The eastern abacus (simplified Chinese: 算盘; traditional Chinese: 算盤; pinyin: suànpán, Japanese: そろばん soroban, simply the abacus in this textbook), as an abacus of fixed beads sliding on rods, originated in China at an uncertain date, but by the late 16th century its use had entirely displaced counting rods as a computing tool in its home country. From China its use spread to other neighboring countries, especially Japan, Korea and Vietnam, remaining as the main calculation instrument until modern times. The way in which it was used, the “Traditional Method”, remained stable for at least four centuries until the end of the 19th century, when an evolution began towards what we will call the “Modern Method”, that makes use of a “Modern Abacus”. This textbook is intended as an introduction to the traditional method, and is aimed at people who already know how to use a modern abacus using the modern method.
Modern abacus is of the 4+1 type, i.e. it has four beads on the lower deck and one on the upper deck.
4+1 abacus number representation
0
1
2
3
4
5
6
7
8
9
This is all that is needed to be able to perform decimal arithmetic with the abacus. However, traditional abacuses had additional beads, the most frequent being the 5+2 type (although the 5+1 type were also popular in Japan) and occasionally the 5+3 type.
Traditional 5+2 Chinese abacus illustrating the suspended bead use
With three upper beads we can store up to 20 on a single rod, which is convenient for traditional division and multiplication techniques. With two upper beads we can achieve the same by using the suspended bead technique (懸珠, Xuán zhū in Chinese[1], kenshu in Japanese), a kind of simulated or virtual upper third bead for the rare occasions when this third bead is required (see figure from 15 to 20).
5+2 abacus number representation
0
1
2
3
4
5
5
6
7
8
9
10
10
11
12
13
14
15
15
16
17
18
19
20
With a lower fifth bead, we have two different ways to represent the numbers 5, 10 and 15. This means that we have options from which we can choose the one that is most convenient for us. In the case of addition and subtraction, the possibility of choosing between two representations for 5 and 10 will allow us to simplify the calculations somewhat.
Traditional techniques can be used on any type of abacus, with only the obvious exception of the use of the lower fifth bead on a 4+1 abacus, the difference between having or not additional upper beads is more a matter of comfort and reliability than of efficiency or capabilities.
Traditional method was used for at least four centuries, covering Ming and Qing dynasties in China and Edo period in Japan. Beginning with the Meiji Restoration in Japan, students of the abacus changed in the sense that they already knew written mathematics before they began to study the abacus, whereas students of earlier times did not know anything about mathematics previously. For most, the abacus was the only form of math they were going to know. This caused a slow adaptation of the teaching and the methods of the abacus to the new times and circumstances, giving rise, after several decades, to what we now call the Modern Method, in fact, a simplified method.
In the English language, the following two works by Takashi Kojima are frequently cited in reference to the modern method:
Several editions of these books can still be found, including e-books formats, and the first one can be consulted at archive.org. In this wikibook, the reader is assumed to be familiar with the content of at least the first of these works.
Today, the modern method may seem optimal in many ways and we may think that some "oddities" of the traditional method, especially the way of organizing the division on the abacus, lack practical sense; but if the traditional method remained stable for centuries despite millions of users, including great figures of mathematics like Seki Takakazu who was a great promoter of the use of the soroban abacus in Japan, it can only be because it was also considered optimal by its users. Only the optimality criterion of the ancients differed from the one we may have today.
Unfortunately, no one in the past bothered to write why things were done that way, they just wrote about how to do things, and we can only speculate on the reasons underlying some of these ancient techniques.
Main differences between traditional and modern methods
These are the three most important points that differentiate traditional techniques from modern ones:
The use of the fifth lower bead in addition and subtraction to simplify both operations a bit, which extends to everything that can be done with the abacus since everything ultimately depends on addition and subtraction.
The use of a division method using a division table that eliminates the mental effort required to determine the quotient figure. This method (kijohou, guīchúfǎ 帰除法) first described in the Mathematical Enlightenment (Suànxué Qǐméng, 算學啟蒙) by Zhū Shìjié 朱士傑 (1299)[4] using counting rods superseded the old division method based on the multiplication table and whose origin dates back to at least the 3rd to 5th centuries AD, to the book The Mathematical Classic of Master Sun (Sūnzǐ Suànjīng 孫子算經)[5][6]. This old method, being the basis of the short and long methods of written division, has in turn replaced the traditional method of dividing in modern times. That is, modern times have taken us back to the old!
Traditional and modern methods also differ in the way the division operation was organized on the abacus. The traditional division arrangement is somewhat more compact than the modern one and also more problematic as it requires (or benefits from) the use of additional, higher beads. This arrangement of the division in turn conditions the way in which multiplication and roots are organized.
As mentioned above, no author in the past has written about why things were done this way, only about how to do things; so we can only guess to try to understand why. But the reader will see throughout this book that the traditional techniques suppose, by comparison to the modern ones, a reduction of the mental effort necessary to use the abacus. This is especially clear in the case of division that uses a division table, but also in the rest of the techniques that will be described since they effectively involve a reduction in the number and/or the extent of "gestures" required to complete an operation. We call gesture here to:
finger or bead mouvements
hand displacements
changes of direction
skipping rods (i.e. changing hand position from a starting rod to other non-adjacent rod)
and each of these gestures,
as a physical process, takes a time to complete,
as governed by our brains, requires our attention, consuming (mental or biochemical) energy,
as done by humans (not machines), has a chance to be done in the wrong way, introducing mistakes.
so that we can expect, by reducing the number and extension of these gestures, a somewhat faster, more relaxed and reliable calculation.
In view of the above, one is tempted to think that by adopting this principle of minimum effort, traditional techniques evolved in the sense of making life with the abacus easier, which could explain its validity throughout the centuries, but it is nothing more than a conjecture without documentary support.
If we think of the modern method, polarized towards simplicity, speed and effectiveness, we could say that it is the sprinter method while the traditional method is the Marathon runner method.
The reader, after following this textbook, will be able to draw their own conclusions about it.
As usual, in this book we will use tables to describe procedures on the abacus, for example:
Example of abacus procedure table
Abacus
Comment
ABCDEFGHIJKLM
896 412
This time the divisor goes to the left and the dividend to the right
896 512
Column E: rule 4/8>5+0, change 4 in E into 5, add 0 to F
896 512
cannot subtract E×B=5×9=45 from FG,
-1
revise down E: subtract 1 from E,
+8
add 8 to F
896 492
etc.
etc.
Where, on the left, either the digit by digit evolution of the state of the abacus or the current addition or subtraction operation is shown along with comments, on the right, about what is being done. The columns of the abacus are labeled with letters at the top (blank spaces represent unused / cleared rods).
This representation, which is perfect for the modern abacus, needs a couple of refinements to adapt it to the traditional abacus.
A column of a traditional abacus can contain a number greater than 9 and it is not possible to write its two digits in our table without disturbing its vertical alignment. To get around this, we will use underline notation for values between 10 and 19 and the first digit (one) will be represented by an underline on the preceding column (see chapter Dealing with overflow for a reason). For example, the situation represented below occurs shortly after starting the traditional division of 998001 by 999
A
B
C
D
E
F
G
H
I
K
J
L
M
9
18
9
0
0
1
0
0
0
0
9
9
9
and is represented in procedure table as
Using underline notation
Abacus
Comment
ABCDEFGHIJKLM
988001 999
Column B value is 18
Notation related to the use of the fifth lower beadAs seen above, numbers 5, 10 and 15 have two possible representations: using or not the 5th lower bead. When it is pertinent to distinguish between the two, we will use the following codes:
F: to denote a lower five (five lower beads activated) as opposed to:
5:upper five (one upper bead activated).
T: ten on a rod (one upper bead and five lower beads activated). On 5+2 type abacus, it is also a lower ten as opposed to t an upper ten (two upper beads activated).
Q:lower fifteen on a rod (two upper beads and five lower beads activated) as opposed to q upper fifteen (suspended upper bead on the 5+2, three upper beads activated on the 5+3).
With any abacus type, addition is simulated by gathering the sets of counters representing the two addends, while subtraction is simulated by removing from the set of counters representing the minuend a set of counters representing the subtrahend. Addition and subtraction are the only two possible operations on any type of abacus. Everything else has to be decomposed into a sequence of addition and subtraction.
There is hardly any difference between addition and subtraction with a modern abacus or a traditional one, if the reader already knows how to perform these two operations fluently with a modern abacus, he will also do well with a traditional one. The only two additional points to consider are:
use of the lower fifth bead to simplify the operations.
alternating rightward and leftward operation to save hand displacements.
The lower fifth bead can be used in addition and subtraction operations just like its companions. Its use is demonstrated in some ancient books such as: Computational Methods with the Beads in a Tray (Pánzhū Suànfǎ 盤珠算法) by Xú Xīnlǔ 徐心魯 (1573)[1], but over time it ceased to appear in the manuals, perhaps as a non-fundamental technique it was no longer explained in the concise books of the past but surely it continued to be taught verbally, as a trick to abbreviate the operations. We dedicate the following chapter: Use of the 5th lower bead to this subject.
Some old books on the abacus, for instance, Mathematical Track (Shùxué Tōngguǐ 數學通軌) by Kē Shàngqiān (柯尚遷) (1578)[2], demonstrate the addition using an alternating direction of operation with the obvious intention of saving hand movements. If the reader has already studied the modern abacus he knows for sure why it is preferable to operate from left to right, and this is not only a question of the use of the abacus. In the 19th century, the well-known Canadian-American astronomer Simon Newcomb, a renowned human computer, recommended the practice of adding and subtracting from left to right using pencil and paper in the introduction to his tables of logarithms[3].
Therefore, the alternation of direction of operation should be considered a secondary matter. If it is mentioned here, it is because regardless of its limited usefulness it is a very interesting exercise that can be difficult at first, resulting in a small challenge that can lead the reader to interesting reflections on the order of movement of the fingers; in particular, on whether carries and borrows should be done before or after.
It may be of interest to know that in the past people learned the abacus without having prior knowledge of mathematics, in particular without knowing anything like an addition or subtraction table; instead they memorized a series of mnemonic rules, verses or rhymes, short phrases in Chinese that indicated which beads had to be moved to result in the addition or subtraction of one digit to/from another digit[4][5][6]. We have an example in English of what these types of rules were like thanks to the booklet: The Fundamental Operations in Bead Arithmetic, How to Use the Chinese Abacus by Kwa Tak Ming[7], Printed in Hong Kong (unknown publisher and date), a work aimed to English-speaking Filipinos according to the author. Here are rules/rhymes/verses that appear on it and whose interpretation is left to the reader:
Addition rules
Subtraction rules
One; lower five, cancel four
One; cancel five, return four
Two; lower five, cancel three
Two; cancel five, return three
Three; lower five, cancel two
Three; cancel five, return two
Four ; lower five, cancel one
Four; cancel five, return one
One ; cancel nine, forward ten. (i.e. carry one to the left column)
One ; cancel ten (i.e. borrow one from the left column), return nine
Two; cancel eight, forward ten
Two; cancel ten, return eight
Three ; cancel seven, forward ten
Three; cancel ten, return seven
Four ; cancel six, forward ten
Four ; cancel ten, return six
Five; cancel five, forward ten
Five; cancel ten, return five
Six; cancel four, forward ten
Six; cancel ten, return four
Seven ; cancel three, forward ten
Seven; cancel ten, return three
Eight ; cancel two, forward ten
Eight; cancel ten, return two
Nine ; cancel one, forward ten
Nine ; cancel ten, return one
Six ; raise one, cancel five, forward ten
Six; cancel ten, return five, cancel one
Seven ; raise two, cancel five, forward ten
Seven ; cancel ten, return five, cancel two
Eight; raise three, cancel five, forward ten
Eight; cancel ten, return five, cancel three
Nine; raise four, cancel five, forward ten
Nine; cancel ten, return five, cancel four
Clearly, the table above does not contain the trivial rules ; eg. "to add two, activate two lower beads" or "to subtract 6, deactivate both an upper and a lower bead". In the event that we cannot proceed with such rules because we do not have the necessary beads at our disposal, then, we use the non-trivial rules listed in the table.
Once the students learned to add and subtract with these types of rules, they began to memorize the multiplication and division tables also in the form of verses or rymas. In total, learning the basics of the abacus required memorizing about 150 rules that had to be recited or sung while applied.
We will have a chance to see more rules by studying the traditional division in this book.
5+1 and 5+3 abacuses, Japan - Ridai Museum of Modern Science, TokyoFixed bead abacus, an inheritance from counting rods
It is a mystery why traditional Chinese and Japanese abacuses had five beads in their lower deck as only four are required from the point of view of decimal numbers representation. As no extant ancient document seems to explain it, this mystery will probably last forever and we are limited to conjectures to try to understand its origin. In this line, we could think that, when they first appeared, fixed beads abacuses were conceived in image and likeness of counting rods, from which they were called to inherit every algorithm. With counting rods, the use of five rods to represent number five was compulsory in order to avoid the ambiguity between one and five, at least initially, when neither a representation for zero nor a checkerboard a la Japanese sangi were used. Furnishing the abacus with five lower beads allowed a parallel or similar manipulations of beads and rods, bringing some kind of hardware and software compatibility to fixed beads abacuses, in fact, the first Chinese books on suanpan also dealt with counting rods, so that both instruments were learned at the same time. We could also invoke a certain desire for compatibility between the abacus and rod numerals that, in one way or another, have been in use until modern times. So, for instance, one would like to change all 5’s to be represented by the five lower beads before writing down a result using rods numerals, in order to avoid very silly and catastrophic transcription mistakes.
