Dealing with overflow

Introduction

Excluding the so-called "special methods", there are two basic ways of arranging general division problems. Not knowing a standard designation for them, I'll use:

• Modern division arrangement (MDA), as explained by Kojima,

• Traditional division arrangement (TDA), as used in ancient books like the Jinkoki (塵劫記), or the Pazhu Suanfa (盤珠算法)

MDA seems perfect for any division method; not just the modern and traditional ones, but also any of the amazing variety of methods one can imagine after reading a page like this and just using the beads of a 1/4 abacus. On the contrary, TDA is problematic with any division method since a collision^[1] between divisor and dividend/remainder frequently occurs and special techniques or abaci are needed to cope with this collision. Even so, TDA has been used for centuries in conjunction with the traditional method of division while MDA seems to have been deprecated until modern times. Why? That remains for me as the greatest of all mysteries of the beads! However, certain advantages to TDA must be recognized, but I wonder if they are enough to justify its historical usage:

• It uses one rod less
• Result does not displace too much to the left as in MDA, which is of interest in the case of chained operations. This and the above points makes TDA more suitable to small rod number abacuses, like the traditional 13-rod suanpan/soroban.
• It saves some finger movements; for instance, in the operation 6231÷93=67 using traditional (chinese) division, I count 14 finger movements with TDA versus 24 with MDA.
• Hand displacements are shorter.
• It is less prone to errors as less rods are skipped.

Regarding the traditional division (Kijohou 帰除法) using TDA, the way to avoid the mentioned collision is to accept that the first column of the dividend/remainder, after the application of Chinese division rules, can overflow and temporarily accept a value greater than 9 (up to 18), while providing some mechanism to deal with such an overflow. Interestingly enough, it seems that no ancient text explains how to do the latter. In the case of a 2/5 or 3/5 abacus we can use the additional upper bead to represent values from 10 to 20 but, What can be done on an 1/5 or 1/4 abacus?.

In a past post, Masaaki presented two examples of traditional division using an apostrophe (‘) to mark the columns or rods that temporarily received a value higher than 9 (overflow). Let us think of this apostrophe as a typographical representation of a small 1 (1), a bead that should be pushed or set somewhere, be it on a real or virtual rod. Note that if we could open or insert a new column in the place of the apostrophe (as it is commonly done in any spreadsheet) all our problems would go away by using the new column to receive the bead, but by doing so we would be using MDA. After a short digression, three alternatives will be described below to stay on TDA.

On geeses and flocks

I'll use the classical exercise  as an example. This exercise is called in chinese: The lone geese return (孤雁歸隊 Gūyàn guīduì). If you enter this division on the abacus, for instance:

    ABCDEFGHIJK

    -----------

    999  998001

and if you have an almighty imagination (not my case :), no doubt, you will identify the lone bead set on K with a lone geese that has just left her flock FGH (you can see the place that she occupied in the lower part of column H). To convince her to rejoin her flock you only have to complete the division!^[2]

First way: Brute force

In principle, we could add the small “1” in any unused column, for example the rightmost one; but this could be annoying and inconvenient because both the hand and the attention would have to be jumping from one place to another on the abacus with the risk of ending up working in the wrong column. Here, without any further consideration, we will simply add the small "1" to the column of the just entered interim quotient digit. This may sound strange or brutal (and indeed it is), but if we can keep the value of the interim digit in memory we can operate as usual and any anomaly will disappear from the abacus in a moment. Let's see it with the   example on an 1/4 abacus:

ABCDEFGHIJK

-----------

999  998001       Chinese rule: 9/9->9+9, remember quotient digit 9!

999 1088001       ("carry run" to the left! Don’t panic!)

     -81          -9*9

999 1007001

      -81         -9*9

999  998901       Chinese rule: 9/9->9+9, remember quotient digit 9!

999 1007901       ("carry run" to the left! Don’t panic!)

      -81         -9*9

999  999801

       -81        -9*9

999  998991       Chinese rule: 8/9->8+8, remember quotient digit 8!

