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^[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

137/2
68.5
ABCD
--+-^b
137
7
+35
3
+15
1
+05
--+-^b
0685

137/4
34.25
ABCDE
--+--^b
137
7
+175
3
+075
1
+025
--+--^b
03425

137/8
17.125
ABCDEF
--+---^b
137
7
+0875
3
+0375
1
+0125
--+---^b
017125

137/16
8.5625
ABCDEF
--+---^b
137
7
+4375
3
+1875
1
+0625
-+----^b
085625

137/32
4.28125
ABCDEFG
--+----^b
137
7
+21875
3
+09375
1
+03125
-+-----^b
0428125

137/64
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:

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

Next Page: Traditional division examples | Previous Page: Special division tables
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↑ 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)

[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)

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