Counting rods, by the way the most versatile and powerful abacus ever, had a flaw: it is extremely slow to manipulate. It is not a surprise that ancient Chinese mathematicians invented the multiplication table to speed up multiplication and that they also discovered the use of this multiplication table to also speed up division. Nor is it a surprise that they also discovered that, by using the abacus fifth bead, addition and subtraction operations could be somewhat simplified. They really had to be very sensitive to slowness.
Yifu Chen, as part of his doctoral thesis[1], has systematically analyzed 16 classic works on the abacus: twelve Chinese books from the late Ming and Qing dynasties and four Japanese books from the Edo period in which addition and subtraction are studied. As a result, Chen finds four different modes of using the fifth bead in addition and two in subtraction. These modes range from intensive or systematic use of the fifth bead to sporadic use or no use at all. From all those texts, only one Chinese book from the late Ming dynasty make full use of the fifth bead, belonging to Chen’s Mode 1 in both addition and subtraction: Computational Methods with the Beads in a Tray (Pánzhū Suànfǎ 盤珠算法) by Xú Xīnlǔ 徐心魯 (1573)[2], by the way, the oldest extant book entirely devoted to the abacus. This fact should not lead us to the erroneous conclusion that the use of the fifth bead was a rarity of old times, since the books analyzed are neither treatises nor compendia on state-of-the-art abacus computation, but introductory manuals or textbooks for learning its use. Rather, it seems that the use or not of the fifth bead in these books corresponds more precisely to the didactic objective pursued by the authors, and that only Xu Xinlu considered it interesting to demonstrate it thoroughly from the beginning and included it in the syllabus of his course. It is mainly thanks to this work that we can rescue the traditional use of the fifth bead to simplify operations.
In what follows a small set of rules for the use of the fifth bead is presented along with their rationale and scope of use. These rules are not explicitly stated in any of the classical works, but can be inferred from the addition and subtraction demonstrations present in them, (especially in the Panzhu Suanfa) as is done in Chen's thesis
In what follows we will use these concepts and notation in reference to the use (or not) of the lower fifth bead.
Notation related to the use of the fifth lower bead
F: to denote a lower five (five lower beads activated) as opposed to:
5:upper five (one upper bead activated).
T: ten on a rod (one upper bead and five lower beads activated). On 5+2 type abacus, it is also a lower ten as opposed to t an upper ten (two upper beads activated).
Q:lower fifteen on a rod (two upper beads and five lower beads activated) as opposed to q upper fifteen (suspended upper bead on the 5+2, three upper beads activated on the 5+3).
carry: this represents number 1 when it is to be added to a column as a carry from the right (addition).
a1 Never use the 5th bead in addition except in the two cases that follow.
a2 4 + carry = F
a3 9 + carry = T
That is to say, when adding 1 to a rod you act as usual, for instance:
A
A
A + 1 =
4
5
and
A
B
A
B
B + 1 =
0
9
1
0
but when adding one as result of a carry you use the fifth lower bead:
A
B
A
B
B + 5 =
4
6
F
1
text
A
B
A
B
B + 5 =
9
6
T
1
You can see the above addition rules mentioned in a slightly different way in *Chen, Yifu (2018), "The Education of Abacus Addition in China and Japan Prior to the Early 20th Century", Computations and Computing Devices in Mathematics Education Before the Advent of Electronic Calculators, Springer Publishing, ISBN978-3-319-73396-8{{citation}}: Unknown parameter |editor1first= ignored (|editor-first1= suggested) (help); Unknown parameter |editor1last= ignored (|editor-last1= suggested) (help); Unknown parameter |editor2first= ignored (|editor-first2= suggested) (help); Unknown parameter |editor2last= ignored (|editor-last2= suggested) (help).
Rule a1 goal is simply to always leave an unused lower bead at our disposal in case the current column has to accept a future carry from the right, while rules a2 and a3 specify the use of the 5th bead in such a situation. Then, we can expect to obtain:
a reduced number of finger movements because we avoid to deal with both upper and lower beads
to avoid skipping rods and to reduce the left-right hand displacement span
to avoid any “carry run” to the left (think of 99999+1=999T0 instead of 99999+1=100000)
The above advantages are automatically realized by using rules a2 and a3, but rule a1 is of a different nature. Rule a1 is a provision for the future, it will simplify things if a future carry actually falls on the current column (which happens about 50% of the time on average), but it will simplify nothing otherwise. Rule a1 is so a kind of a bet (subtraction rules below are also of the same nature).
Rules a1 to a3 are for columns that can receive a carry, which excludes the rightmost column in normal (rightward) operation.
In inverse (leftward) operation, no column will receive a future carry from the right, so that rule a1 is out of scope and does not operate, but rules a2 and a3 should always be used. (This is mentioned because an ancient technique, now defunct, used leftward operation in alternation with normal operation to avoid long hand displacements. Not of general use but an extremely interesting exercise anyway).
Exceptionally, if you do know that some column will never receive a carry, you are also free of rule a1. (This seems a strange situation, but we need to introduce it to cope with the central part of the Test Drive below).
Both rules tend to leave activated lower beads at our disposal for the case we need to borrow from them in the future (it is like always holding small change in our pocket just in case), saving us some movements and/or wider or more complex hand displacements, such as borrowing from non-adjacent columns or skipping rods.
Is not automatically obtained, it is only fulfilled when we actually need to borrow from the present rod. This is similar to the case of addition rule a1.
Once more, the rightmost column is outside the scope of these rules as we will never borrow from it.
Also, In leftward or inverse operation we will never borrow from the current column, so these rules do not apply (which may be seen as an additional reason to prefer rightward operation in normal use).
Diagrams from Xu Xinlu's Panzhu Suanfa (1573) for the additionDiagrams from Xu Xinlu's Panzhu Suanfa (1573) for subtraction
It was common in ancient books on the abacus to demonstrate addition and subtraction using the well-known exercise that consists of adding the number 123456789 nine times to a cleared abacus until the number 1111111101 is reached, and then erase it again by subtracting the same number nine times (This exercise seems to have the Chinese name: Jiǔ pán qīng 九盤清, meaning something like clearing the nine trays). You can find the sequence of intermediate results of the Panzhu Suanfa in this 1982 article by Hisao Suzuki (鈴木 久男): Chuugoku ni okeru shuzan kagen-hou 中国における珠算加減法 (Abacus addition and subtraction methods in China)[3]. This is a Japanese text (spiced up with some classical Chinese) that deals with addition and subtraction methods as they appear in various Chinese books from the 16th century. In pages 12-17, the Panzhu Suanfa version of the 123456789 exercise is graphically displayed on the upper series of 1:5 diagrams. The short Chinese phrases below each bar specify how the current digit was obtained (Table 1 in the Appendix A below serves a similar purpose but in a different and more convenient way for us).
Using the addition rules explained above, we should get the following sequence of results each time we complete the addition of 123456789 (see Table 1 for more details):
at this point, adding 123456789 once more results in 1111111101, but this number appears in the Panzhu Suanfa as:
TTTTTTTT1
which cannot be obtained by the use of the above rules only. A similar situation occurs when repeating this exercise but starting with 999999999 instead of a cleared abacus (see Table 2), reaching 1TTTTTTTT0. This is why we introduced the last comment on the scope of addition rules above. It might be that, by inspection or intuition, we realize that using the 5th bead here does not generate any carry, so that we can overcome the a1 rule and proceed to this, somewhat theatrical, result ...
As it can be seen here, few F’s and T’s appear on the intermediate results, but a few more appear in the middle of calculation (Table 1), being immediately converted to 4’s and 9’s by borrowing, which is the purpose for which they were introduced. The F’s and T’s remaining on the intermediate results are only the unused ones.
Of course, the rules for addition can also be directly used in multiplication and the rules for subtraction in division, roots, etc.
Additionally, if using traditional division method (see chapter: [[../../Division/Modern and traditional division; close relatives/|Modern and traditional division; close relatives]]) on the 2:5 or 3:5 abacus, we can introduce an additional rule:
k1 Always use lower five’s, ten’s, and fifteen’s (F, T, Q) when adding to the remainder after application of the division rules.
This is so because, although we are adding to a rod, the next thing we will do is start subtracting from it (if the divisor has more than one digit). It is a kind of extension of the first rule for subtraction (s1). For instance, initiating 87÷98:
87÷98
Abacus
Comment
ABCDEFG
87 98
Dividend AB, divisor FG
8Q 98
A: Rule 8/9>8+8
-64
886 98
etc.
Just after application of the division rule 8/9>8+8 we should have:
Write caption here!
A
B
C
D
E
F
G
8
Q
0
0
0
9
8
By the way, you may sometimes find somewhat conflicting the use of the second rule for subtraction (s2) in Chinese division. For instance, 1167/32 = 36,46875
1167/32 = 36,46875
Abacus
Comment
ABCDEFG
32 1167
1/3->3+1 rule
32 3267
-3*2=-6 in f, use 2nd subtraction rule
-6
32 31T7
Now, which rule should be used here? 1/3->3+1 or 2/3->6+2 ? In fact, we can use any of them and revise up as needed, but it is faster to realize that the remainder is actually 3207 so that the second Chinese rule is the appropriate one, so, simply change columns EF to 62 and continue.
Abacus
Comment
ABCDEFG
32 3627
...
Finally, if you are using the traditional Chinese multiplication method or similar on the suanpan, you may face overflow on some columns, so that an additional rule:
It is clear that the use of the 5th bead may reduce the number of bead or finger movements required in some calculations (Think of 99999 + 1 = 999T0 vs. 99999 + 1 = 100000). An estimate based on the 123456789 exercise and some of its derivatives (see the next chapter) leads to a reduction of 10% on average (counting simultaneous movements of upper and lower beads separately). This is a modest reduction, but the advantage of the 5th bead goes beyond simply reducing the number of finger movements, as it also reduces the number and/or the extent of other hand gestures required in calculations (hand displacement, changes of direction, skipping rods,...). As already stated in the [[../../Introduction/|introduction]] to this book, each gesture:
as a physical process, takes a time to complete
as governed by our brains, requires our attention, consuming (mental or biochemical) energy
as done by humans (not machines), has a chance to be done in the wrong way, introducing mistakes
So, under this optic, we can expect that the use of the 5th bead will result in a somewhat faster, more relaxed and reliable calculation by reducing the total number of required gestures. It is not easy to measure this triple advantage using a single parameter.
Skipping columns, as Yifu Chen comments in his two works mentioned above, seems to have traditionally been viewed as something to be avoided as a possible source of errors. Without this concept the subtraction rule (s2) cannot be understood since it does not always lead to a reduction in the number of finger movements, but it always reduces the range of hand movement and the need to skip rods. Have you ever felt insecure with divisors or roots that contain embedded zeros? They force us to skip columns.
In any case, the advantage of using the fifth bead, although not negligible, is only modest, and each one must decide whether it is worth using it or not. After getting used to and becoming fluent in using the 5th bead, there is no better test of its efficiency than using a 4+1 abacus again and being sensitive to the amount of additional work required to complete tasks on it.
Heffelfinger, Totton; Hinkka, Hannu (2011). "The 5 Earth Bead Advantage". 算盤 Abacus: Mystery of the Bead. Archived from the original on August 1, 2021. {{cite web}}: Unknown parameter |accesdate= ignored (|access-date= suggested) (help)
You can practice using the fifth bead online with Soroban Trainer (see chapter: [[../../Introduction#External resources|Introduction]]) using this file 123456789-5bead.sbk that you should download to your computer and then submit it to Soroban Trainer (It is a text file that you can inspect with any text editor and that you can safely download to your computer).
Addition and subtraction/Extending the 123456789 exercise
As we have seen in the previous chapter, the "123456789 exercise", consisting of adding that number nine times to a cleared abacus until reaching the number 1111111101 and then subtracting it nine times until the abacus is cleared again, has been used since ancient times to illustrate and practice addition and subtraction. It is a convenient exercise because:
it is long enough to be a non-trivial exercise
if we do not return to the initial value (zero) it is a sign that we have made a mistake
we do not need a book or exercise sheet
uses many of the elementary cases of addition or subtraction of a digit to/from another digit
but it also has a couple of drawbacks:
it does not use all pairs of digits (ex. a 3 is never added to a 5)
after repeating it several times, one begins to mechanically memorize the exercise, so that we are no longer practicing addition and subtraction
To avoid these two problems we can extend the exercise in several ways.
Instead of using a cleared abacus, we can fill 9 columns with a digit (111111111, 222222222, etc.), this multiplies by 10 the number of exercises at our disposal. Now we are sure to use all possible cases of addition and subtraction digit by digit while mechanical memorization becomes harder.
