## Introduction

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.

of which the first is by far the most important.

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.

## Reverse operation

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.

Chapter Extending the 123456789 exercise proposes its daily use as a way to perfect our understanding of beading.

## Learning the abacus in the past

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:

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.

## Chapters

### Use of the 5th lower bead

The specialized use of the 5th lower bead and the non-unique representation of numbers 5, 10 and 15 to simplify operations.

### Extending the 123456789 exercise

A plethora of addition and subtraction exercises that can be done without an exercise sheet.

## References

1. Xú Xīnlǔ (徐心魯) (1993) [1573]. Pánzhū Suànfǎ (盤珠算法) (in Chinese). Zhōngguó kēxué jìshù diǎnjí tōng huì (中國科學技術典籍通彙). `{{cite book}}`: Unknown parameter `|trans_title=` ignored (`|trans-title=` suggested) (help)
2. Kē Shàngqiān (柯尚遷) (1993) [1578]. Shùxué Tōngguǐ (數學通軌) (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)
3. Newcomb, Simon (c1882), Logarithmic and other mathematical tables with examples of their use and hints on the art of computation, New York: Henry Holt and Company `{{citation}}`: Check date values in: `|year=` (help)
4. Suzuki, Hisao (鈴木 久男) (1982). "Chuugoku ni okeru shuzan kagen-hou 中国における珠算加減法". Kokushikan University School of Political Science and Economics (in Japanese). 57 (3). ISSN 0586-9749 – via Kokushikan. `{{cite journal}}`: Unknown parameter `|trans_title=` ignored (`|trans-title=` suggested) (help)
5. 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, ISBN 978-3-319-73396-8 `{{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)
6. 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). `{{cite book}}`: Unknown parameter `|trans_title=` ignored (`|trans-title=` suggested) (help)
7. Kwa Tak Ming (1920?), The Fundamental Operations in Bead Arithmetic, How to Use the Chinese Abacus (PDF), San Francisco: Service Supply Co. `{{citation}}`: Check date values in: `|year=` (help)