Traditional Abacus and Bead Arithmetic/Division/Guide to traditional division (帰除法)

Introduction

Traditional division method (TD), kijohou, guī chúfǎ (帰除法), is one of the two main methods of division used with the abacus. This method makes use of both the multiplication table and a specific division table and has been the standard method studied with the abacus for at least 4 centuries, losing popularity in the 1930s. As a digit-by-digit or slow division algorithm, has been introduced in the previous chapter, where its special characteristic is revealed: it does not require thinking but only following some rules. This document is an introduction to its use on the abacus and it is assumed that the reader is already proficient in the modern division (MD) method.

The division table

In the previous chapter Modern and traditional division; close relatives, the following division table has been introduced

Division Table (八算, *Hassan*, *Bā suàn*)
1/9>1+1	2/9>2+2	3/9>3+3	4/9>4+4	5/9>5+5	6/9>6+6	7/9>7+7	8/9>8+8	9/9>9+9
1/8>1+2	2/8>2+4	3/8>3+6	4/8>5+0	5/8>6+2	6/8>7+4	7/8>8+6	8/8>9+8
1/7>1+3	2/7>2+6	3/7>4+2	4/7>5+5	5/7>7+1	6/7>8+4	7/7>9+7
1/6>1+4	2/6>3+2	3/6>5+0	4/6>6+4	5/6>8+2	6/6>9+6
1/5>2+0	2/5>4+0	3/5>6+0	4/5>8+0	5/5>9+5
1/4>2+2	2/4>5+0	3/4>7+2	4/4>9+4
1/3>3+1	2/3>6+2	3/3>9+3
1/2>5+0	2/2>9+2
1/1>9+1

where in each cell the result of the Euclidean division $(10\times a)/b=q,r$ ( $q$ : quotient, $r$ : remainder, $a,b$ digits from 1 to 9) is expressed in the form $a/b>q+r$ for reasons that we will see below. This means that the following hold:

$10a=q\cdot b+r$

The table has three zones corresponding to the following: If the divisor has $n$ figures and we compare it with the first n digits (from the left) of the dividend, with added trailing zeros if necessary, three cases can occur:

the dividend is greater than or equal to the divisor (ex. $770/689$ )
the dividend is less than the divisor and the first digit of the divisor is equal to the first digit of the dividend (ex. $670/689$ )
the dividend is less than the divisor and the first digit of the divisor is greater than the first digit of the dividend (ex. $570/689$ )

The blank cells below the diagonal of the division table above correspond to case 1. They could be filled in the style of the tables that can be seen elsewhere^[1], but we leave them empty for simplicity. If during the division we fall into this zone, we will proceed, for now, simply by revising up the previous quotient digit as we will see in the examples that follow.

The diagonal elements (in gray) correspond to the case 2 and can only occur if the divisor has at least two digits.

Finally, the other non-diagonal elements correspond to the case 3, which can be considered the most important to study.

There is no doubt that memorizing the division table takes time and effort and that you want to know if the traditional method of division is right for you before investing so much time and effort. Fortunately, the division by nine, five, and two tables are remarkably simple and can be memorized almost instantly (see below), as well as diagonal elements for multi digit divisors. This means that we can learn this traditional technique using divisors that start with only 9, 5 or 2 without much effort and thus be able to decide whether it is worth spending time learning the whole table or not. In what follows we will use examples based on such divisors.

Easy-to-memorize division rules
Diagonal	Divide by 9	Divide by 5	Divide by 2
1/1>9+1	1/9>1+1	1/5>2+0	1/2>5+0
2/2>9+2	2/9>2+2	2/5>4+0
3/3>9+3	3/9>3+3	3/5>6+0
4/4>9+4	4/9>4+4	4/5>8+0
5/5>9+5	5/9>5+5
6/6>9+6	6/9>6+6
7/7>9+7	7/9>7+7
8/8>9+8	8/9>8+8
9/9>9+9

Why do the division rules include remainders?