The following table contains the intermediate values for reference. Such values are traversed from top to bottom in addition and from bottom to top in subtraction.
Table 1: The 123456789 exercise with background
+1..9
0
1
2
3
4
+1..9
0
000000000
111111111
222222222
333333333
444444444
0
1
123456789
234567900
345679011
456790122
567901233
1
2
246913578
358024689
469135800
580246911
691358022
2
3
370370367
481481478
592592589
703703700
814814811
3
4
493827156
604938267
716049378
827160489
938271600
4
5
617283945
728395056
839506167
950617278
1061728389
5
6
740740734
851851845
962962956
1074074067
1185185178
6
7
864197523
975308634
1086419745
1197530856
1308641967
7
8
987654312
1098765423
1209876534
1320987645
1432098756
8
9
1111111101
1222222212
1333333323
1444444434
1555555545
9
Table 1: The 123456789 exercise with background (continuation)
Additionally, instead of using the number 123456789 we can think of using any permutations of these digits that we are able to comfortably remember; for example, 987654321, the only one we will consider here. This gives us 10 other independent exercises for addition and subtraction practice. The following table shows us the intermediate values of this new series of exercises using a background.
In total, we already have 20 different exercises.
Table 2: The 987654321 exercise with background starting with addition
+9..1
0
1
2
3
4
+9..1
0
000000000
111111111
222222222
333333333
444444444
0
1
987654321
1098765432
1209876543
1320987654
1432098765
1
2
1975308642
2086419753
2197530864
2308641975
2419753086
2
3
2962962963
3074074074
3185185185
3296296296
3407407407
3
4
3950617284
4061728395
4172839506
4283950617
4395061728
4
5
4938271605
5049382716
5160493827
5271604938
5382716049
5
6
5925925926
6037037037
6148148148
6259259259
6370370370
6
7
6913580247
7024691358
7135802469
7246913580
7358024691
7
8
7901234568
8012345679
8123456790
8234567901
8345679012
8
9
8888888889
9000000000
9111111111
9222222222
9333333333
9
Table 2: The 987654321 exercise with background starting with addition (continuation)
If you still do not have enough with the 20 previous exercises, you can count on another 20 independent exercises just start by subtracting 123456789 or 987654321 from the background nine times, after which we will return the abacus to its original state by adding the number nine times. In doing so, sooner or later we will find negative numbers that we can handle on "the other side" of the abacus; that is, by borrowing 1 from a real or imaginary column located further to the left. Before ending the exercise, that borrowed 1 will be returned with a carry to its real or imaginary column, and we will be able to finish the exercise with the abacus in its original state.
Table 3: The 123456789 exercise with background starting with subtraction
-1..9
0
1
2
3
4
-1..9
0
000000000
111111111
222222222
333333333
444444444
0
1
9876543211
9987654322
98765433
209876544
320987655
1
2
9753086422
9864197533
9975308644
86419755
197530866
2
3
9629629633
9740740744
9851851855
9962962966
74074077
3
4
9506172844
9617283955
9728395066
9839506177
9950617288
4
5
9382716055
9493827166
9604938277
9716049388
9827160499
5
6
9259259266
9370370377
9481481488
9592592599
9703703710
6
7
9135802477
9246913588
9358024699
9469135810
9580246921
7
8
9012345688
9123456799
9234567910
9345679021
9456790132
8
9
8888888899
9000000010
9111111121
9222222232
9333333343
9
Table 3: The 123456789 exercise with background starting with subtraction (continuation)
-1..9
5
6
7
8
9
-1..9
0
555555555
666666666
777777777
888888888
999999999
0
1
432098766
543209877
654320988
765432099
876543210
1
2
308641977
419753088
530864199
641975310
753086421
2
3
185185188
296296299
407407410
518518521
629629632
3
4
61728399
172839510
283950621
395061732
506172843
4
5
9938271610
49382721
160493832
271604943
382716054
5
6
9814814821
9925925932
37037043
148148154
259259265
6
7
9691358032
9802469143
9913580254
24691365
135802476
7
8
9567901243
9679012354
9790123465
9901234576
12345687
8
9
9444444454
9555555565
9666666676
9777777787
9888888898
9
Table 4: The 987654321 exercise with background starting with subtraction
-9..1
0
1
2
3
4
-9..1
0
000000000
111111111
222222222
333333333
444444444
0
1
9012345679
9123456790
9234567901
9345679012
9456790123
1
2
8024691358
8135802469
8246913580
8358024691
8469135802
2
3
7037037037
7148148148
7259259259
7370370370
7481481481
3
4
6049382716
6160493827
6271604938
6382716049
6493827160
4
5
5061728395
5172839506
5283950617
5395061728
5506172839
5
6
4074074074
4185185185
4296296296
4407407407
4518518518
6
7
3086419753
3197530864
3308641975
3419753086
3530864197
7
8
2098765432
2209876543
2320987654
2432098765
2543209876
8
9
1111111111
1222222222
1333333333
1444444444
1555555555
9
Table 4: The 987654321 exercise with background starting with subtraction (continuation)
This is the most interesting proposal in the context of traditional methods. The forty exercises above can be performed using the lower 5th bead, as explained in detail in the previous chapter: Use of the 5th lower bead, which will allow you to master this traditional technique.
And finally, why not? Even if only for the pleasure of overcoming a different difficulty, we can combine the previous exercises with an alternating direction of operation, from left to right and from right to left, as explained in the introductory chapter to [[../../Addition and subtraction#Reverse operation/|addition and subtraction]].
Example
Abacus
Comment
ABCDEFGHIJ
Cleared abacus
+1
+2
+3
+4
+5
+6
+7
+8
+9
123456789
First step completed
+9
+8
+7
+6
+5
+4
+3
+2
+1
246913578
Second step completed
etc.
With this, you could go one step further in your understanding of bead mechanics.
With the 160 exercises presented here, you no longer have an excuse, you can practice addition and subtraction for hours, without exercise sheets, while comfortably seated on your sofa with only your abacus resting on your knees.
Sunzi division of 309 by 7 using counting rods309/7 = 441/7
Of the four fundamental arithmetic operations, division is probably the most difficult to learn and perform. Being basically a sequence of subtractions, there are a large number of algorithms or methods to carry it out and many of these methods have been used with the abacus[1][2]. Of these, two stand out for their efficiency and should be considered the main ones:
The modern division method (MD), shojohou in Japanese, shāng chúfǎ in Chinese (商除法); the oldest of the two, its origin dates back to at least the 3rd to 5th centuries AD, as it is cited in the book: The Mathematical Classic of Master Sun (Sūnzǐ Suànjīng 孫子算經). If we call it modern it is because it is the one taught today because it is the most similar to the division with paper and pencil. This method of division is based on the use of the multiplication table. During the Edo period it was introduced to Japan by Momokawa Jihei[3], but it did not gain popularity[4] until the 20th century with the development of what we have been calling the Modern Method.
The traditional division method (TD), kijohou in Japanese, guī chúfǎ in Chinese (帰除法), first described in the Mathematical Illustration (Suànxué Qǐméng, 算學啟蒙) by Zhū Shìjié 朱士傑 (1299)[5]. Its main peculiarity is that it uses a division table in addition to the multiplication table, which saves the mental effort of determining what figure of the quotient we have to try. In addition, we can design custom division tables for multi-digit dividers that save us the use of the multiplication table.
Both methods were first used in China with Counting rods.
In this Part of the book we deal primarily with the traditional method of division while assuming that the reader already has experience with the modern method of division.
In this chapter we try to show how modern and traditional methods, apparently so different, are actually closely related, while trying to justify why this method was invented.
How to cope with the traditional division arrangement (TDA) using different types of abacuses, especially the modern 4+1 and the traditional Japanese 5+1.
As explained in the previous chapter, there are two main methods of division used with the abacus: the modern division and the traditional division. The modern method of division (MD), shojohou in Japanese, shāng chúfǎ in Chinese (商除法) , is actually the oldest, dating back to approx. 200 CE and only makes use of the multiplication table. By comparison, the traditional method (TD), kijohou, guī chúfǎ (帰除法), is more recent but also very old, dating back to the times of counting rods, at least from the 13th century. This method makes use of both the multiplication table and a specific division table. TD has been the standard method studied with the abacus for at least 4 centuries[1][2], losing popularity in the 1930s. The reason for this is that modern abacus students already know how to divide with pencil and paper before embarking on the study of the abacus and, having a very tight study program or being very busy, it is not a question of spending time learning a new method of division and memorizing a new table, but of taking advantage of what is already known; MD is the closest thing to written long division that can be done on the abacus.
It would be difficult to say which of the two division methods is more efficient, Kojima[3] does not dare to say it, what does seem generally accepted is that the traditional method is more comfortable or relaxed since one does not have to think about anything, just follow the rules. From what follows, one can think that MD is somewhat more efficient (faster) than TD, one could say that while MD is for the sprinter, TD is for the Marathon runner; i.e. for those who have to spend many hours a day doing divisions…
At first glance it may seem that these two methods of division are very different from each other, we will show in what follows that the two methods are so similar and related that they can be considered close relatives (with MD being the older brother and TD the youngest) to the point that if you are already skilled in MD you are also skilled in TD! … although you don't know it yet and you are still far from getting all the power of TD.
For this purpose, we will go back to older division methods in order to position MD and TD within the framework of the Chunking methods[4] (sometimes also called the partial quotients method or the hangman method) which will allow us to show the extreme similarity of both approaches. After this, we will delve into the hidden beauty of TD and understand why it simplifies life with the abacus. In what follows we assume that you already know all about the modern division with the abacus, how to revise up and down, etc. as explained, for instance, by Kojima.
Let's take 1225÷35 = 35 as an example. There is no simpler way to proceed than by repeated subtraction and since 35 is greater than the first two digits of the dividend, we will start subtracting 35 from 122 using a column from the abacus as a counter.
1225÷35 = 35
Abacus
Comment
ABCDEFGHI
35 1225
Start, counter in D,
35 1 875
subtract 35 from GH, add 1 to counter D,
35 2 525
subtract 35 from GH, add 1 to counter D,
35 3 175
subtract 35 from GH, add 1 to counter D,
35 31 140
subtract 35 from HI, add 1 to counter E,
35 32 105
subtract 35 from HI, add 1 to counter E,
35 33 70
subtract 35 from HI, add 1 to counter E,
35 34 35
subtract 35 from HI, add 1 to counter E,
35 35 00
subtract 35 from HI, add 1 to counter E.
35 35
No remainder. Done, quotient is 35!
That was easy but a little long. If we can easily double the divisor and retain it in memory, we can shorten the operation by subtracting one or two times the divisor chunks.
times
chunks
1
35
2
70
1225÷35 = 35
Abacus
Comment
ABCDEFGHI
35 1225
Start, counter in D,
35 2 525
subtract 70 from GH, add 2 to counter D,
35 3 175
subtract 35 from GH, add 1 to counter D,
35 32 105
subtract 70 from HI, add 2 to counter E,
35 34 35
subtract 70 from HI, add 2 to counter E,
35 35 00
subtract 35 from HI, add 1 to counter E.
35 35
No remainder. Done, quotient is 35!
Or even better if we can build a table like the one below by doubling the divisor three times[5]:
times
chunks
1
35
2
70
4
140
8
280
1225÷35 = 35
Abacus
Comment
ABCDEFGHI
35 1225
Start, counter in D,
35 2 525
subtract 70 from GH, add 2 to counter D,
35 3 175
subtract 35 from GH, add 1 to counter D,
35 34 35
subtract 140 from HI, add 4 to counter E,
35 35 0
subtract 35 from HI, add 1 to counter E.
35 35
No remainder. Done, quotient is 35!
which is somewhat shorter and, clearly, nothing could be faster than having a complete multiplication table of the divisor
Multiplication table by 35
times
chunks
1
35
2
70
3
105
4
140
5
175
6
210
7
245
8
280
9
315
then
1225÷35 = 35
Abacus
Comment
ABCDEFGHI
35 1225
Start, counter in D,
35 3 175
subtract 105 from GH, add 3 to counter D,
35 35 00
subtract 175 from HI, add 5 to counter E.
35 35
No remainder. Done, quotient is 35!
There is no doubt, this is an optimal division method, nothing can be faster and more comfortable ... once we have a chunk table like the one above. But calculating the chunk table is time consuming and requires paper and pencil to write it and this extra work would only be justified if we have a large number of divisions to do with the same common divisor.
In 1617 John Napier, the father of logarithms, presented his invention to alleviate this problem consisting of a series of rods, known as Napier's Bones, with the one-digit multiplication table written on them and that could be combined to get the multiplication table of any number. For example, in our case
Chinese set of Napier's bones
Multiplication table by 35 using Napier's bones
1
35
2
70
3
105
4
140
5
175
6
210
7
245
8
280
9
315
There is no doubt that such an invention spread to the East and was used in conjunction with the abacus, but this use must be considered as exceptional; not everyone had Napier bones close at hand. Another tool is needed and that tool is the multiplication table learned by heart.
