## 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 ${\displaystyle 1/n,2/n,\cdots 9/n}$ tabulated (or memorized) where ${\displaystyle n}$ is one of such powers of two or five.

For instance, given

${\displaystyle 137=1\cdot 10^{2}+3\cdot 10^{1}+7\cdot 10^{0}}$

Then

${\displaystyle {\frac {137}{8}}={\frac {1\cdot 10^{2}+3\cdot 10^{1}+7\cdot 10^{0}}{8}}={\frac {1}{8}}\cdot 10^{2}+{\frac {3}{8}}\cdot 10^{1}+{\frac {7}{8}}\cdot 10^{0}=}$

${\displaystyle =(0.125)\cdot 10^{2}+(0.375)\cdot 10^{1}+(0.875)10^{0}=12.5+3.75+0.875=17.125}$

Which can easily be done on the abacus by working from right to left. For each digit of the numerator:

1. Clear the digit
2. 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
130875
3 clear 3 in B
104625
1 clear 1 in A
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[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

Power of two fractions
D D/2 D/4 D/8 D/16a D/32a D/64a
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

^a Unit rod left displacement.

### Examples of use

68.5 ABCD --+-b 137 7 +35 3 +15 1 +05 --+-b 0685
34.25 ABCDE --+--b 137 7 +175 3 +075 1 +025 --+--b 03425
17.125 ABCDEF --+---b 137 7 +0875 3 +0375 1 +0125 --+---b 017125
8.5625 ABCDEF --+---b 137 7 +4375 3 +1875 1 +0625 -+----b 085625
4.28125 ABCDEFG --+----b 137 7 +21875 3 +09375 1 +03125 -+-----b 0428125
 2.140625 ABCDEFGH --+-----b 137 7 Clear 7 in C +109375 3 Clear 3 in B +046875 1 Clear 1 in A +015625 -+------b 02140625

^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[4] 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:

1. clearing the digit
2. 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

Next Page: Traditional division examples | Previous Page: Special division tables
1. Williams, Samuel Wells; Morrison, John Robert (1856), A Chinese commercial guide, Canton: Printed at the office of the Chinese Repository, p. 298
2. 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)
3. Kwa Tak Ming (1922), The Fundamental Operations in Bead Arithmetic, How to Use the Chinese Abacus (PDF), San Francisco: Service Supply Co.
4. 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)