Suppose we are going to divide 35 by 9, the 3/9>3+3 rule tells us that we must use 3 as an interim quotient and the next step will be to subtract the chunk 3✕9=27 from 35, leaving a remainder of 8. If we also memorize the remainders, we can save this multiplication step as follows: we cancel, clear or erase the first digit of the dividend, in this case 3, then we add the remainder (3) to the next figure (5) of the dividend. In this way, we obtain the same result but without using the multiplication table. With one-digit divisors we will never have to resort to the multiplication table, and in the case of divisors with several figures, proceeding in the same way, we will save one of the necessary multiplications. We will see it on the abacus below, but first we need a few words about how we are going to arrange the division on the abacus.

Modern Division Arrangement (MDA)

It is am assumed throughout this textbook that the reader has already studied the modern abacus method, as typified in the work of Takashi Kojima^[2]. In the following examples we will illustrate traditional division using the same division layout that you are already familiar with so that you can more easily follow them and use your usual 4+1 type abacus if you want. We will call this layout Modern Division Arrangement (MDA), but this is not the way division was traditionally organized on the abacus. Later, I will introduce the Traditional Division Arrangement (TDA) which, as we will see, it has some advantages and some disadvantages, including the need (or at least the convenience) of using a specialized abacus with additional upper beads.

While using MDA you can use the same rules you already know about the unit rod if you need them.

Let us see the 35÷9 case from the above section, first without using the (rule) remainders

35÷9 without using the (rule) remainders
Abacus	Comment
ABCDEFGH
9 35	Divisor in A, dividend in GH, rule: 3/9>3+3
+3	enter quotient 3 in E
9 335
-27	subtract chunk 3✕9=27 from GH
9 3 8	new remainder/dividend in H
...	...

And now using the remainders

35÷9 using the (rule) remainders
Abacus	Comment
ABCDEFGH
9 35	Divisor in A, dividend in GH, rule: 3/9>3+3
+3	enter quotient 3 in E
9 335
-3	clear first dividend digit in G
9 3 5
9 +3	add remainder 3 to H
9 3 8	new remainder/dividend in H
...	...

That is:

When using MDA, the rule a/b>q+r must be read: “write q as interim quotient digit to the left, clear a and add r to the right”

One digit divisors

The number 123456789 has traditionally been used to demonstrate the use of multiplication and division tables in ancient Chinese^[3] and Japanese works^[4]^[5]. Here we will use it with the “easy divisors” 9, 5 and 2.

Example 123456789÷9=13717421

123456789÷9=13717421
Abacus	Comment
ABCDEFGHIJ	(Divisor not indicated)
123456789	Rule 1/9>1+1
+1	enter quotient 1 into A
-1	clear B
+1	add remainder 1 to adjacent digit
1 33456789	Rule 3/9>3+3
13 6456789	Rule 6/9>6+6
1361056789
+1-9	revising up
137 156789	Rule 1/9>1+1
1371 66789	Rule 6/9>6+6
1371612789
+1-9	revising up
13717 3789	Rule 3/9>3+3
1371731089
+1-9	revising up
137174 189	Rule 1/9>1+1
1371741 99
+1-9	revising up
1371742 9
+1-9	revising up
13717421	Done!

Example 123456789÷5=24691357.8

123456789÷5=24691357.8
Abacus	Comment
ABCDEFGHIJ	(Divisor not indicated)
123456789	Rule 1/5>2+0
2 23456789	Rule 2/5>4+0
24 3456789	Rule 3/5>6+0
246 456789	Rule 4/5>8+0
2468 56789
+1-5	revising up
2469 6789
+1-5	revising up
24691 1789	Rule 1/5>2+0
246912 789
+1-5	revising up
246913 289	Rule 2/5>4+0
2469134 89
+1-5	revising up
2469135 39	Rule 3/5>6+0
24691356 9
+1-5	revising up
24691357 4	Rule 3/5>6+0
246913578	Done!

Example 123456789÷2=61728394.5

123456789÷2=61728394.5
Abacus	Comment
ABCDEFGHIJ	(Divisor not indicated)
123456789	Rule 1/2>5+0
5 23456789
+1-2	revising up
6 3456789
+1-2	revising up
61 1456789	Rule 1/2>5+0
615 456789
+2-4	revising up twice
617 56789
+2-4	revising up twice
6172 16789	Rule 1/2>5+0
61725 6789
+3-6	revising up three times
61728 789
+3-6	revising up twice
617283 189	Rule 1/2>5+0
6172835 89
+4-8	revising up four times
6172839 9
+4-8	revising up four times
61728394 1	Rule 1/2>5+0
617283945	Done!

