View source for 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 <math>1/n,2/n,\cdots 9/n</math> tabulated (or memorized) where <math>n</math> is one of such powers of two or five. 

For instance, given

<math display = block> 137=1\cdot 10^2+3\cdot 10^1+7\cdot 10^0 </math>

Then 

<math display = block> \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= </math>

<math display = block> = (0.125)\cdot 10^2+(0.375)\cdot 10^1+(0.875)10^0=12.5+3.75+0.875=17.125 </math>

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 

{| class="wikitable" style="font-family:Courier; line-height: 1.10em;"
|+ Division 137/8 using fractions
|-
! Abacus !! Comment
|-
| &nbsp;'''ABCDEF'''
| 
|-
| &nbsp;--+---
| Unit rod
|-
| &nbsp;137
| enter 137 on A-C as a guide
|-
| &nbsp;&nbsp;&nbsp;<s>7</s>
| clear 7 in C
|-
| &nbsp;&nbsp;+0875
| add 7/8 to C-F
|-
| &nbsp;130875
| 
|-
| &nbsp;&nbsp;<s>3</s> 
| clear 3 in B
|-
| &nbsp;+0375
| add 3/8 to B-E
|-
| &nbsp;104625
| 
|-
| &nbsp;<s>1</s>
| clear 1 in A
|-
| +0125
| add 1/8 to A-D
|-
| &nbsp;&nbsp;17125
| Done!
|-
| &nbsp;--+---
| 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<ref>{{Citation
 |author=Williams, Samuel Wells
 |author2=Morrison, John Robert
 |title=A Chinese commercial guide
 |year=1856
 |publisher=Canton: Printed at the office of the Chinese Repository
 |url=https://archive.org/details/chinesecommercia00willuoft/page/298/mode/2up
 |page=298
}}</ref><ref>{{cite web
 |title=Specially Crafted Division Tables
 |year=2020
 |author=Murakami, Masaaki
 |work=算盤 Abacus: Mystery of the Bead
 |url=http://totton.idirect.com/soroban/masaaki/Specially%20Crafted%20Division%20Tables.pdf
 |format=PDF
 |accesdate=August 13, 2021
|archiveurl=https://web.archive.org/web/20210801083501/http://totton.idirect.com/soroban/masaaki/Specially%20Crafted%20Division%20Tables.pdf
 |archivedate=August 1, 2021
}}</ref><ref>{{Citation
 |author=Kwa Tak Ming
 |title=The Fundamental Operations in Bead Arithmetic, How to Use the Chinese Abacus
 |place=San Francisco
 |year=1922
 |publisher=Service Supply Co.
 |url=https://archive.computerhistory.org/resources/access/text/2016/12/B1671.01-05-01-acc.pdf
 |format=PDF
}}</ref>, 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 ===

<div style="font-family:Courier">
{| class="wikitable" style=text-align:center
|+ Power of two fractions
|-
!D
!D/2
!D/4
!D/8
!D/16{{ref|reference_name_A|a}}
!D/32{{ref|reference_name_A|a}}
!D/64{{ref|reference_name_A|a}}
|-
!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
|-
|
|
|
|
| colspan="3" |Unit rod left displacement
|}</div>

{{note|reference_name_A|a}}Unit rod left displacement.

