Traditional Abacus and Bead Arithmetic, Division, Division by powers of two

Introduction

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 {}1 / n , 2 / n , ⋯ 9 / n tabulated (or memorized) where n is one of such powers of two or five.

For instance, given

137 = 1 ⋅ 1 {0}^{2} + 3 ⋅ 1 {0}^{1} + 7 ⋅ 1 {0}^{0}

Then

\frac{137}{8} = \frac{1 ⋅ 1 {0}^{2} + 3 ⋅ 1 {0}^{1} + 7 ⋅ 1 {0}^{0}}{8} = \frac{1}{8} ⋅ 1 {0}^{2} + \frac{3}{8} ⋅ 1 {0}^{1} + \frac{7}{8} ⋅ 1 {0}^{0} =

= ( 0.125 ) ⋅ 1 {0}^{2} + ( 0.375 ) ⋅ 1 {0}^{1} + ( 0.875 ) 1 {0}^{0} = 12.5 + 3.75 + 0.875 = 17.125

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

We only need to have the corresponding fractions tabulated or memorized, as in the table below.

Powers of two

In the past, both in China and in Japan, monetary and measurement units were used that were related by a factor of 16[1][2][3], 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.

Table of fractions

^a Unit rod left displacement.

Examples of use

^b "+" indicates the unit rod position.

Division by 2 in situ

The fractions for divisor 2 are easily memorizable and this method corresponds to the division by two "in situ" or "in place" explained by Siqueira4] 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: [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

For instance, 123456789/2:

The unit rod does not change in this division.

Powers of five

Table of fractions

• ↑ Williams, Samuel Wells; Morrison, John Robert (1856), A Chinese commercial guide, Canton: Printed at the office of the Chinese Repository, p. 298{{citation}}: CS1 maint: publisher location (link)
• ↑ Murakami, Masaaki (2020). "Specially Crafted Division Tables" (PDF). 算盤 Abacus: Mystery of the Bead. Archived from the original (PDF) on August 1, 2021. Retrieved August 13, 2021.
• ↑ Kwa Tak Ming (1922), The Fundamental Operations in Bead Arithmetic, How to Use the Chinese Abacus (PDF), San Francisco: Service Supply Co.
• ↑ Siqueira, Edvaldo; Heffelfinger, Totton. "Kato Fukutaro's Square Roots". 算盤 Abacus: Mystery of the Bead. Archived from the original on August 1, 2021. Retrieved August 10, 2021.

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