unit rod, switch to the tens rod, then switch back to the unit rod to undo the excess.
This could also be thought of as adding 10 and a -6 (or subtracting 6). According to the little book, the key to doing this quickly is to think of numbers in terms of "complementary" pairs: 9 and 1, 8 and 2, 7 and 3, 4 and 6, 5 and 5. If there is power-of-ten overflow or underflow involving one of the numbers in a pair, just do the opposite operation with the other number in the pair. It gets easier with practice, believe me. The same strategy applies to adding or subtracting a 5-bead when you run out of 1-beads (e.g., 3+3), except there are only two pairs: 4 and 1, 3 and 2. The real kicker is when you have a problem like 13 - 6, in which you need to add the "tens complement" of 6 (i.e.,4) to the unit rod, but you can't do that unless you move a 5-bead and subtract the "fives complement" (i.e.,1) from the unit rod. You convert a subtraction to an addition to a subtraction as far as the 1-beads on the unit rod are concerned. Have I mentioned that effective use of the abacus takes some concentration at first until it becomes "automatic"? Try not to overthink it. There are also techniques for multiplication, division, and roots, but I've only skimmed those so far. It appears that the comparison to the paper method is again apt, as the simplifying principle is the distributive property: reducing a complex multiplication to a sum of one-digit multiplications.

The freaky conclusion of abacus training is the operator becoming able to do abacus manipulations on an imaginary abacus, enabling savant-like mental calculation. I don't plan to reach that point for a looooong time, but here is an incredible account that I'm not sure I believe. If the story's completely true, I might like to hire him as my mentat (I'll get that Leto!). Here are some of the more useful links I've found.

1. Denton Zone10:43 PM

Hey Hey Art: How about a copy of Venetians Blind for the Festivious Holidays??

2. Thanks for you kind words about Abacus: Mystery of the Bead. You can print up the manual if you want by following the Abacus.pdf file link on the title page.

The abacus does indeed lead to an amazing ability to do mental arithmetic, especially if one begins soroban at an early age. Here's a link to a small movie on Youtube.com that shows children doing just this.