999  999791

       -72        -8*9

999  999071

        -72       -8*9

999  998999       finally, revising up

999  999          done!

On a 1/5 abacus, things are easier. We can use the 5th bead to avoid carry runs.

ABCDEFGHIJK

-----------

    ...

999  998901       Chinese rule: 9/9->9+9, remember quotient digit 9!

999  9T7901      

      -81         -9*9

999  999801

    ...           ...etc.

As we can see, we can do things this way but it does not seem like a very attractive method; we need memorization and a lot of attention to avoid making mistakes. So let's find a more convenient way.

Second way: Suspended lower beads

If we use a 1/5, instead of pushing the bead all the way up, effectively adding the small “1” to the interim quotient digit as in the previous case, it seems more reasonable to push it only halfway, leaving a suspended lower bead as illustrated at the top of the image below. This suspended bead will represent the overflow while respecting the integrity of the quotient digit.

This seems like a perfect method to deal with the overflow, both in division and multiplication, everything remains under our eyes and nothing has to be memorized. In fact, when using suspended lower beads there is no need for additional upper beads, and the 1/5 abacus becomes as powerful as the 2/5  or 3/5 instruments. This might help explain why the 1/5 abacus was so popular in the past and why the 5th bead survived for so long. Note in the bottom half of the figure that, with some complication, this method can also be extended to the 1/4 abacus. From here on,  I will use underlined digits to represent the overflow according to the figure, since the underline reminds me of what the suspended bead looks like and they don't mess up abacus diagrams typed with monospaced fonts.

1/5 abacus

Let us repeat the above exercise with this technique. The divisor is no longer represented and I some more details are also introduced to additionally illustrate how the fifth lower bead may be used in subtraction to somewhat simplify the operation (T is 10, 1 upper bead + 5 lower beads set)

1/4 abacus

And now on a 1/4 abacus. We need to use the suspended group of four lower beads as a code for 9:

If you have tried this, you have noticed that the group of four suspended beads behaves the same as the  suspended upper bead used on the 2/5 abacus; i.e. with "inverse arithmetic"^[3].

Third Way: Minimal memorization

I said above that using suspended lower beads seems a perfect method… but in fact it is somewhat annoying due to its inherent slowness. It is always difficult to suspend a bead, especially the small ones of modern soroban with little free space left on the rods, and this despite the silly trick of pinching the bead with two fingers and then retiring the hand as if taking a flower. I said that with an 1/5 abacus there is no need of additional upper beads, but no doubt, if you have a lot of multiplications or divisions to do, you will prefer the speed that those additional beads provide, since one rarely need to suspend a bead on the 2/5, and never on the 3/5.

Rather than physically moving/suspending the overflow bead, it is enough to think that the bead has been already suspended on the quotient rod, or pushed on an imaginary rod flying around your abacus, or simply remember that the “overflow status” has been set to ON and that it needs to be unset back to OFF as soon as possible. This last way is similar to the process of setting flags ON/OFF in old electronic calculators programming. Obviously, moving no bead is faster than moving any bead, so nothing can be faster than this alternative. Nevertheless, we should expect to need some practice to get used to this method and prepare to make some more mistakes due to memorization. However, memorizing a digit, as in the brute force method, is worse than simply memorizing an alert condition as required here.

No need for a new example. The previous ones can be followed under this new view simply by interpreting the underlines as something like OverflowFlag: ON.

Conclusion

We have seen here three techniques to deal with overflow that pushes the small “1” up on the interim quotient digit:

1. All way, effectively adding it as a carry to the quotient
2. Only half way, leaving a suspended lower bead
3. Nothing at all

These methods bring us the possibility of using traditional techniques and arrangements on any abacus type by simply adapting the mechanics to the presence/absence of additional beads. This is an advantage if you like traditional techniques.

I suspect that any or both of the two last methods could be the never-told way to deal with overflow used in ancient times. Consider that the second method can be demonstrated to others in just seconds, and that once seen, it is neither forgotten nor requires further explanation; It is so obvious. So there is not much need to write long texts to convey that knowledge.