It should be noted that the above procedures do not exhaust the possibilities of the chunking methods. If you read The Definitive Higher Math Guide on Integer Long Division[4] article, you will be amazed at the variety of division methods that can be performed. Both MD and TD used in the abacus belong to this category, as we are going to see.
One of the key points of learning abacus is to be aware that this instrument allows us to correct some things very quickly and without leaving traces and this is specially useful in the case of division. So if we have to divide 634263÷79283, instead of busting our brain trying to find the correct quotient figure, we simply choose an approximate provisional or interim figure by simplifying the original problem to 63÷7 and test it by trying to subtract the chunk (interim quotient digit)✕79283 from the dividend; one of the following will occur:
The interim quotient digit is correct
It is excessive and we must revise it down
It is insufficient and we must revise it up
Let's see it applied to our previous example. Instead of directly trying to solve 1225÷35 we simplify and try to solve 12÷3 using the memorized multiplication by 3 table.
Instead of directly trying to solve the original problem 1225÷35 or the approximation used in MD 12÷3, we simplify still more and try to solve 10÷3; that is, we use a cruder approach to the original problem by ignoring the second digit of the dividend, so we must prepare to revise the interim quotient more frequently. By this change of focus from 12÷3 to 10÷3 we are adopting the philosophy of TD; it is only a slight variation of the chunking technique used in MD. This is why we can consider both division mechanisms as close relatives, members of the chunking methods family of division algorithms… and this is also why it can be said that if you are already proficient in modern division you are also already proficient in traditional division! but let us follow...
Continuing with our example
1225÷35
Abacus
Comment
ABCDEFGHI
35 1225
10÷3↦3 from multiplication table
+3
enter interim quotient in E
35 31225
Try to subtract chunk 3✕35 from FGH,
-09
first 3✕3 from FG
35 3 325
-15
then 3✕5 from GH
35 3 175
ok.
35 3 175
10÷3↦3
+3
enter interim quotient in F
35 33175
Try to subtract 3✕35 from GHI,
-09
first 3✕3 from GH
35 33 85
-15
then 3✕5 from HI
35 33 70
remainder greater than divisor (35)
+1-35
Revising up
35 34 35
remainder equal to divisor (35)
+1-35
Revising up again
35 35
No remainder, done! 1225÷35 = 35
Note that MD and TD, as explained so far, can be freely intermixed during the same division problem. This is an interesting and recommended exercise that allows you to compare both strategies side by side.
TS uses a simpler and lower approach to the original problem than MD, so that we can foresee some pros and cons
Pros
Some may consider this approach simpler
It will be necessary to revise down less frequently (revising down is usually more difficult and prone to mistakes than revising up)
Cons
We need to revise the interim quotient more frequently, which is an efficiency issue.
The previous two pros probably played a role in the development of the sophisticated technique we know as traditional division, but understanding why it was the preferred method for centuries, despite the above con, requires reflecting on the origin of the mental effort made during division and discovering the hidden beauty of TD.
When we learn the multiplication table we memorize a sequence of phrases like:
“nine times nine , eighty-one”
“nine times eight, seventy-two”
...
The order in which these phrases are learned can vary, but the structure of the phrases is similar in all languages, at least it is in Chinese and Japanese. It consists of a label that contains the two factors to be multiplied followed by the product. As soon as we think of the label, it, acting as an invocation, calls to our consciousness the value of the product. Let us represent it in the following way (read ➡ as the invocation):
Language
Label
Product
English
nine times nine
➡
eighty-one
Chinese
九九
➡
八十一
Japanese
くく
➡
はちじゅういち
Symbolic
9✕9
➡
81
How do we use this multiplication table during division? Let's think about our example above using shojohou or modern division method: 17÷3↦5, from the multiplication by three table we need the largest product that can be subtracted from 17. We need to scan in our memory (represented by ⤷) at least a few lines of said table and for each product rescued, see if it is less than 17 and choose the maximum of those less than 17. A complicated process that can be represented as:
3✕1
➡
3
3✕2
➡
6
⤷
3✕3
➡
9
yes
⤷
3✕4
➡
12
yes
⤷
3✕5
➡
15
yes
select this one!
⤷
3✕6
➡
18
no
3✕7
➡
21
3✕8
➡
24
3✕9
➡
27
This process is time and energy consuming. Computer specialists might find a similarity between this process and searching a relational database table on a non-indexed column; the inefficiency of such a search is well known. Creating a new index or key for that table based on the column and the search criteria can improve things drastically. Can we do something similar in our case to make the division more comfortable?
Indexing the multiplication table (division table)
To do something similar to indexing the multiplication table in terms of the products to facilitate the search, we should memorize new phrases that contain those products as labels; that is, phrases that begin with them; for instance:
Label
Quotient
3/3
1
6/3
2
9/3
3
12/3
4
15/3
5
18/3
6
21/3
7
24/3
8
27/3
9
That is, we have to memorize a division table, which is a hard work. Also think that the table above is not optimal in the sense that much of the numbers between 1 and 29 are missing; perhaps we should memorize a table of the following style instead:
Label
Quotient
Remainder
1/3
0
1
2/3
0
2
3/3
1
0
4/3
1
1
5/3
1
2
…
…
…
27/3
9
0
28/3
9
1
29/3
9
2
where the third column contains the remainders of the euclidean division. You will probably agree that memorizing such a table is out of the reach of ordinary humans (think of the table for 9!).
If we dedicate a lifetime to dividing with the abacus using MD method we would end up facing all elementary divisions of the type ab÷c where where a, b and c are digits and ab < c0, about 360 in total. However, if we were to use TD, we would be faced with all elemental divisions of the type a0÷c or (10✕a)÷c, only 36 in total! ... and this makes the memorization of a division table viable. In fact, to divide by 3 it is enough to memorize:
Label
Quotient
Remainder
10/3
3
1
20/3
6
2
or, in a more compact symbolic form
Rule
1/3 > 3+1
2/3 > 6+2
that we can use directly to solve our example without any thinking by simply choosing the figure suggested by the rule as the interim quotient.
1225÷35 = 35
Abacus
Comment
ABCDEFGHI
35 1225
Use rule 1/3 > 3+1
+3
enter interim quotient in E
35 31225
Try to subtract chunk 3✕35 from FGH,
-09
first 3✕3 from FG
35 3 325
-15
then 3✕5 from GH
35 3 175
ok.
35 3 175
Use rule 1/3 > 3+1
+3
enter interim quotient in F
35 33175
Try to subtract 3✕35 from GHI,
-09
first 3✕3 from GH
35 33 85
-15
then 3✕5 from HI
35 33 70
remainder greater than divisor (35)
+1-35
Revising up
35 34 35
remainder equal to divisor (35)
+1-35
Revising up again
35 35
No remainder, done! 1225÷35 = 35
but we have not yet made use of the remainder that appears in the rules after the plus sign; that and other issues will be covered in the next chapter.
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.
where in each cell the result of the Euclidean division (: quotient, : remainder, digits from 1 to 9) is expressed in the form for reasons that we will see below. This means that the following hold:
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. )
the dividend is less than the divisor and the first digit of the divisor is equal to the first digit of the dividend (ex. )
the dividend is less than the divisor and the first digit of the divisor is greater than the first digit of the dividend (ex. )
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.
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.
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”
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.
Consider, for example, , 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, , where is the divider (9) and 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 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 and add the remainder (from the rule) to the adjacent column
Then we subtract the chunk from the remainder if we can; otherwise we have to revise down and restore 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 below). Let us see the above case
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”
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:
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!
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.
↑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, ISBN978-981-15-6525-0
The division table contains 45 rules, including the 9 diagonal elements for multi-digit divisors.
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
The same number of independent elements that we find in the multiplication table (given the commutativity of this operation) whose memorization was one of the feats of our childhood in school. Memorizing the division table is therefore a similar task to learning the multiplication table.
These rules:
From an operational point of view, these rules should be read or interpreted slightly differently depending on whether we use the traditional (TDA) or the modern (MDA) division arrangement.
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”
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”
From a theoretical point of view, each rule expresses the result of a Euclidean division: (: quotient, : remainder, digits from 1 to 9) or, equivalently
If we think about this last point, in fact there is no need to memorize the division rules since we can obtain them in situ, when we need them, by a simple mental process. But then we would be making a mental effort similar to that required with the modern method of division and we would be moving away from the philosophy of the traditional method. There is no doubt, the efficiency and goodness of the traditional method is only achieved by memorizing the rules and we should only resort to the aforementioned mental process during the learning phase, when some rule resists coming to memory.
Fortunately, a series of patterns that appear in the division table come to our aid making it easier for us to learn it, leaving only 14 hard rules out of a total of 45.
In the chapter: Guide to traditional division (帰 除法) we already mentioned that the division rules by 9, 5 and 2, as well as the diagonal rules, have a particularly simple structure that allows almost immediate memorization.
Easy 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
For this reason, the examples presented in that chapter only made use of divisors starting with 2,5 and 9. If you practice several examples with such divisors, it will not be difficult for you to memorize these 22 rules (almost half of the total!); which is a drastic reduction in the work to be done and not the only one.
Of the remaining rules, the division by 8 series is the longest but not the most difficult, since it has an internal structure:
Division by 8 rules
1/8>1+2
5/8>6+2
2/8>2+4
6/8>7+4
3/8>3+6
7/8>8+6
4/8>5+0
Leaving aside 4/8>5+0 (think of this as 8x5 = 40), the two sub-series 1, 2, 3 and 5, 6, 7 have the same remainders and the quotients are as simple as 1, 2, 3 and 6, 7, 8; so, without a doubt, this will not be the series that will be the most difficult for you to learn.
Finally, as a last resort for learning, note the following series of terms adjacent to the diagonal of the table.
Subdiagonal rules
4/5>8+0
5/6>8+2
6/7>8+4
7/8>8+6
8/9>8+8
There are really only two new rules here, but grasping the structure of the table above will also help you memorize the rules for divisors 5, 8, and 9.
In summary, of the 45 rules included in the division table, 31 fall within one of the previous patterns (grayed)
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
and we are left with only 14 "hard" rules to memorize with no other help. This is no longer a huge job. Cheer up and don't give up! with some effort and practice, the greatest of the arcane mysteries of Traditional Bead Arithmetic will be yours!
What follows is a simple historical note with little or no practical relevance.
The multiplication table in the English language contains all the 81 two-digit products in any order; that is, it includes both 8x9 = 72 and 9x8 = 72, which is unnecessary given the commutativity of the multiplication. On the contrary, in Chinese it only contained one of the terms of these pairs 8x9 = 72; always with the first factor less than or equal to the second[1][2]. On the other hand, the division rules were enunciated by giving first the divisor that is always greater than the dividend, with the exception of the rules that we have called diagonals in which it is equal. This allows a combined multiplication-division table to be conceived that covers the entire "space" of pairs of digits as operands:
Combined table of multiplication and division rules
9✕9 81
9\8 8+8
9\7 7+7
9\6 6+6
9\5 5+5
9\4 4+4
9\3 3+3
9\2 2+2
9\1 1+1
8✕9 72
8✕8 64
8\7 8+6
8\6 7+4
8\5 6+2
8\4 5+0
8\3 3+6
8\2 2+4
8\1 1+2
7✕9 63
7✕8 56
7✕7 49
7\6 8+4
7\5 7+1
7\4 5+5
7\3 4+2
7\2 2+6
7\1 1+3
6✕9 54
6✕8 48
6✕7 42
6✕6 36
6\5 8+2
6\4 6+4
6\3 5+0
6\2 3+2
6\1 1+4
5✕9 45
5✕8 40
5✕7 35
5✕6 30
5✕5 25
5\4 8+0
5\3 6+0
5\2 4+0
5\1 2+0
4✕9 36
4✕8 32
4✕7 28
4✕6 24
4✕5 20
4✕4 16
4\3 7+2
4\2 5+0
4\1 2+2
3✕9 27
3✕8 24
3✕7 21
3✕6 18
3✕5 15
3✕4 12
3✕3 9
3\2 2+6
3\1 3+1
2✕9 18
2✕8 16
2✕7 14
2✕6 12
2✕5 10
2✕4 8
2✕3 6
2✕2 4
2\1 5+0
1✕9 9
1✕8 8
1✕7 7
1✕6 6
1✕5 5
1✕4 4
1✕3 3
1✕2 2
1✕1 1
Where we have altered the writing of our division rules to adapt them to the order of arguments used in Chinese. To highlight this fact we have replaced "/" by "\", so that the division rules as they appear in the above table must be interpreted in the form: Read a\b c+d: as: a divide into b0c times leaving d as remainder.
The combined table has 81 elements or rules, to which we must add the diagonal rules
Diagonal
1/1>9+1
2/2>9+2
3/3>9+3
4/4>9+4
5/5>9+5
6/6>9+6
7/7>9+7
8/8>9+8
9/9>9+9
and the rules for revising down given in the previous chapter.
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
that were studied separately. This adds up to a total of 99 rules to which we can add the approximately 50 addition and subtraction rules. The traditional learning of the abacus consisted fundamentally of the memorization and practice of these 150 rules.