Multi Digit divisors

Consider, for example, $359936/9728=37$ , in this case it is convenient to think of the divisor as made up of a divider, the first digit, followed by a multiplier, the rest of the digits of the divisor, that is, $9728=dmmm$ , where $d$ is the divider (9) and $mmm$ is the multiplier (728). The Chinese and Japanese names for this division method (帰除 Guīchú in Chinese, 帰除法 Kijohou in Japanese) refer to this: 帰, Guī, Ki is the header and 除, chú, jo is the multiplier^[6].

In this case, the way to act is as follows:

First we consider only the divider $d$ and do exactly the same as in the case of the single digit divisor i.e. we follow the division rule: get the interim quotient $q$ and add the remainder (from the rule) to the adjacent column
Then we subtract the chunk $q\times {\text{multiplier}}$ from the remainder if we can; otherwise we have to revise down $q$ and restore $d$ to the remainder using the following rules:

Rules to revise down (two-digit divisors)
While dividing by	Revise q to	Add to remainder
1	q-1	+1
2	q-1	+2
3	q-1	+3
4	q-1	+4
5	q-1	+5
6	q-1	+6
7	q-1	+7
8	q-1	+8
9	q-1	+9

These rules are for two-digit divisors, for divisors with more digits things may be more complicated, as in MD (see example $23712/5928$ below). Let us see the above case

Example 359936÷9728=37

359936÷9728=37
Abacus	Comment
ABCDEFGHIJKLM
9728 359936	Rule 3/9>3+3
9728 3 89936	enter 3 to G, clear H and add 3 to I
-2184	subtract chunk 3✕multiplier 3✕728=2184^a from I-L
9728 3 68096	Rule 6/9>6+6
9728 3614096	enter 6 to H, clear I and add 6 to J
-4368	subtract chunk 6✕multiplier 6✕728=4368 from J-M
9728 36 9728	revising up
+1-9728
9728 37	Done!

Note: ^a This is an abbreviated notation meaning that 3✕7, 3✕2 and 3✕8 have to be subtracted from IJ, JK, and KL respectively.

Example 235÷59=3.98…

235÷59=3.98…
Abacus	Comment
ABCDEFGHIJ
59 235	Rule 2/5>4+0
59 4 35	enter 4 to E, clear F and add 0 to G
-36	cannot subtract chunk 4✕multiplier 4✕9=36 from GH!
-1+5	revise down following above rules
59 3 85
-27	subtract chunk 3✕multiplier 3✕9=27 from GH
59 3 58	Rule 5/5>9+5
59 3913	enter 9 to F, clear G and add 5 to H
-81	subtract chunk 9✕multiplier 9✕9=81 from HI
59 39 49	Rule 4/5>8+0
...	etc.

Example 23711÷5928=3,9998…

Caption text
Abacus	Comment
ABCDEFGHIJKLMN
5928 23711	Rule 2/5>4+0
5928 4 3711	enter 4 to G, clear H and add 0 to I
-36	subtract 4✕9=36 from IJ
5928 4 111
-8	subtract 4✕2=8 from JK
5928 4 31
-32	cannot subtract 4✕8=32 from KL!
-1+592	revise down and restore the subtracted excess to IJK
5928 3 5951
-24	continue normally, subtract 3✕8=24 from KL
5928 3 5927	Rule 5/5>9+5
...	etc.

Traditional Division Arrangement (TDA)

As commented above, there are two basic ways of arranging general division problems. Let us see them side by side:

Modern division arrangement (MDA), as explained by Kojima^[2],

MDA 25÷5=5
Abacus	Comment
ABCDEF
5 25	Dividend starting in E
5 5	After division quotient begins in D

Traditional division arrangement (TDA), as used in ancient books since the times of counting rods^[7] to the first part of the 20th century^[8],

TDA 25÷5=5
Abacus	Comment
ABCDEF
5 25	25÷5=5 Dividend starting in E
5 5	After division quotient begins in E

So far we have used MDA with the traditional division without any problem. TDA, however, is problematic with any division method, the traditional one included. This troublesome nature is due to a collision between the divisor and the dividend/remainder that occurs frequently (that is, both require the simultaneous use of the same column), and special techniques or abaci are needed to deal with this collision. Despite this, the TDA has been used for centuries in conjunction with the traditional method of division, at least since the 13th century, while the MDA has been shelved until modern times. It is clear that certain advantages can be recognized to TDA, but it is not so clear that they are enough to justify its historical use:

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 abacuses with a small number of rods, 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.

