Traditional Abacus and Bead Arithmetic/Division/Division by powers of two

Introduction

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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   tabulated (or memorized) where   is one of such powers of two or five.

For instance, given

 

Then

 

 

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

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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

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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

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137/2
 ABCD
 --+-b
 137
   7
  +35
  3
 +15
 1
+05
 --+-b
 0685
68.5
137/4
 ABCDE
 --+--b
 137
   7
  +175
  3
 +075
 1
+025
 --+--b
 03425
34.25
137/8
 ABCDEF
 --+---b
 137
   7
  +0875
  3
 +0375
 1
+0125
 --+---b
 017125
17.125
137/16
 ABCDEF
 --+---b
 137
   7
  +4375
  3
 +1875
 1
+0625
 -+----b
 085625
8.5625
137/32
 ABCDEFG
 --+----b
 137
   7
  +21875
  3
 +09375
 1
+03125
 -+-----b
 0428125
4.28125
137/64
 ABCDEFGH
 --+-----b
 137
   7 Clear 7 in C
  +109375
  3 Clear 3 in B
 +046875
 1 Clear 1 in A
+015625
 -+------b
 02140625
2.140625

^b "+" indicates the unit rod position.

Division by 2 in situ

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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

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Table of fractions

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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


  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)