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:

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
+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/16Template:Ref	D/32Template:Ref	D/64Template:Ref
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

Template:NoteUnit rod left displacement.

Examples of use

137/2

ABCD
--+-Template:Ref
137
7
+35
3
+15
1
+05
--+-Template:Ref
0685
68.5

137/4

ABCDE
--+--Template:Ref
137
7
+175
3
+075
1
+025
--+--Template:Ref
03425
34.25

137/8

ABCDEF
--+---Template:Ref
137
7
+0875
3
+0375
1
+0125
--+---Template:Ref
017125
17.125

137/16

ABCDEF
--+---Template:Ref
137
7
+4375
3
+1875
1
+0625
-+----Template:Ref
085625
8.5625

137/32

ABCDEFG
--+----Template:Ref
137
7
+21875
3
+09375
1
+03125
-+-----Template:Ref
0428125
4.28125

137/64

ABCDEFGH
--+-----Template:Ref
137
7	Clear 7 in C
+109375
3	Clear 3 in B
+046875
1	Clear 1 in A
+015625
-+------Template:Ref
02140625
2.140625

Template:Note"+" 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: [[../../Roots/Square root/|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

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