Suanpan showing 8 to 20 illustrating the use of "extra" and "suspended" beads. The values represented in each column range from 8 on the left to 20 on the right

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, but we will do it in chapter: Dealing with overflow!.

In the case of a 5+2 or 5+3 abacus we can use the additional upper bead(s) to represent values from 10 to 20, using the suspended bead (懸珠 xuán zhū in Chinese, kenshu in Japanese) in the 5+2 case .

The third or suspended bead is expected to be used only in about 1% of cases, which justifies the adoption of the 5+2 model as standard instead of the 5+3. (If you are interested in using TDA on any abacus, head over to the Dealing with overflow chapter to see how)

When using TDA, the rule a/b>q+r must be read: “change a into q as interim quotient digit and add r to the right”

For examples of TD using TDA, refer to the Traditional division examples chapter.

About the efficiency of TD

As you can see in the examples with single digit divisors, TD efficiency deteriorates as the divisor starts with lower figures in the sense that we have to revise up more frequently. We can say that the efficiency is zero when the divisor starts with 1; in fact, we don't even have division rules except 1/1>9+1 (which is statistically excessive, see chapter: Learning the division table). For this last case, the trick is to divide by 2 in situ (chapter: Division by powers of two) both divisor and dividend, which is very fast, and proceed to divide both results normally; now the divisor begins with a digit between 5 and 9. for example: $128/16$

Caption text
Abacus	Comment
ABCDEFGHI
16 128	Divide in situ by 2
8 64	Rule 6/8>7+4
8 7 8
+1-8	revising up
8 8	Done!

$128/16=(128/2)/(16/2)=64/8=8$

In other cases, our intuition and experience with MD could help us.

This lower efficiency of TD compared to MD is the price to pay to save us the mental work of deducting the interim quotient figure that we have to try.

References

↑ "割り算九九". Japanese Wikipedia. {{cite web}}: Unknown parameter |Language= ignored (|language= suggested) (help); Unknown parameter |accesdate= ignored (|access-date= suggested) (help); Unknown parameter |trans_title= ignored (|trans-title= suggested) (help)
↑ ^a ^b Kojima, Takashi (1954), The Japanese Abacus: its Use and Theory, Tokyo: Charles E. Tuttle Co., Inc., ISBN 978-0-8048-0278-9
↑ Xú Xīnlǔ (徐心魯) (1993) [1573]. Pánzhū Suànfǎ (盤珠算法) (in Chinese). Zhōngguó kēxué jìshù diǎnjí tōng huì (中國科學技術典籍通彙). {{cite book}}: Unknown parameter |trans_title= ignored (|trans-title= suggested) (help)
↑ Yoshida, Mitsuyoshi (吉田光由) (1634). Jinkoki (塵劫記) (in Japanese). {{cite book}}: Unknown parameter |trans_title= ignored (|trans-title= suggested) (help)
↑ Shinoda, Shosaku (篠田正作) (1895). Jitsuyo Sanjutsu (実用算術) (in Japanese). {{cite book}}: Unknown parameter |trans_title= ignored (|trans-title= suggested) (help)
↑ Lisheng Feng (2020), "Traditional Chinese Calculation Method with Abacus", in Jueming Hua; Lisheng Feng (eds.), Thirty Great Inventions of China, Jointly published by Springer Publishing and Elephant Press Co., Ltd, ISBN 978-981-15-6525-0
↑ Zhū Shìjié 朱士傑 (1993) [1299]. Suànxué Qǐméng (算學啟蒙) (in Chinese). Zhōngguó kēxué jìshù diǎnjí tōng huì (中國科學技術典籍通彙). {{cite book}}: Unknown parameter |trans_title= ignored (|trans-title= suggested) (help)
↑ Kwa Tak Ming (1922), The Fundamental Operations in Bead Arithmetic, How to Use the Chinese Abacus (PDF), San Francisco: Service Supply Co.