What follows is a matter that arises from practice, not from any book in the past. The diagonal rules for divisors 1 and 2
Diagonal rules for 1 and 2
2/2>9+2
1/1>9+1
are excessive in the sense that we are often forced to revise up the divisor several times. In practice the following two statistical rules (to give them a name) behave better allowing a faster calculation.
Excluding the so-called "special methods", there are two basic ways of arranging general division problems. Not knowing a standard designation for them, we have called them in the chapter: Guide to traditional division:
Modern division arrangement (MDA), as explained by Kojima[3],
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 like the Jinkoki (塵劫記)[4], or the Panzhu Suanfa (盤珠算法)[5]
TDA 25÷5=5
Abacus
Comment
ABCDEF
5 25
25÷5=5 Dividend starting in E
5 5
After division quotient begins in E
Sunzi division (MD) traditionally used three horizontal rows of digits
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: The Definitive Higher Math Guide on Integer Long Division[6], and just using the beads of a 4+1 (modern) abacus. On the contrary, TDA is problematic with any division method since a collision between divisor and dividend/remainder frequently occurs, that is, both require the simultaneous use of the same column and, as this is not possible in principle, for example, in the case of modern division we would be forced to postpone the entry of the interim quotient digit in the abacus until the corresponding column be cleared by subtraction. As a result, 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 and the adoption of the modern abacus, even though MDA is the first idea that would occur to us if we tried to adapt the old division method used with counting rods (paradoxically MD!) to a single row instead of the usual three. Why? that could remain a mystery forever. However, certain advantages to TDA must be recognized:
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, one can 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.
Are they enough to justify its historical usage?
Regarding the traditional division (Guī chúfǎ, 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. This is not a problem with a traditional 5+2 or 5+3 abacus; As already explained, the additional upper beads can be used to store values as high as 20 in one column of the abacus. The problem arises when we think that 5+1 type abaci were popular in Japan during the Edo period and it seems that no ancient Japanese text explains how to deal with overflow. This is the question: What can be done on an 5+1 or 4+1 abacus?.
In a post to Soroban and Abacus Group, a member presented two examples of traditional division using an apostrophe (‘) to mark the columns or rods that temporarily received a value higher than 9 (overflow)[7].
Example from Soroban and Abacus Group
Abacus
Comment
ABC abcdef
898 888122
見八無頭作九八（Div. table）...
898 9'68122
九九八十一引（Mul. table）...
...
...
The apostrophe has the hindrance of breaking the vertical alignment of the columns of the abacus in the procedure tables, but let us think of this apostrophe as a typographical representation of a small 1 (¹), a bead that should be pushed, set or activated somewhere, be it on a real or imaginary column. 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.
The lone goose returns to its flock. An illustration for a traditional multiplication / division exercise with abacus.
We will use the classical exercise 998001÷999=999 as an example to illustrate the three mentioned alternatives. 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:
Abacus
ABCDEFGHIJK
999 998001
and if you have an almighty imagination, 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 and obtain 999!
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 998001999=999 example on an 4+1 abacus:
998001÷999 = 999; Brute force method on 4+1 abacus
Abacus
Comment
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 5+1 abacus, things are easier. We can use the 5th bead to avoid carry runs.
998001÷999 = 999; Brute force method on 4+1 abacus (2nd quotient digit)
Abacus
Comment
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 as we need memorization and a lot of attention to avoid making mistakes. So one should not attempt this method except as an exercise in concentration.
Suspended lower beads on 5+1 and 4+1 type abacuses (note the underline notation).
If we use a 5+1, 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 to the right. 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 5+1 abacus becomes as powerful as the 5+2 or 5+3 instruments. This might help explain why the 5+1 abacus was so popular in the past and why the 5th lower 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 4+1 abacus. From here on, We will use underlined digits to represent the overflow according to the figure, since the underline reminds us of what the suspended bead looks like and they don't mess up abacus procedure tables typed with monospaced fonts as the apostrophe does.
Let us repeat the above exercise with this technique. The divisor is no longer represented and some more details are also introduced to additionally illustrate how the fifth lower bead may be used in subtraction to somewhat simplify the operation (as usual, T is 10, 1 upper bead + 5 lower beads activated)
On an 5+1 abacus
Abacus
Comment
ABCDEF
998001
988001
Chinese rule: 9:9 > 9+9
-8
Subtract 81 from BC
9T8001
-1
9T7001
-8
Subtract 81 from CD
999001
-1
998901
997901
Chinese rule: 9:9 > 9+9
-8
Subtract 81 from CD
999901
-1
999801
-8
Subtract 81 from DE
998T01
-1
998991
998791
Chinese rule: 8:9 > 8+8
-7
Subtract 72 from DE
998T91
-2
998T71
-7
Subtract 72 from EF
9989T1
-2
998999
Revising up
-9
(from right to left to save a hand displacement)
998990
-9
998900
-9
998000
+1
999000
Done!
See also division examples for illustrations of this division on 5+1, 5+2 and 5+3 type abacuses.
And now on a 4+1 abacus. We need to use the suspended group of four lower beads as a code for 9:
On an 4+1 abacus
Abacus
Comment
ABCDEF
998001
988001
Chinese rule: 9:9 > 9+9
-81
Subtract 81 from BC
987001
-81
Subtract 81 from CD
998901
997901
Chinese rule: 9:9 > 9+9
-81
Subtract 81 from CD
999801
-81
Subtract 81 from DE
998991
998791
Chinese rule: 8:9 > 8+8
-72
Subtract 72 from DE
998071
-72
Subtract 72 from EF
998999
Revising up
999000
Done!
If you have tried this, you have probably noticed that the group of four suspended beads behaves the same as the suspended upper bead used on the 5+2 abacus; i.e. with "inverse arithmetic", if you move the suspended bead toward the abacus bean you are subtracting instead of adding!.
It has been 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 abacus 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. It is true that with an 5+1 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 additional beads provide, since one very seldom need to suspend a bead on the 5+2, and never on the 5+3.
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, around you..., 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.
We have seen here three techniques to deal with overflow on 4+1 and 5+1 abacuses that pushes the small “1” up on the interim quotient column:
All way, effectively adding it as a carry to the quotient
Only half way, leaving a suspended lower bead
Nothing at all (but in our minds)
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 finally end up convinced by traditional techniques.
It has been mentioned that no ancient Japanese text explains how to deal with overflow with a 5+1 abacus. Most likely the form used was one of the last two methods introduced here. 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.
The number 123456789 has also been used to demonstrate multiplication and division in many ancient books on the abacus. Some, like the Panzhu Suanfa[8], start with the traditional multiplication (see chapter: [[../../Multiplication/|Multiplication]]) of this number by a digit and use the division to return the abacus to its original state; others, like the Jinkoki[9], do it the other way around, starting with division and ending the exercise with multiplication. The latter is what we do here.
The number 123456789 is divisible by 3, 9 and 13717421, so divisions by 2, 3, 4, 5, 6, 8 and 9 have results with finite decimal expansion (2 and 5 are divisor of the decimal basis or radix 10 ). Only division by 7 leads to a result with an infinite number of decimal places, so here we will cut it off and give a remainder.
Unfortunately, this exercise does not use all the division rules, but it is a good start and allows you to practice without a worksheet.
You can practice traditional division online with Soroban Trainer (see chapter: [[../../Introduction#External resources|Introduction]]) using this file kijoho-1digit.sbk that you should download to your computer and then submit it to Soroban Trainer (It is a text file that you can inspect with any text editor and that you can safely download to your computer).
Suppose we have to perform a large number of divisions by 36525, which could be the case if we do calendar calculations. Then we can simplify the task by creating a specialized division table for this divisor. Following what is stated in the chapter: Guide to traditional division, we will start by calculating the following three Euclidean divisions:
Creating a special division table for 36525
100000÷36525
200000÷36525
300000÷36525
Quotient
Remainder
Quotient
Remainder
Quotient
Remainder
2
26950
5
17375
8
07800
Which can be summarized in the following specialized division table:
Division by 36525 table
1/36525>2+26950
2/36525>5+17375
3/36525>8+07800
And now we can use this table to do divisions without touching the multiplication table. For example, how many Julian centuries of 36525 days can fit in 1 000 000 days?
1000000/36525
Abacus
Comment
ABCDEFGHIJKLM
36525 1000000
Use rule: 1/36525>2+26950 on column G
36525 2000000
change 1 in G into 2
+26950
add 26950 to H-L
36525 2269500
Use rule: 2/36525>5+17375 on column H
36525 2569500
change 2 in H into 5
+17375
add 17375 to I-M
36525 2586875
revise up
+1
-36525
36525 2650350
revise up
+1
-36525
36525 2713825
Done! 1000000÷36525=27, remainder 13825
And we have done a division by a five-digit divisor without using the multiplication table!
Dividing by numbers that start with 1 is awkward, the following table may be used to divide by 19[2].
Division by 19 table
19
1
5+05
Division by 99 table
99
1
1+01
2
2+02
3
3+03
4
4+04
5
5+05
6
6+06
7
7+07
8
8+08
9
9+09
9801÷99
Abacus
Comment
ABCDEFGHI
9801 99
Dividend AD, divisor HI
9891 99
A: Rule 9/99>9+09
9899 99
B: Rule 8/99>8+08
+1
revising up
-99
99 99
Done! No remainder, quotient: 99
Dividing by 𝝅 is common in applications, here are the tables for two approximations of this irrational number.
Division by 𝝅 tables
314
31416
1
3+058
1
3+05752
2
6+116
2
6+11504
3
9+174
3
9+17256
Finally, the division by 666 table.
Division by 666 table
666
1
1+334
2
3+002
3
4+336
4
6+004
5
7+338
6
9+006
However, It is not advisable to divide by this number; results can be unpredictable… and uncontrollable! In any case, remember the advice:
I say to you againe, doe not call up Any that you can not put downe; by the Which I meane, Any that can in Turne call up somewhat against you, whereby your Powerfullest Devices may not be of use.
A fraction whose denominator only contains 2 and 5 as divisors has a finite decimal representation. This allows an easy division by powers of two or five if we have the fractions tabulated (or memorized) where is one of such powers of two or five.
For instance, given
Then
Which can easily be done on the abacus by working from right to left. For each digit of the numerator:
Clear the digit
Add the fraction corresponding to the working digit to the abacus starting with the column it occupied
Division 137/8 using fractions
Abacus
Comment
ABCDEF
--+---
Unit rod
137
enter 137 on A-C as a guide
7
clear 7 in C
+0875
add 7/8 to C-F
130875
3
clear 3 in B
+0375
add 3/8 to B-E
104625
1
clear 1 in A
+0125
add 1/8 to A-D
17125
Done!
--+---
unit rod
We only need to have the corresponding fractions tabulated or memorized, as in the table below.
In the past, both in China and in Japan, monetary and measurement units were used that were related by a factor of 16[3][4][5], a factor that begins with one which makes normal division uncomfortable. For this reason, it was popular to use the method presented here for such divisions.
The fractions for divisor 2 are easily memorizable and this method corresponds to the division by two "in situ" or "in place" explained by Siqueira[6] as an aid to obtaining square roots by the half-remainder method (半九九法, hankukuho in Japanese, Bàn jiǔjiǔ fǎ in Chinese, see Chapter: [[../../Roots/Square root/|Square root]]), it is certainly a very effective and fast method of dividing by two. Fractions for other denominators are harder to memorize.
Being a particular case of what was explained in the introduction above, to divide in situ a number by two we proceed digit by digit from right to left by:
clearing the digit
adding its half starting with the column it occupied
Let's take an example: . We do this multiplication by adding the 12 partial products that result from the expansion:
That is, all the products listed in this table:
Partial products of 345✕6789
✕
6000
700
80
9
300
1800000
210000
24000
2700
40
240000
28000
3200
360
5
30000
3500
400
45
But these products can be added in any of the (12 factorial) ways of ordering them, so we could say that there are, at least, almost 500 million ways to calculate the product of the two given numbers.
But it is clear that, of this immense number of ways of sequentially adding partial products, only a few can be efficiently and safely generated and followed by the human brain. But these few are still a lot ... especially if we think that we can also choose whether or not to enter multiplicand and multiplier in the abacus and where to start adding the partial products with respect to said operands. In what follows we will focus on this last aspect.
Addition and subtraction are inverse operations in the sense that each undoes the effect of the other by reverting the result to the first operand; for example: and now subtracting . On the abacus:
Reverting addition by subtraction
Abacus
Comment
ABC
422
+3
Add 313 to ABC
+1
+3
735
Result
-3
Subtract 313 to ABC
-1
-3
422
Result reverted
and, as we can see, we not only obtain the starting value but also obtain it in the original position. In turn, multiplication and division are also inverse operations; i.e: if
where is the quotient of dividing by and is the remainder, we can reverse the operation in the form:
for example: where 65 is the quotient and 47 the remainder and we can reverse the operation in the form . On the abacus, using modern division and multiplication methods:
4727÷72, modern method
Abacus
Comment
ABCDEFGHI
4727÷72
72 4727
Dividend:F-I, divisor:AB
72 64727
Try 6 as interim quotient
-42
Subtract 6✕7=42 from FG
-12
Subtract 6✕2=12 from GH
72 6 407
72 65407
Try 5 as interim quotient
-35
Subtract 5✕7=35 from GH
-10
Subtract 5✕2=10 from HI
72 65 47
Stop: quotient=65, remainder=47
72 65 47
Reverting by multiplication
+35
Add 5✕7=35 to GH
+10
Add 5✕2=10 to HI
72 65407
72 6 407
Clear F
+42
Add 6✕7=42 to FG
+12
Add 6✕2=12 to GH
72 64727
Clear E
72 4727
Done!
