Traditional Abacus and Bead Arithmetic/Introduction

Forewords edit

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 and traditional abacus edit

Modern 4+1 abacus
5+3 and 5+1 abaci

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.

Modern and traditional methods edit

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:

  • The Japanese Abacus: its Use and Theory (1954)[2]
  • Advanced Abacus: Theory and Practice (1963)[3]

Several editions of these books can still be found, including e-books formats, and the first one can be consulted at 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 edit

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.

The principle of least effort edit

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.

Abacus procedure tables, some terms and notation edit

As usual, in this book we will use tables to describe procedures on the abacus, for example:

Example of abacus procedure table
Abacus Comment
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
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
988001    999 Column B value is 18
    Notation related to the use of the fifth lower bead
    As 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).

External resources edit

Soroban Trainer edit

Soroban Trainer showing a 5+2 type abacus using the suspended upper bead.

If you are interested in traditional techniques but do not have a traditional abacus yet, you can use the JavaScript application

 Soroban Trainer 

References edit

  1. Chen, Yifu (2013). L’étude des Différents Modes de Déplacement des Boules du Boulier et de l’Invention de la Méthode de Multiplication Kongpan Qianchengfa et son Lien avec le Calcul Mental (PhD thesis) (in French). Université Paris-Diderot (Paris 7). p. 40. {{cite book}}: Unknown parameter |trans_title= ignored (|trans-title= suggested) (help)
  2. Kojima, Takashi (1954), The Japanese Abacus: its Use and Theory, Tokyo: Charles E. Tuttle Co., Inc., ISBN 978-0-8048-0278-9
  3. Kojima, Takashi (1963), Advanced Abacus: Theory and Practice, Tokyo: Charles E. Tuttle Co., Inc., ISBN 978-0-8048-0003-7
  4. Zhū Shìjié 朱士傑 (1993) [1299]. Suànxué Qǐméng (算學啟蒙) (in Chinese). Zhōngguó kēxué jìshù diǎnjí tōng huì (中國科學技術典籍通彙). {{cite book}}: Unknown parameter |trans_title= ignored (|trans-title= suggested) (help)
  5. Ang Tian Se; Lam Lay Yong (2004), Fleeting Footsteps; Tracing the Conception of Arithmetic and Algebra in Ancient China (PDF), World Scientific Publishing Company, ISBN 981-238-696-3
  6. Sunzi 孫子 (3rd to 5th centuries AD). 孫子算經 (in Chinese). {{cite book}}: Check date values in: |year= (help); Unknown parameter |trans_title= ignored (|trans-title= suggested) (help)