Mechanical Shortcuts
2025-02-20
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Last year I bought a Japanese abacus, called a soroban (そろばん). I had played with slide rules extensively at that point, and I wanted to learn about other forms of mechanical calculation. The model I purchased was a simple 17-rod educational tool, analogous to a a basic four-function pocket calculator one might find at a convenience store. But the beauty of mechanical calculators abacus is that the cheapest and the most expensive models work just as efficiently as each other.
The standard soroban contains rods held inside a frame. The rods are divided into top and bottom sections, with one bead in the top and four beads in the bottom. This creates a "bi-quinary" system: the four beads on the bottom represent units, and the top bead represents fives, allowing any number from 0 to 9 to be encoded onto one rod.
To learn mathematics on a soroban, I have been using the book "The Japanese Abacus: Its Use and Theory" by Takashi Kojima. A product of the 1950s, the book is very dense but extremely informative, and is still considered one of the definitive beginner's English texts on using a soroban.
A passage in the addition section of the book stood out to me:
The fundamental principle which makes abacus operation simple and speedy is mechanization. ...
To give an example, in adding 7 to 9, the student accustomed to the Western mode of calculation will probably form 16 on the board as a result of mental calculation to the effect that 9 and 7 is 16. But such procedure is in every way inferior to the above-mentioned mechanical one. Not only does this Western method require mental exertion and time but it is liable to cause perplexity and errors.
When a problem of addition and subtraction is worked on the board, the procedure is very simple. Addition and subtraction, which involve two rods, are simplified by means of a complementary digit, that is, the digit necessary to give the sum 10 when added to a given digit. For instance, suppose we have to add 7 on a rod where there is 9; then we think or say, "7 and 3 is 10," and subtract 3 from the rod in question, and add 1 to next rod on the left. When we have to subtract 7 from 16, we think or say, "7 from 10 leaves 3," and subtract 1 from the next rod on the left, and add 3 to the rod in question. This means that 10 is always reduced to 1, and added or subtracted on the tens’ rod. Therefore, after recalling the complementary digit, the operator has simply to perform either of the two mechanical operations: subtracting the complementary digit and adding 1 on the tens’ rod (in addition) or subtracting 1 on the tens’ rod and adding the complementary digit (in subtraction). The result then will naturally form on the board. No matter how many digits may be contained in the numbers to be added or subtracted, the entire operation is performed by applying this mechanical method to each digit in turn.
In modern terms, calculation on a soroban relies on complements, specifically fives complement and tens complement. To add numbers, it's usually sufficient to move a complement of five or ten, then move the top bead or next bead to the left as the complement requires. By focusing only on that complement, the answer appears on the board automatically.
This aspect of sorobans fascinates me. When I wrote scripts and programs, I often use shortcuts to reduce the number of computations that need to be performed, but those shortcuts are abstract. It's entirely different when I can use a shortcut when manipulating a tangible object, and the result of that shortcut causes the result to pop right out. The feeling is nothing like the feeling of working out a math problem with paper and pencil--it's even different from the feeling of using a slide rule.
I practiced with my soroban for a short time after getting it, but I soon moved on to other projects, and now I've forgotten some of the basics. I'm going back now and re-training myself so I can use it anytime, even if I'm not very fast at it.
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[Last updated: 2025-02-20]