# 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 $1/n,2/n,\cdots 9/n$ tabulated (or memorized) where $n$ is one of such powers of two or five.
For instance, given
$137=1\cdot 10^{2}+3\cdot 10^{1}+7\cdot 10^{0}$
Then
${\frac {137}{8}}={\frac {1\cdot 10^{2}+3\cdot 10^{1}+7\cdot 10^{0}}{8}}={\frac {1}{8}}\cdot 10^{2}+{\frac {3}{8}}\cdot 10^{1}+{\frac {7}{8}}\cdot 10^{0}=$
$=(0.125)\cdot 10^{2}+(0.375)\cdot 10^{1}+(0.875)10^{0}=12.5+3.75+0.875=17.125$
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
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.

## 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, 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

Power of two fractions :
| D | D/2 | D/4 | D/8 | D/16 | D/32 | D/64 |
| 1 | 05 | 025 | 0125 | 0625 | 03125 | 015625 |
| 2 | 10 | 050 | 0250 | 1250 | 06250 | 031250 |
| 3 | 15 | 075 | 0375 | 1875 | 09375 | 046875 |
| 4 | 20 | 100 | 0500 | 2500 | 12500 | 062500 |
| 5 | 25 | 125 | 0625 | 3125 | 15625 | 078125 |
| 6 | 30 | 150 | 0750 | 3750 | 18750 | 093750 |
| 7 | 35 | 175 | 0875 | 4375 | 21875 | 109375 |
| 8 | 40 | 200 | 1000 | 5000 | 25000 | 125000 |
| 9 | 45 | 225 | 1125 | 5625 | 28125 | 140625 |
| | | | | 1 | 1 | 1 |
| | | | | Unit rod left displacement | Unit rod left displacement | Unit rod left displacement |
^a Unit rod left displacement.

### Examples of use

137/2 :
* ABCD
* 137
* 7
* +35
* 3
* +15
* 1
* +05
* 0685
* 68.5
137/4 :
* ABCDE
* 137
* 7
* +175
* 3
* +075
* 1
* +025
* 03425
* 34.25
137/8 :
* ABCDEF
* 137
* 7
* +0875
* 3
* +0375
* 1
* +0125
* 017125
* 17.125
137/16 :
* ABCDEF
* 137
* 7
* +4375
* 3
* +1875
* 1
* +0625
* 085625
* 8.5625
137/32 :
* ABCDEFG
* 137
* 7
* +21875
* 3
* +09375
* 1
* +03125
* 0428125
* 4.28125
137/64 :
| ABCDEFGH |
| 137 |
| 7 | Clear 7 in C |
| +109375 |
| 3 | Clear 3 in B |
| +046875 |
| 1 | Clear 1 in A |
| +015625 |
| 02140625 |
| 2.140625 |
^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 Siqueira 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:
123456789÷2 in situ :
| Abacus | Comment |
| ABCDEFGHIJ | |
| 123456789 | |
| 9 | Clear 9 in I |
| +45 | Add its half to IJ |
| 1234567845 | |
| 8 | Clear 8 in H |
| +40 | Add its half to HI |
| 1234567445 | |
| 7 | Clear 7 in G |
| +35 | Add its half to GH |
| 1234563945 | |
| 6 | Clear 6 in F |
| +3 | Add its half to FG |
| 1234533945 | |
| 5 | Clear 5 in E |
| +25 | Add its half to EF |
| 1234283945 | |
| 4 | Clear 4 in D |
| +2 | Add its half to DE |
| 1232283945 | |
| 3 | Clear 3 in C |
| +15 | Add its half to CD |
| 1217283945 | |
| 2 | Clear 2 in B |
| +1 | Add its half to BC |
| 1117283945 | |
| 1 | Clear 1 in A |
| +05 | Add its half to AB. |
| 617283945 | Done! |
The unit rod does not change in this division.

## Powers of five
### Table of fractions

Power of five fractions :
| D | D/5 | D/25 | D/125 | D/625 |
| 1 | 0.2 | 0.04 | 0.008 | 0.0016 |
| 2 | 0.4 | 0.08 | 0.016 | 0.0032 |
| 3 | 0.6 | 0.12 | 0.024 | 0.0048 |
| 4 | 0.8 | 0.16 | 0.032 | 0.0064 |
| 5 | 1 | 0.2 | 0.04 | 0.008 |
| 6 | 1.2 | 0.24 | 0.048 | 0.0096 |
| 7 | 1.4 | 0.28 | 0.056 | 0.0112 |
| 8 | 1.6 | 0.32 | 0.064 | 0.0128 |
| 9 | 1.8 | 0.36 | 0.072 | 0.0144 |
* ↑ Williams, Samuel Wells; Morrison, John Robert (1856), A Chinese commercial guide, Canton: Printed at the office of the Chinese Repository, p. 298
* ↑ Murakami, Masaaki (2020). "Specially Crafted Division Tables" (PDF). 算盤 Abacus: Mystery of the Bead. Archived from the original (PDF) on August 1, 2021. {{cite web}}: Unknown parameter |accesdate= ignored (|access-date= suggested) (help)
* ↑ 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. {{cite web}}: Unknown parameter |accesdate= ignored (|access-date= suggested) (help)