We have reversed the operation and returned the abacus to its original state. Note the relative position of operands and results using the modern method:
Relative position of operands and results (modern method)
Abacus
Comment
ABCDEFGHI
4727÷72
72 4727
Divisor & dividend
72 65 47
Divisor: AB, quotient: EF & remainder: HI
Now let's try the same with the traditional method of division (TD) and the traditional division arrangement (TDA).
4727÷72, traditional method
Abacus
Comment
ABCDEFGHI
4727÷72
72 4727
Dividend:F-I, divisor:AB
72 5227
Rule: 4/7>5+5 (overflow!)
-10
Subtract 5✕2=10 from GH
72 5127
+1
Revise up F
-72
Subtract 72 from GH
72 6407
72 6557
Rule: 4/7>5+5
-10
Subtract 5✕2=10 from HI
72 6547
Stop: quotient=65, remainder=47
now the relative position of operands and results using the traditional method is different:
Relative position of operands and results (traditional method)
Abacus
Comment
ABCDEFGHI
4727÷72
72 4727
Divisor & dividend
72 6547
Divisor: AB, quotient: FG & remainder: HI
If we want to reverse the operation by multiplication, we could first proceed by memorizing the digit of the multiplicand to use and clearing it, then we would proceed to add the partial products:
Reverting traditional division using TDA
Abacus
Comment
ABCDEFGHI
72 6547
Reverting by multiplication
72 6 47
Clear G and remember 5
+35
Add 5✕7=35 to GH
+10
Add 5✕2=10 to HI
72 6407
72 407
Clear F and remember 6
+42
Add 6✕7=42 to FG
+12
Add 6✕2=12 to GH
72 4727
Done!
and we have also reversed the operation and returned the abacus to its original state. In this way we proceed exactly the same as with modern multiplication, previously freeing up and reusing the space occupied by the digit in use of the multiplicand. However, memorizing and keeping something in memory while working with the abacus opens a door for errors and it is desirable to minimize this possibility by trying to keep the digit in memory for as little time as possible. This is achieved by altering the order in which we add the partial products:
Introducing traditional multiplication
Abacus
Comment
ABCDEFGHI
72 6547
Reverting by multiplication
+10
Add 5✕2=10 to HI
+35
Clear G and add 5✕7=35 to GH
72 6407
72 407
Clear F and remember 6
+12
Add 6✕2=12 to GH
+42
Clear F and add 6✕7=42 to FG
72 4727
Done!
As we can see, we have delayed clearing the digit in use until the last possible moment. This is the basis of the traditional method of multiplication.
The traditional method of multiplication was first introduced using counting rods[7] and the best way to introduce it to the modern abacist is to consider that a multi-digit multiplier consists of a head (the first digit from the left) and a body (the rest of the digits); for example: 4567✕23, considering 4567 as the multiplier, its head is 4 and the body 567. So, for each digit of the multiplicand (from right to left):
proceed as in modern multiplication with the product of the digit of the multiplicand by the body of the multiplier
then clear the digit of the current multiplicand and add its product by the head of the multiplier to the column just cleared and the one adjacent to its right
note: ^a Result is 10F041 if you use the 5th lower bead,105041 otherwise.
But things are not always as simple as in the previous example; if both the multiplicand and the multiplier contain high digits (7, 8, 9) we may have overflow problems and need to deal with them (see chapter: Dealing with overflow), as in the case 999✕999=998001:
999✕999 Traditional method
Abacus
Comment
ABCDEFGHIJK
Multiplicand:A-C, Multiplier: I-K
999 999
Head: I (9), Body: JK (99)
+81
Add 9✕9=81 to DE
+81
Add 9✕9=81 to EF
+81
Clear C and add 9✕9=81 to CD
998991 999
+81
Add 9✕9=81 to CD
+81
Add 9✕9=81 to DE
+81
Clear B and add 9✕9=81 to BC
988901 999
(overflow!)
+81
Add 9✕9=81 to BC
+81
Add 9✕9=81 to CD
+81
Clear A and add 9✕9=81 to AB
888001 999
(double overflow!)
998001 999
Normalizing result, done!
The most convenient, as in the case of division, is to have additional upper beads, that is, a 5+2 type abacus or a 5+3 if we are lucky enough. For the 4+1 and 5+1 abaci, it may be best to use the following fallback to the method outlined in the previous section, clearing the current digit of the multiplicand at the beginning (or when necessary) to have space to hold the partial results; for instance:
999✕999 Traditional method (fallback for 4+1 and 5+1 abaci)
Abacus
Comment
ABCDEFGHIJK
Multiplicand:A-C, Multiplier: I-K
999 999
+81
Clear C, remember 9 and add 9✕9=81 to CD
+81
Add 9✕9=81 to DE
+81
Add 9✕9=81 to EF
998991 999
+81
Clear B, remember 9 and add 9✕9=81 to BC
+81
Add 9✕9=81 to CD
+81
Add 9✕9=81 to DE
998901 999
+81
Clear A, remember 9 and add 9✕9=81 to AB
+81
Add 9✕9=81 to BC
+81
Add 9✕9=81 to CD
998001 999
If you practice the previous example together with the two traditional exercises: 898✕989, using 898 both as multiplier and multiplicand, you will be prepared for any traditional multiplication problem.
↑Siqueira, Edvaldo; Heffelfinger, Totton. "Kato Fukutaro's Square Roots". 算盤 Abacus: Mystery of the Bead. Archived from the original on August 1, 2021. {{cite web}}: Unknown parameter |accesdate= ignored (|access-date= suggested) (help)
↑Volkov, Alexei (2018), "Visual Representations of Arithmetical Operations Performed with Counting Instruments in Chinese Mathematical Treatises", Researching the History of Mathematics Education - An International Overview, Springer Publishing, ISBN978-3-319-68293-8{{citation}}: Unknown parameter |editor1first= ignored (|editor-first1= suggested) (help); Unknown parameter |editor1last= ignored (|editor-last1= suggested) (help); Unknown parameter |editor2first= ignored (|editor-first2= suggested) (help); Unknown parameter |editor2last= ignored (|editor-last2= suggested) (help)
Kojima, Takashi (1963), "III Other multiplication methods", Advanced Abacus: Theory and Practice, Tokyo: Charles E. Tuttle Co., Inc., ISBN978-0-8048-0003-7
Obtaining square and cubic roots are the most complex operations studied within Elemental Arithmetic. The eastern abacus is very well adapted to obtaining square roots by a direct and efficient procedure, but unfortunately the same cannot be said with reference to cube roots which, although possible, require a tortuous path full of comings and goings that it is prone to errors.
Cargill Gilston Knott (1856 - 1922), one of the fathers of modern seismology, was a Scottish physicist and mathematician who served nine years as a professor of mathematics, acoustics and electromagnetism at the Imperial University of Tokyo; after which he was awarded the Order of the Rising Sun by Emperor Meiji in 1891. During his stay in Japan he came into contact with the Japanese abacus which he studied in depth and without a doubt he used professionally in his own work as a teacher and researcher. The result of this study was a famous 55 pages article[1] written in 1885 about it, which for a long time has been the best informed account in English, and obligatory reference, on the history and foundations of soroban. The next two chapters in this book develop and expand on Knott's vision of the traditional methods of obtaining square and cube roots, the vision of a western scientist and mathematician, offering both a theoretical and practical approach illustrated with several examples.
Obtaining square and cubic roots with the abacus can be a somewhat long process and during the learning phase it is interesting to have a tool that allows us to control whether we are doing it correctly.
For square roots you can try the excellent Murakami's Square root tutor with Kijoho, a JavaScript application that you can run directly in your browser or download to your computer from its GitHub repository. You just have to enter the root in the small input box on the left and repeatedly press the "next" button on the screen to see the development of the process step by step.
And mainly for cube roots, the following BC code may help, copy and paste it to a text file and call it knott.bc:
/* Functions to help to learn/verify square and cube roots a la Knott with the abacus, soroban, suanpan. See: https://jccabacus.blogspot.com/2021/06/roots-la-knott.html as a reference. Jesús Cabrera, June 2021 CC0 1.0 Universal (CC0 1.0) Public Domain Dedication Use at your oun risk!*/define int(x){
# Integer part of x
auto os,r
os=scale;scale=0
r=x/1scale= os
return(r)}define cbrt(x){
# Cube root of x
return(e(l(x)/3))}define knott2(r0, y0, alpha){/* Square root following Cargill G. Knott steps See example of use in file sr200703.bc use: $ sr200703.bc |bc -l knott.bc*/auto so, div
so =scale;/* Store old scale value */scale=1
a =10*y0
div =100*r0+ alpha/2print"New dividend: ",div/1,"\n"
b = int(div/(a))
tf = div -b*a -b^2/2if(tf<0){
b=b-1;print"Revising down, b = ",b,"\n"
tf = div -b*a -b^2/2}print"New root: ", a+b,", New half-remainder: ", tf/1print"\n==================\n\n"scale= so;/* restore old scale value */return}define knott3(r0, y0, alpha){/* Cube root following Cargill G. Knott steps See example of use in file cr488931400152.bc use: $ cat cr488931400152.bc |bc -l knott.bc*/auto so, div, ta, tb, tc, td, te
so =scale;/* Store old scale value */scale=0
a =10*y0
div =1000*r0+ alpha
print"New dividend: ",div,"\n\n"
ta = div/y0; rem1= div % y0print"a) /a: ", ta," rem1: ", rem1,"\n"
tb =(10*ta)/3; rem2=(10*ta)%3print"b) /3: ", tb," rem2: ", rem2,"\n"
b = tb/(100*a)print" b = ",b,"\n"
tc = tb - b*(a+b)*100print"d) : ",tc,"\n"
b = tb/(100*(a+b))print" b = ",b,"\n"
tc = tb - b*(a+b)*100print"d) : ",tc,"\n"if(b==10){/* Trick to avoid some problems */
b =9print"b: ",b,"\n"
tc = tb - b*(a+b)*100print"d) tc: ",tc,"\n"}
td = tc*3+rem2print"e) *3: ",td,"\n"
te =(td/10)*y0+rem1print"f) *a: ",te,"\n"
tf = te - b^3print"g) -b^3: ",tf,"\n"print"\nNew root: ",(a+b)," New remainder: ",tf,"\n\n"print"==================\n\n"scale= so;/* restore old scale value */return}
/*
Example: square root of 200703
Use:
$ cat sr200703.bc |bc -l knott.bc
or
$ bc -l knott.bc < sr200703.bc
*/
print "\nSquare root of ", 200703, " = ", sqrt(200703), "\n\n"
/*
Decompose in pairs of digits (will be alpha): 20, 07, 03
Initialize (first step)
*/
alpha = 20
b = int(sqrt(alpha))
r0 = alpha - b^2
a = 0
tf = r0/2
print "First root: ", b, ", First half-remainder: ", tf, "\n"
print "==================\n\n"
/*
Main:
Repeat for each pair of digits (alpha)...
*/
alpha =07
mute=knott2(tf, a+b, alpha)
alpha =03
mute=knott2(tf, a+b, alpha)
/*
For additional digits continue with alpha = 00
*/
alpha =00
mute=knott2(tf, a+b, alpha)
alpha =00
mute=knott2(tf, a+b, alpha)
alpha =00
mute=knott2(tf, a+b, alpha)
alpha =00
mute=knott2(tf, a+b, alpha)
Output:
Square root of 200703 = 447.99888392718122931160
First root: 4, First half-remainder: 2.00000000000000000000
==================
New dividend: 203.5
Revising down, b = 4
New root: 44, New half-remainder: 35.5
==================
New dividend: 3551.5
Revising down, b = 7
New root: 447, New half-remainder: 447.0
==================
New dividend: 44700.0
Revising down, b = 9
New root: 4479, New half-remainder: 4429.5
==================
New dividend: 442950.0
New root: 44799, New half-remainder: 39799.5
==================
New dividend: 3979950.0
New root: 447998, New half-remainder: 395998.0
==================
New dividend: 39599800.0
New root: 4479988, New half-remainder: 3759928.0
==================
Let be the number of which we want to obtain the square root ; Let's consider its decimal expansion, for example: . Let's separate its digits into groups of two around the decimal point in the following way
or, in other words, let's define the sequence of integers
and let's build the sequence recursively from
and let be the integer part of the square root of
i.e. is the largest integer whose square is not greater than . Finally, let us call remainders to the differences
For our example we have:
0
0
0
0
1
4
4
2
0
2
56
456
21
15
3
78
45678
213
309
4
90
4567890
2137
1121
5
12
456789012
21372
26628
⋯
Let's see that, by construction, grows as (two more digits in each step), in fact the sequence : (0, 400, 456, 456.78, 456.7890, etc.) tends to or . By comparison, , as the integer part of the square root of , grows only as (one digit more each step). As is the largest integer whose square is not greater than we have as above but
by definition of , or
multiplying by
but as grows only as , the second term tends to zero as . With this
and so that we have
For other numbers, the above factors are: and , where is the number of two-digit groups to the left of the decimal point, negative if it is followed by 00 groups (ex. for , for , etc.).