===Examples of use ===

<div style="display:inline-table; vertical-align:top;">
{| class="wikitable" style="font-family:Courier; line-height: 1.10em;"
|+ 137/2
|-
| &nbsp;'''ABCD'''
|-
| &nbsp;--+-{{ref|reference_name_B|b}}
|-
| &nbsp;137
|-
| &nbsp;&nbsp;&nbsp;<s>7</s>
|-
| &nbsp;&nbsp;+35
|-
| &nbsp;&nbsp;<s>3</s>
|-
| &nbsp;+15
|-
| &nbsp;<s>1</s>
|-
| +05
|-
| &nbsp;--+-{{ref|reference_name_B|b}}
|-
| &nbsp;0685
|-
! 68.5
|}</div>
<div style="display:inline-table; vertical-align:top;">
{| class="wikitable" style="font-family:Courier; line-height: 1.10em;"
|+ 137/4
|-
| &nbsp;'''ABCDE'''
|-
| &nbsp;--+--{{ref|reference_name_B|b}}
|-
| &nbsp;137
|-
| &nbsp;&nbsp;&nbsp;<s>7</s>
|-
| &nbsp;&nbsp;+175
|-
| &nbsp;&nbsp;<s>3</s>
|-
| &nbsp;+075
|-
| &nbsp;<s>1</s>
|-
| +025
|-
| &nbsp;--+--{{ref|reference_name_B|b}}
|-
| &nbsp;03425
|-
! 34.25
|}
</div>
<div style="display:inline-table; vertical-align:top;">
{| class="wikitable" style="font-family:Courier; line-height: 1.10em;"
|+ 137/8
|-
| &nbsp;'''ABCDEF'''
|-
| &nbsp;--+---{{ref|reference_name_B|b}}
|-
| &nbsp;137
|-
| &nbsp;&nbsp;&nbsp;<s>7</s>
|-
| &nbsp;&nbsp;+0875
|-
| &nbsp;&nbsp;<s>3</s>
|-
| &nbsp;+0375
|-
| &nbsp;<s>1</s>
|-
| +0125
|-
| &nbsp;--+---{{ref|reference_name_B|b}}
|-
| &nbsp;017125
|-
! 17.125
|}
</div>
<div style="display:inline-table; vertical-align:top;">
{| class="wikitable" style="font-family:Courier; line-height: 1.10em;"
|+ 137/16
|-
| &nbsp;'''ABCDEF'''
|-
| &nbsp;--+---{{ref|reference_name_B|b}}
|-
| &nbsp;137
|-
| &nbsp;&nbsp;&nbsp;<s>7</s>
|-
| &nbsp;&nbsp;+4375
|-
| &nbsp;&nbsp;<s>3</s>
|-
| &nbsp;+1875
|-
| &nbsp;<s>1</s>
|-
| +0625
|-
| &nbsp;-+----{{ref|reference_name_B|b}}
|-
| &nbsp;085625
|-
! 8.5625
|}
</div>
<div style="display:inline-table; vertical-align:top;">
{| class="wikitable" style="font-family:Courier; line-height: 1.10em;"
|+ 137/32
|-
| &nbsp;'''ABCDEFG'''
|-
| &nbsp;--+----{{ref|reference_name_B|b}}
|-
| &nbsp;137
|-
| &nbsp;&nbsp;&nbsp;<s>7</s>
|-
| &nbsp;&nbsp;+21875
|-
| &nbsp;&nbsp;<s>3</s>
|-
| &nbsp;+09375
|-
| &nbsp;<s>1</s>
|-
| +03125
|-
| &nbsp;-+-----{{ref|reference_name_B|b}}
|-
| &nbsp;0428125
|-
! 4.28125
|}
</div>
<div style="display:inline-table; vertical-align:top;">
{| class="wikitable" style="font-family:Courier; line-height: 1.10em;"
|+ 137/64
|-
| &nbsp;'''ABCDEFGH'''
|-
| &nbsp;--+-----{{ref|reference_name_B|b}}
|-
| &nbsp;137
|-
| &nbsp;&nbsp;&nbsp;<s>7</s>
| Clear 7 in C
|-
| &nbsp;&nbsp;+109375
|-
| &nbsp;&nbsp;<s>3</s>
|Clear 3 in B
|-
| &nbsp;+046875
|-
| &nbsp;<s>1</s>
|Clear 1 in A
|-
| +015625
|-
| &nbsp;-+------{{ref|reference_name_B|b}}
|-
| &nbsp;02140625
|-
! 2.140625
|}
</div>
{{note|reference_name_B|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<ref>{{cite web
 |title=Kato Fukutaro's Square Roots
 |year=
 |author=Siqueira, Edvaldo
 |author2=Heffelfinger, Totton
 |work=算盤 Abacus: Mystery of the Bead
 |url=http://totton.idirect.com/soroban/katoSq/
 |accesdate=August 10, 2021
 |archiveurl=https://web.archive.org/web/20210801083040/http://totton.idirect.com/soroban/katoSq/
 |archivedate=August 1, 2021
}}</ref> 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:
# clearing the digit
# adding its half starting with the column it occupied

For instance, 123456789/2:

{| class="wikitable" style="font-family:Courier; line-height: 1.10em;"
|+ 123456789÷2 ''in situ''
|-
! Abacus !! Comment
|-
| '''&nbsp;ABCDEFGHIJ'''
| 
|-
| &nbsp;123456789
| 
|-
| &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<s>9</s>
| Clear 9 in I
|-
| &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;+45
| Add its half to IJ
|-
| &nbsp;1234567845
| 
|-
| &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<s>8</s>
| Clear 8 in H
|-
| &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;+40
| Add its half to HI
|-
| &nbsp;1234567445
| 
|-
| &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<s>7</s>
| Clear 7 in G
|-
| &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;+35
| Add its half to GH
|-
| &nbsp;1234563945
| 
|-
| &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<s>6</s>
| Clear 6 in F
|-
| &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;+3
| Add its half to FG
|-
| &nbsp;1234533945
| 
|-
| &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<s>5</s>
| Clear 5 in E
|-
| &nbsp;&nbsp;&nbsp;&nbsp;+25
| Add its half to EF
|-
| &nbsp;1234283945
| 
|-
| &nbsp;&nbsp;&nbsp;&nbsp;<s>4</s>
| Clear 4 in D
|-
| &nbsp;&nbsp;&nbsp;+2
| Add its half to DE
|-
| &nbsp;1232283945
| 
|-
| &nbsp;&nbsp;&nbsp;<s>3</s>
| Clear 3 in C
|-
| &nbsp;&nbsp;+15
| Add its half to CD
|-
| &nbsp;1217283945
| 
|-
| &nbsp;&nbsp;<s>2</s>
| Clear 2 in B
|-
| &nbsp;+1
| Add its half to BC
|-
| &nbsp;1117283945
| 
|-
| &nbsp;<s>1</s>
| Clear 1 in A
|-
| +05
| Add its half to AB.
|-
| &nbsp;&nbsp;617283945
| Done!
|}

The unit rod does not change in this division.

== Powers of five ==
=== Table of fractions ===
{{incomplete}}
{| class="wikitable"
|+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
|}

<noinclude>
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</noinclude>