Fig 1. A meaningless arrangement of the digit abacus.
Please contact me if you find any problems with or want to see anything added to this tutorial.
Warning: This document is a work-in-progress, and is currently incomplete.
Yes, abacus lessons may seem like an odd thing to put up on the 'Net. After all, is not the computer you're viewing this page on capable of doing anything you can do on an abacus in a millionth or billionth the time?
While this is true, I feel that abacuses are important. I learned to use one as a young child, and a lot of what I know about algorithms stems back to the techniques you use on an abacus. Moreover, they can be a great reliever of stress, as well as a functional alternative to hand-calculators in some situations, and one of the few hobbies you can perform in the dark.
Often abacuses are more usable than other devices. Believe it or not, to the trained hand, it is simply faster to add two numbers on an abacus than to enter them into a calculator or computer, and have it add them for you, because the abacus is designed in a way that is very natural for the human hand to use quickly and without errors, more so than your typical numeric keypad.
Also, abacuses are historically important; for thousands of the years they were the most sophisticated calculating device available, and performed quite well in many diverse societies. By understanding abacuses, you can share a little bond with the many who have gone before you.
Many abacus tutorials already exist. Unfortunately, they suffer from two main problems. One is that they're often poorly translated from Asian documents, and thus completely useless. The other is that they're not usually in-depth enough.
Many people believe abacuses are only capable of adding. This is completely wrong. Others know that you can also subtract, multiply, and divide on them, and occasionally you hear about being able to take square roots on them, but there stops the speculation. To say this is its limit, however, is to say that these are also the limits of a computer, which has built-in circuits only to perform these same operations.
In the advanced sections of this guide, I will demonstrate how to go beyond all traditional usage, and compute sines, cosines, fractional exponents, logs, and other functions never thought calculatable using an abacus. This is what makes this tutorial unique.
Feel free to skip this section if you really don't care what the correct pluralisation of abacus is.
The plural of abacus has a been a matter of heated debate for some time. Webster's Dictionary, the equivocal American reference, gives abaci as the plural. This was done in analogy to words descending from Latin such as cactus and fungus, since in Latin there is a noun form with singular ending -us and plural ending -i. This has been one of the few Latin pluralisations to persist where ones like formulae have fallen into disuse.
But on the other hand, the equivocal British reference, the Oxford Dictionary, gives the plural of abacus to be abacuses. As odd as this sounds, it is more in tune to the actual roots of the word, which are not at all Latin, but Arabic. Although the Arabic pluralisation is not used, I believe it's reasonable to say that it doesn't make sense to pluralise a non-Latin word as though it were Latin, masking its etymological roots. For this reason, I choose this plural in this document.
Abacuses are not difficult to obtain. Often, however, they must be ordered from overseas, or from importers of Asian souvenirs. I give several links below, but don't click on anything before making sure you're getting the right kind.
As a beginner, you will want a Chinese abacus, with seven beads on each rod, not a Japanese one with only five beads on each rod. The lessons will explain why these two types exist and help you decide if you really want to move to a Japanese abacus. The Chinese abacus is also helpful for doing base 16 (hexadecimal) calculations, as will be explained later.
Ideally, the beads should be large enough that you can easily slide them along their rods using either your thumb or index finger without affecting other beads or much precision of movement. Material is less important; I tend to find abacuses with wooden beads and metal rods the most durable and easy-to-use, since the beads are light and have a pleasing appearance and the device can resist pressure, but wooden rods suffice, as do aluminum or plastic beads.
Because abacuses are such simple devices mechanically, literally anyone can make their own abacus with a few simple tools. I don't specialise in this, but essentially all you need is a drill to put the rods through the center bar, some short wooden planks for the frame, some drilled beads that fit snugly on the rods but are free to move, and a few nails to hold the frame together.
Order an abacus at any of these on-line sites:
AsianIdeas Inc. - Chinese
Tiger Gifts - Chinese
Soroban - G4725, PatiQ, 84251, assembly kit - Japanese
Dragon-Gate - too tiny to use but pretty - Chinese
Jesse Dilson - comes with a book - Chinese
If you find good abacuses that I don't know about, even if it's just at a local store, please notify me!
Luis Fernandes' simple Chinese abacus
A simple, usable Chinese abacus. Quite sufficient for this tutorial.