This is the basis for traditional manual square root methods.
For and , it is trivial to find such that its square does not exceed through the use of the following table of squares that is easily retained in memory as it is only a subset of the multiplication table. In the case of the example we find .
For , we have as defined above and we try to build in the way:
where is a one-digit integer ranging from 0 to 9. To obtain we have to choose the greatest digit from 0 to 9 such that:
or
if we write . Expanding the binomial we have
or
The left side of the above expression may be seen as simply the previous remainder with the next two-digits group appended to it, and the parenthesis of the last term as twice the previous root with digit b appended to it. In our example, for we have 56 on the left and the above expression is
which holds only for or so that 1 is our next root digit but, how can we proceed in the general case without having to systematically explore every possibility ()?
Here Knott[1] distinguishes two different approaches:
And it is the strategy usually used with paper and pencil and can also be implemented, of course, on the abacus. In the above expression, if we see the left part as the dividend and the parenthesis as the divisor, b is the first digit of the division
but since we don't know b yet, we approximate it using only the main part of the divisor
This gives us a guess as to what the value of b might be, but we need:
Verify that the value thus obtained is correct, or, where appropriate, correct it up or down as needed.
Obtain the next remainder to prepare the calculation of the next digit of the root.
Both steps require subtracting or and from ; checking that we are not going to negative values and that what remains is less than (otherwise we would have to revise up or down). If we do this correctly, what we are left with is the new remainder . It should be noted that, as we proceed in the calculations ( increasing) is progressively a smaller and smaller contribution to the divisor ; so the process indicated above will look more and more like a mere division.
This is the method proposed by Takashi Kojima in his second book: Advanced Abacus - Theory and Practice[2], and that you can see described in Square roots as solved by Kojima[3] in Totton heffelfinger’s website, works to which I refer the reader to see explanations and examples. What follows here, for purposes of illustration, is a sketch of how the calculation might be started in our example:
Napier's bones arranged to help with the third root digit of the example
First 3 digits of preparing the divisor
Abacus
Comment
ABCDEFGHIJKLM
4567890123
Entering radicand starting in CD (first group)
2
First root digit in B
-4
Subtract square of B from first group
2 567890123
Null remainder
4 567890123
Doubling B. Appending next group (56)to remainder
41 567890123
5/4≈1, try 1 as next root digit
-4
Continue division by 41, subtract 1✕41 from EF
-1
41 157890123
15 as remainder
42 157890123
Double second root digit
42 157890123
Append next group (78)
423157890123
157/42≈3, try 3 as next root digit
-12
Continue division by 423, subtract 3✕423 from E-H
-06
-09
423 30990123
309 as remainder
426 30990123
Double third root digit
426 30990123
Append next group (90)
etc.
As can be seen, twice the root grows to the left of the abacus to the detriment of the radicand / remainder and the groups of two figures still unused. This is contrary to what happens with the rest of the elementary operations on the abacus, where the result sought replaces the operand (or one of them). For this reason, the traditionally preferred method for obtaining square roots seems to have been the following, where we will see the root appear directly on the abacus, not its double .
This modified expression will allow us to directly obtain the square root in the abacus following practically the same procedure as above with only keeping half-remainders on our instrument. Here, with square roots, the change is almost trivial, but it will be more important when dealing with cube roots. As can be seen in the above expression, neglecting the term we obtain a guess of by simply dividing the extended half-remainder by the previous root (in fact ). After that, we need again:
Verify that the value thus obtained is correct, or, where appropriate, correct it up or down as needed.
Obtain the next half-remainder to prepare for the next digit of the root.
This is done by subtracting and from the half remainder, and this makes it convenient to memorize the following table of semi-squares:
Semi-squares table
1
0.5
2
2
3
4.5
4
8
5
12.5
6
18
7
24.5
8
32
9
40.5
Fortunately, since 2 is a divisor of our base (10), the decimal fractions in the table have a finite expression; which will not happen when we try to extend this procedure to cube roots and we have to deal with thirds of cubes. According to Knott, this makes cube roots a problem that is not well suited to being treated with abacus.
In this example we have two groups of two figures: 09 and 61. The first group informs us that the first digit of the root is 3.
There are two ways to start square roots:
Aligning the groups to the left of the abacus from column B and using the traditional division to obtain the semi-remainder.
A
B
C
D
E
...
0
9
6
1
This is the form used in the old books and also the one used in Murakami's Square root tutor with Kijoho (see below External resources).
Using traditional division
Abacus
Comment
ABCDE
0961
Align the radicand with B
30961
Enter first root digit in A
-9
Subtract the square of first root digit (9)
30061
30305
Divide the remainder B-E by 2 (帰除法)
Aligning the groups to the left of the abacus from column A and using [[../../Division/Division by powers of two#Division by 2 in situ|in-situ]] division, as explained in chapter: [[../../Division/Division by powers of two|Division by powers of two]], to obtain the semi-remainder.
A
B
C
D
E
...
0
9
6
1
This method is somewhat faster.
Using division in situ
Abacus
Comment
ABCDE
0961
Align the radicand with A
-9
Subtract the square of first root digit (9)
0061
0305
Divide in situ the remainder by 2
30305
Enter first root digit in A
From here, the state of the abacus coincides and we can continue:
The root 2137… (21.37…) is appearing on the left. See chapter: [[../../Abbreviated operations#Square root|Abbreviated operations]] to see how to quickly approximate the next 4 digits.
The method explained as: Preparing the dividend is known as 半九九法 (Hankukuhou in Japanese, Bàn jiǔjiǔ fǎ in Chinese) that we freely translate here as the Half-remainder method and is by far the most convenient to use, at least for two reasons:
The root, and not its double, replaces the operand (radicand) as in the rest of operations on the abacus.
(The most important) Dividing by numbers that start with 1 is awkward. Think of the first group of two digits, its value is between 1 and 99 and determines the first figure of the square root. For values of the first pair between 25 and 99 (75% of the cases) the first digit of the root is between 5 and 9 and its double begins with one! Therefore, if we use the method preparing the divisor, we will be dividing by numbers that begin with 1 in 75% of the cases. On the contrary, if we use the method preparing the dividend, only in the case that the first group is 1, 2 or 3 (3% of the cases) will we have to divide by numbers that start with one.
The superiority of the half-remainder or preparing the dividend method is undeniable.
Siqueira, Edvaldo; Heffelfinger, Totton. "Kato Fukutaro's Square Roots". 算盤 Abacus: Mystery of the Bead. Archived from the original on August 1, 2021. {{cite web}}: Unknown parameter |accesdate= ignored (|access-date= suggested) (help)
Murakami's Square root tutor with Kijoho, a JavaScript application that you can run directly in your browser or download to your computer from its GitHub repository. You just have to enter the root in the small input box on the left and repeatedly press the "next" button on the screen to see the development of the process step by step. So you can generate as many examples or exercises as you want.
Let be the number of which we want to obtain the cube root ; Let's consider its decimal expansion, for example: . Let's separate its digits into groups of three around the decimal point in the following way
or, in other words, let's define the sequence of integers
and let's build the sequence recursively from
and let be the integer part of the cube root of
i.e. is the largest integer whose cube is not greater than . Finally, let us call remainders to the differences
For our example we have:
0
0
0
0
1
456
456
7
113
2
789
456789
77
256
3
012
456789012
770
256012
4
300
456789012300
7701
78119199
5
000
456789012300000
77014
6949021256
⋯
Let's see that, by construction, grows as (three more digits in each step), in fact the sequence , i.e. 0, 400, 456, 456.789, 456.789012, etc. tends to (). By comparison, , as the integer part of the cube root of , grows only as (one digit more each step). As is the largest integer whose square is not greater than we have as above but
by definition of , or
multiplying by
but as grows only as , the second term tends to zero as .
and so that we have
For other numbers, the above factors are: and , where is the number of three-digit groups to the left of the decimal point, negative if it is followed by 000 groups (ex. for , for , etc.).
This is the basis for traditional manual cube root methods.
For . It is trivial to find such that its square does not exceed through the use of the following table of cubes that can be easily retained in memory. In the case of the example it is
For , we have as defined above and we try to build in the way:
where is a one-digit integer ranging from 0 to 9. To obtain we have to choose the greatest digit from 0 to 9 such that:
or
if we write . Expanding the binomial we have
or
The left side of the above expression may be seen as simply the previous remainder with the next three-digits group appended to it. If we evaluate the term on the right for each value of and compare with the left term we have:
0
0
≤ 113789
1
14911
≤ 113789
2
30248
≤ 113789
3
46017
≤ 113789
4
62224
≤ 113789
5
78875
≤ 113789
6
95976
≤ 113789
7
113533
≤ 113789
⬅
8
131552
> 113789
9
150039
> 113789
and it is clear that the next figure of our root is a 7 but, how can we proceed in the general case without having to systematically explore every possibility ()?
Here Knott[1] distinguishes two different approaches:
And it is the strategy usually used with paper and pencil and can also be implemented, of course, on the abacus. In the above expression, if we see the left part as dividend and the parentese as divisor, is the first digit of the division
but since we don't know b yet, we approximate it using only the main part of the divisor
This gives us a guess as to what the value of might be, but we need:
Verify that the value thus obtained is correct, or, where appropriate, correct it up or down as needed.
Obtain the next remainder to prepare the calculation of the next digit of the root.
we prepare the dividend by dividing (the next three-digits group appended to the previous remainder) by
As usual, we don't know and we can't evaluate the parentheses on the right, but we can get a clue about by approximating the parentheses by its main part and use it as a trial divisor.
so that
After that, we need again:
Verify that the value thus obtained is correct, or, where appropriate, correct it up or down as needed.
Obtain the next remainder to prepare for the obtention of the next digit of the root by evaluating .
Please note that:
Divisor 3 is involved in the prepared dividend and this leads to non-finite decimal fractions.
The division by not only worsens the above, but also makes the prepared dividend specific to the current step, since the value of evolves with the calculation of the different figures of the result.
This did not occur in the calculation of square roots and, as a consequence, the process of obtaining cube roots is much more complicated and requires a complex cycle of preparation-restoration of the dividend that, following Knott, can be represented by the following scheme:
a) Divide by .
b) Divide by 3.
c) Obtain as the first digit of the division of the above by .
d) Subtract (Equivalent to subtracting and in ).
e) Multiply by 3.
f) Multiply by .
g) Subtract .
In our example (), using traditional division and traditional division arrangement (like Knott does), working the two first digits:
Cube root of 456.7890123
Abacus
Comment
ABCDEFG
456789
Enter number aligning first group with B
-343
-7^3=343
113789
First remainder
7113789
Enter 7 in A as first root digit and append second group
^1 a) It is unnecessary to extend the division by 7 beyond the current three-digit group. The 4 in G is a division remainder meaning 4/7.
^2 b) The same can be said of division by 3. It is carried out up to column F and the remainder (1) is temporarily added to column G. The value (5) in said column is a strange hybrid meaning 1/3 and 4/7 . It does not matter, it will be reabsorbed in steps e) and f).
Members of the Soroban & Abacus Group modified the technique described by Knott to adapt it to modern soroban use[3]. The result is allegedly faster at the expense of being less compact and requiring an abacus with more rods to store intermediate data. The simplicity of having the result directly substituting the radicand is also lost.
You can also find a compilation of modern methods for both square and cube roots in Tone Nikki (とね日記)[2] by a Japanese blogger (Author's name does not appear to be available).
The following examples are all worked using traditional division and traditional division arrangement. Components of the dividend preparation-restoration cycle are labelled with a), b), etc as detailed above.
Now we continue using [[../../Abbreviated operations/|Abbreviated operations]]. We need to divide the remainder (7497) by three times the square of the current root ()
The first triplet 110 is between 64 and 125, so that the cube root of 110 591 is between 40 and 50. First root digit is 4
First digit:
Cube root of 110591: First digit
Abacus
Comment
ABCDEFG
Cube root of 110591
110591
Enter number aligning first group with B
-64
Subtract 6^3=216 from BCD
46591
46: first remainder
46591
Enter 4 in A as first root digit and append second group
4 46591
OK 1st digit!
