## Introduction

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.

## Chapters

### Modern and traditional division; close relatives

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.

### Guide to traditional division (帰除法)

Here, we will see how to use the traditional method.

### Learning the division table

It contains some indications that may make it easier for you to memorize the division table.

### Dealing with overflow

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.

### Special division tables

Division tables can be coined for multi-digit divisors, allowing dividing by them without resorting to the multiplication table.

### Traditional division examples

A basic set of examples to illustrate all of the above.

### Division by powers of two

Another traditional division method different from 帰除法 based in fractions; a form of division in situ.

## References

1. Suzuki, Hisao (鈴木 久男) (1980). "Chūgoku ni okeru josan-hō no kigen (1 ) 中国における除算法の起源（1)". Kokushikan University School of Political Science and Economics (in Japanese). 55 (2). ISSN 0586-9749 – via Kokushikan. `{{cite journal}}`: Unknown parameter `|trans_title=` ignored (`|trans-title=` suggested) (help)
2. Suzuki, Hisao (鈴木 久男) (1981). "Chūgoku ni okeru josan-hō no kigen (2 ) 中国における除算法の起源（2)". Kokushikan University School of Political Science and Economics (in Japanese). 56 (1). ISSN 0586-9749 – via Kokushikan. `{{cite journal}}`: Unknown parameter `|trans_title=` ignored (`|trans-title=` suggested) (help)
3. Momokawa, Jihei (百川治兵衛) (1645). Kamei Zan (亀井算) (in Japanese). `{{cite book}}`: Unknown parameter `|trans_title=` ignored (`|trans-title=` suggested) (help)
4. Smith, David Eugene; Mikami, Yoshio (1914), A history of Japanese mathematics, Chicago: The Open court publishing company, p. 43-44
5. 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)