Sarat Chandran and David A. Bagley's Morphing Java Abacus
This is the do-it-all abacus applet, able to simulate Chinese, Japanese, and Russian abacuses, and even able to generalise them to arbitrary bases. Be careful, however; the number they display as being on the abacus considers the ones column to be the third column from the left, and the rightmost two columns represent two decimal columns, so do not use them. It also insists on dominating your entire screen, which is rather annoying.
When you look at your abacus, it should have a rectangular frame, with a wooden bar through the middle horizontally, and several thin rods passing vertically through the middle bar from top to bottom. The frame will be called the frame, the bar passing through the middle I will call the centre, and the rods I will call rods. On a Chinese abacus, each rod should be threaded through seven beads, two above the centre and five below.
The beads below the centre are called earth beads, because they are lower down than the other beads, while the beads above the centre are similarly called heaven beads.
This is all you need to get started.
To simplify things, I'm going to start out with a very simple abacus I call the digit abacus. It has only one rod and looks like this:
The above is a meaningless arrangement of the abacus, because every bead must be either pushed as far up as it can go or as far down as it can go. Beads are not allowed to "float" in the middle of a rod like the one above.
The digit abacus can represent the numbers 0 through 9. To accomplish this with only seven beads, it distinguishes between heaven beads and earth beads. Each of the earth beads is given a value of 1, but it only contributes this value when it is pushed against the centre. Beads that are pushed away from the centre contribute no value at all. Thus, this is how you represent 1:
Similarly, if more than one earth bead is pushed against the centre, their values add. Thus, this is how you represent 3:
Heaven beads on the other hand, besides just being physically higher, are worth much more as well. Each is worth 5. Just like the earth beads, they only contribute their value when pushed towards the centre. This is how we represent 5:
By combining the right number of heaven and earth beads, we can achieve any value between 0 and 9, such as 8:
Because we have 1 heaven bead, worth 5, and 3 earth beads, each worth 1, we have a total value of 5+1+1+1=8. When no beads are contributing any value, we achieve zero:
Before continuing, make sure you can figure out how to represent each of the ten digits on some rod of your own abacus. With practice, you will eventually learn to recognise the patterns for each of them very quickly.
Now, you may think it's terribly limiting to only be able to represent these ten digits, and no other numbers. Indeed, we'd have the same problem when writing down numbers on paper if it wasn't for the place-value system, which enables us to write multiple digits to represent a single larger number.
When you write the numeral 123 down on a piece of paper, the three digits represent the three numbers 100, 20, and 3 respectively, because the 3 is in the rightmost ones place, the 2 is in the tens place, and the 1 is in the hundreds place. Their position implies a multiplier for their value, and we add these values together to obtain the true number being represented: 100+20+3 = 123.
Similarly, on a real abacus with more than one rod, each rod represents a place. The rightmost rod represents the ones place, the next rod to its left the tens place, and the next rod to its left the hundreds place, and so on. Since you know how to put any digit on each rod, you can write out larger numbers as several digits in a row, each on one rod, just as you do on paper. For example, here is the number 4,377,084 on a real abacus:
You can easily verify that the beads on each rod add up to the correct digit.
The act of entering the number zero on the abacus is particularly important, because you do it at the beginning of each calculation (similarly to pressing the clear button a calculator). Different people have different ways of doing this. One convenient way is to tilt the abacus up towards you, causing all the beads to fall down, and then laying it flat and sweeping the heaven beads up with a straightened hand. In other cases where the abacus is already mostly clear, you may wish to simply sweep both the earth beads down and the heaven beads up. Either way, this is called clearing the abacus, and the result should like this:
As an exercise, try and enter each of these numbers into your abacus:
Click on each number for its solution.Note for the advanced: The last two are, of course, dirty tricks. You cannot normally enter negative or fractional numbers on an abacus. Decimal numbers like 0.5 can easily be handled however, by relabeling the rods so that the ones column is somewhere in the middle, and columns to the right of it are the tenths columns, hundredths column and so on. Be careful to remember where your decimal point is if you do this. Negative numbers can also be represented by simply remembering that your current value is negative, possibly using a bead or two in the leftmost column to remind you of this. When I discuss calculation techniques I'll put in some more of these notes for the advanced to explain how to handle non-positive or non-integer numbers.
Equally important is the ability to read off a number from the abacus, and put it back in a form you can use. This is how you will obtain your answers when you later compute them. To practice this, write down or read off each the numbers being displayed on each of these abacuses:
Click on each abacus for its solution.
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