Second digit:
Cube root of 110591: Second digit
Abacus
Comment
ABCDEFG
4 46591
a) Divide B-F by 4 (A)
4116473
b) Divide B-F by 3
4388234
c) Divide B by 4 (A)
4868234
d) Subtract BxB=64 from CD
48 4234
e) Multiply C-F by 3 in C-G
48 1273
f) Multiply C-F by 4 (A) in C-G
48 511
g) Cannot subtract 8^3=512 from EFG! Going back (See note at the end)
48 511
-f) Divide C-F by 4 (A)
48 1273
-e) Divide C-F by 3
48 4234
-d) Add 8x8=64 in CD
4868234
-c) Revise down B
-1
+4
47T8234
d) Subtract BxB=49 from CD (T=10)
4759234
e) Multiply C-F by 3 in C-G
4717773
f) Multiply C-F by 4 (A) in C-G
47 7111
g) Subtract B^3=343 from EFG
47 6768
OK 2nd digit! Remainder 6768
Third digit:
Cube root of 110591: Third digit
Abacus
Comment
ABCDEFGHIJ
47 6768000
Append 000 to previous remainder
47 6768000
a) Divide C-H by 47 (AB)
4714400000
b) Divide C-H 3
4748000000
c) Divide C by 47 (AB)
4795700000
d) Subtract C^2=81 from EF
4794890000
e) Multiply D-H by 3 in D-I
4792298300
f) Multiply D-H by 47 (AB) in D-J
479 689490
g) Subtract C^3=729 from HIJ
479 688761
OK 3rd digit! Remainder 688761
Fourth digit:
Cube root of 110591: Fourth digit
Abacus
Comment
ABCDEFGHIJKLM
479 688761000
Append 000 to previous remainder
479 688761000
a) Divide D-J by 479
4791437914194
b) Divide D-J by 3
4794793046394
c) Divide D by 479 1d
4799482046394
d) Subtract 9^2=81 from GH
4799473946394
e) Multiply E-J by 3 in E-K
4799142184194
f) Multiply E-J by 479 in E-M
4799 68106330
g) Subtract -D^3=729 from KLM
4799 68105601
Ok 4th digit! Remainder 68105601
Now we finish the calculation using [[../../Abbreviated operations/|abbreviated operations]]. We need to divide the remainder (68105601) by three times the square of the current root (4799). The first four digits of the result are appended after the ones already obtained; for instance:
Cube root of 110591: Continue using abbreviated operations
Abacus
Comment
ABCDEFGHIJKLM
4799 68105601
Divide E-M by 4799
479914191623
Divide E-M by 4799
47992957204
Divide E-M by 3
47999857
Compare this to
As we can see, we have obtained a result with 7 correct figures.
Note: We found above that with root 48 we could not subtract , or we had a negative remainder (-1). This might seem unfortunate since it forced us to undo part of the work and correct the new root figure downwards, but in practice what we find is a fortunate result: the small remainder (-1) tells us that 48 was a excellent approximation (by excess) to the root, opening a new way to solve the problem. In fact, what we have is:
or
where we can use
so that
compare to . We could have thus achieved great precision with little effort!
The abacus is currently studied as a traditional art or as a means to develop numerical and cognitive skills in general, it is not expected that in the computer age it will be used as a calculator to solve real world problems. But if that were the case and you had to solve a large number of cube roots (something unusual) you might want to move from traditional methods or basic arithmetic to modern numerical analysis methods and try the Newton-Raphson method. You can find an adaptation of this method to the abacus in jccAbacus[4]...
↑Baggs, Shane; Heffelfinger, Totton (2011). "Cube Roots". 算盤 Abacus: Mystery of the Bead. Archived from the original on August 1, 2021. {{cite web}}: Unknown parameter |accesdate= ignored (|access-date= suggested) (help)
This chapter is special in the sense that its content is not directly related to the traditional method of the abacus, but it is an interesting resource to shorten arithmetic operations, both with the abacus and in written calculation. We include it in this book because we make sporadic use of these abbreviated operations throughout it.
Some arithmetic books of the pre-computer era included a chapter on abbreviated operations. The motivation is the following. Suppose we measure the side of a square and we obtain and we want to calculate its area
a result with 6 digits, but if we have measured the side of the square with a measuring tape that only appreciates millimeters, what we can say is that the value of the side is between and , that is:
So that will be a value between and . This means that we only know with certainty the first two digits of the result S (74) and that the third digit is probably a 6; the rest of the digits of the multiplication are meaningless (we say they are not significant) and we should not include them in our result. We should write:
being the significant figures of our result. So if only three of the six figures in the product are significant, why calculate all six? This is what abbreviated operations are for.
In this chapter we will follow the examples that appear in Matemáticas by Antonino Goded Mur[1] hereinafter simply Matemáticas, a small Spanish manual, and see the way these calculations can be done with the abacus.
Write the product of the multiplicand by the first figure of the multiplier, write below the product of the multiplicand without its last figure by the second of the multiplier, below the product of the multiplicand without its last two figures by the third of the multiplier and so on.
—Translated from Matemáticas
Example from Matemáticas:
6665
x 1375
———————
33325
46655
19995
6665
———————
9164375
6665
x 1375
————
6665
1999
466
33
————
9163
Normal operation
Abbreviated operation
On the abacus this problem can be dealt with in several ways, for example, using Kojima’s Multiplication Beginning with the Highest Digits of the Multiplier and Multiplicand, explained in his second book[2], where he says:
As the operation starts by multiplying the first digits of the multiplier and multiplicand, it is convenient for approximations.
We can also try multifactorial multiplication[3] or the like; for instance
6666x1375
Abacus
Comment
ABCDEFGHIJKLM
6665 1375
Setup problem
. .
Unit rods
-1
Clear J
+6665
Add 1✕6665 to G-J
6665 6665375
+18
Add 3✕6 to GH
+18
Add 3✕6 to HI
+18
Add 3✕6 to IJ
-3
Clear K
+15
Add 3✕5 to JK
6665 8664575
666 8664575
Clear D
+42
Add 7✕6 to HI
+42
Add 7✕6 to IJ
+42
Add 7✕6 to JK
-7
Clear L
666 91307 5
66 91307 5
Clear C
+30
Add 5✕6 to IJ
+30
Add 5✕6 to JK
-5
Clear M
66 91637
Result
. .
Unit rods
9164
Result after rounding to 4 figures
But we can also use multiplication methods starting with the last multiplicand digits as modern multiplication:
6666x1375 Modern multiplication
Abacus
Comment
ABCDEFGHIJKLM
6665 1375
Setup problem
+330
Add 5✕66 to K-M
-5
Clear J
6665 137 330
+4662
Add 7✕666=4662 to J-M
-7
Clear I
6665 13 4992
+19995
Add 3✕6665=19995 to I-M
-3
Clear H
6665 1 24987
+6665
Add 1✕6665=6665 to H-L
-1
Clear G
6665 91637
Result
6665 9164
Result after rounding to 4 figures
And even traditional multiplication by clearing first and then adding the partial products shifted one column to the left
The first digit of the quotient is found as usual, the remainder is divided by the divisor without its last digit, the new remainder by the divisor without its last two digits and so on.
As can be seen, the potentially infinite sequence of long division steps in which a new quotient figure is obtained is replaced by a finite sequence of divisions by a shrinking divisor in which we obtain only one digit of the quotient. This can be done using our favorite division method; for example, using traditional division and traditional division arrangement:
The current method is followed until half the figures of the root have been exceeded, obtaining the next digits by dividing the remainder followed by the periods not used by the double of the root found, followed by as many zeros as periods have been added.
Withouth going into details, this way of shortening the square root obtention can be justified in several ways, for example using Taylor series development or Newton's method, perhaps not the simplest way but that is interesting to mention especially for what comes below about cubic roots.
In what follows the process will be illustrated using the Half-remainder method (半九九法) as explained in chapter: Square root, which requires changing remainder into half-remainder and double of the root into simply the root in the Matemáticas paragraph above. Note that the second phase, the division, can be done in the form of an abbreviated division since it only makes sense to obtain a limited number of figures from its quotient. As a consequence, obtaining the last figures from the root costs progressively less work and time; so we can call this division the accelerated phase of root extraction.
The current method is followed until half the figures in the root have been exceeded, obtaining the next digits by dividing the remainder followed by the periods not used by the triple of the square of the root followed by as many zeros as the periods have been added.
This abbreviation can also be justified in several ways, including Newton's method, by the way and by far the best approximation to obtain cube roots with the abacus[4] even though it is not a traditional technique, it is much more efficient than any traditional method and, if we use it, we can say that, in a certain sense, we are using an abbreviated method from the beginning. Nevertheless, here is an example using a traditional method: the cube root of 666. We will follow here the method explained by Cargill G. Knott[5] (see chapter: Cube root).
Obviously, the cube root of 666 is between 8 and 9 because the number is in the range 512-728.
Cube root of 666
Abacus
Comment
ABCDEFG
666
Enter 666 in BCD
+
(Unit rod)
-512
Subtract 83=512 from BCD
154
8154
Enter 8 in A. Divide B-F by 8 (A)
8192500
Divide B-F by 3
8641662
Divide B by 8 (A)
8781662
Subtract B2=49 from CD
8732662
Multiply C-F by 3 in C-G
87 9800
Multiply C-F by 8 (A) in C-G
87 7840
Subtract B3=343 from EFG
87 7497
Root 8.7, Remainder 7.497
So we have obtained 8.7 as the root so far, leaving a remainder of 7.497. To apply the shortcut we need to form the divisor ; We will use Newton's binomial expansion to form the square and we will multiply it by three by adding twice the value obtained.
Caption text
Abacus
Comment
ABCDEFGHIJKLM
87 7497
Squaring 8.7
+49
-112
+64
87 7497 7569
Multiplying it by 3
+14
+10
+12
+18
87 7497 22707
Proceed to divide 7.497 by 227.07 (can be abbreviated!)
8733----22707
obtain only two digits of quotient
Alternatively, you can also divide twice by 8.7 and then by 3 to get the same result. Compare the result 8.733 to 3666=8,7328917
What follows is a completely different type of abbreviated calculation that may prove useful in practice. They are all a consequence of Taylor's theorem.
↑Kojima, Takashi (1963), Advanced Abacus: Theory and Practice, Tokyo: Charles E. Tuttle Co., Inc., ISBN978-0-8048-0003-7
↑Tejón, Fernando; Heffelfinger, Totton (2005). "Multifactorial Multiplication". 算盤 Abacus: Mystery of the Bead. Archived from the original on August 1, 2021. {{cite web}}: Unknown parameter |accesdate= ignored (|access-date= 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, ISBN978-981-15-6525-0
Heffelfinger, Totton; Hinkka, Hannu (2011). "The 5 Earth Bead Advantage". 算盤 Abacus: Mystery of the Bead. Archived from the original on August 1, 2021. {{cite web}}: Unknown parameter |accesdate= ignored (|access-date= suggested) (help)
Heffelfinger, Totton; Tejón, Fernando (2005). "Multifactorial Multiplication". 算盤 Abacus: Mystery of the Bead. Archived from the original on August 1, 2021. {{cite web}}: Unknown parameter |accesdate= ignored (|access-date= suggested) (help)
Siqueira, Edvaldo; Heffelfinger, Totton. "Kato Fukutaro's Square Roots". 算盤 Abacus: Mystery of the Bead. Archived from the original on August 1, 2021. {{cite web}}: Unknown parameter |accesdate= ignored (|access-date= suggested) (help)
Baggs, Shane; Heffelfinger, Totton (2011). "Cube Roots". 算盤 Abacus: Mystery of the Bead. Archived from the original on August 1, 2021. {{cite web}}: Unknown parameter |accesdate= ignored (|access-date= suggested) (help)
On a 5+2 abacus
On a 5+1 abacus
On a 5+3 abacus
![{\displaystyle y={\sqrt[{3}]{x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be50c0a49b200fb46800951d0268b0a9d4e3fdda)
![{\displaystyle y_{i}=\lfloor {\sqrt[{3}]{x}}_{i}\rfloor }](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c6b74c3db5ed9d039940654a5f656b7a4a6324a)
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![{\displaystyle {\sqrt[{3}]{666}}=8.7328917}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5729ad823780f5c1171e0ce28be7d60240908ce)
![{\displaystyle {\sqrt[{3}]{110591}}=47.9998553236\cdots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/58759cfd56f5119a684ff8c3f2a44f644f9eb5ee)
![{\displaystyle {\sqrt[{3}]{110591}}=48{\sqrt[{3}]{1-{\frac {1}{48^{3}}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9dfc416a4ded744ab4cef22da345dd34f371ff44)
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![{\displaystyle {\sqrt[{3}]{110591}}=48{\sqrt[{3}]{1-{\frac {1}{48^{3}}}}}\approx 48\left(1-{\frac {1}{3\cdot 48^{3}}}\right)=48-{\frac {1}{3\cdot 48^{2}}}=47.999855324}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0886d419a1837fa53ce9da9caed1312594d81565)
![{\displaystyle {\sqrt[{3}]{110591}}=47.999855323638}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9b059f89e7e920214a3c8a3da6d8fa20f8bf068d)
![{\displaystyle {\sqrt[{3}]{250047}}=60+3=63}](https://wikimedia.org/api/rest_v1/media/math/render/svg/94b9cd87a0ad9a4aedc9f6022f2257d3303482db)
![{\displaystyle {\sqrt[{n}]{1\pm a}}\approx 1\pm a/n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b0bf4bd0251f9272482556100461b4b94471c92)
![{\displaystyle {\sqrt[{3}]{1.006}}\approx 1.002}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4f7beba0891699e752852c46cf2b43c652da5c7)
