Traditional Abacus and Bead Arithmetic/Division/Traditional division examples

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One-digit divisors (short division)[edit | edit source]

The number 123456789 has also been used to demonstrate multiplication and division in many ancient books on the abacus. Some, like the Panzhu Suanfa[1], start with the traditional multiplication (see chapter: Multiplication) of this number by a digit and use the division to return the abacus to its original state; others, like the Jinkoki[2], do it the other way around, starting with division and ending the exercise with multiplication. The latter is what we do here.

The number 123456789 is divisible by 3, 9 and 13717421, so divisions by 2, 3, 4, 5, 6, 8 and 9 have results with finite decimal expansion (2 and 5 are divisor of the decimal basis or radix 10 ). Only division by 7 leads to a result with an infinite number of decimal places, so here we will cut it off and give a remainder.

Unfortunately, this exercise does not use all the division rules, but it is a good start and allows you to practice without a worksheet.

123456789 divided by 9[edit | edit source]

123456789 divided by 9
Abacus Comment
ABCDEFGHIJKLM Divisor 9 at M
123456789   9 Column A: Apply 1/9>1+1
133456789   9 Change 1 in A into 1 and add 1 to B
136456789   9 Column B: Apply rule 3/9>3+3 Change 3 in B into 3 and add 3 to C
136T56789   9 Column C: Apply rule 6/9>6+6 Change 6 in C into 6 and add 6 to D
136056789   9 (Same as above)
137156789   9 Revise up
137166789   9 Column D: Apply rule 1/9>1+1 Change 1 in D into 1 and add 1 to E
137162789   9 Column E: Apply rule 6/9>6+6 Change 6 in E into 6 and add 6 to F
137173789   9 Revise up
137173089   9 Column F: Apply rule 3/9>3+3 Change 3 in F into 3 and add 3 to G
137174189   9 Revise up
137174199   9 Column G: Apply rule 1/9>1+1 Change 1 in G into 1 and add 1 to H
137174209   9 Revise up
137174210   9 Revise up. Done! 123456789/9=13717421
Traditional division (帰 除法) of 123456789 by 9
Traditional division (帰除法) of 123456789 by 9

123456789 divided by 8[edit | edit source]

123456789 divided by 8
Abacus Comment
ABCDEFGHIJKLM Dividend in A-I, divisor 8 at M
123456789   8
143456789   8 Column A: rule 1/8>1+2, change 1 in A into 1, add 2 to B
153456789   8 Column B: rule 4/8>5+0, change 4 in B into 5, add 0 to C
153T56789   8 Column C: rule 3/8>3+6, change 3 in C into 3, add 6 to D
153056789   8 (Same as above)
154256789   8 Revise up C, add 1 to C, subtract 8 from D
154296789   8 Column D: rule 2/8>2+4, change 2 in D into 2, add 4 to E
154316789   8 Revise up D, add 1 to D, subtract 8 from E
154318789   8 Column E: rule 1/8>1+2, change 1 in E into 1, add 2 to F
154320789   8 Revise up E, add 1 to E, subtract 8 from F
154320849   8 Column G: rule 7/8>8+6, Change 7 in G into 8, add 6 to H
154320969   8 Revise up G, add 1 to G, subtract 8 from H
154320973   8 Column H: rule 6/8>7+4, change 6 in H into 7, add 4 to I
154320985   8 Revise up H, add 1 to H, subtract 8 from I
1543209862  8 Column I: rule 5/8>6+2, change 5 in I into 6, add 2 to J
15432098624 8 Column J: rule 2/8>2+4, change 2 in J into 2, add 4 to K
1543209862508 Column K: rule 4/8>5+0, change 4 in K into 5, add 0 to L.

Done! 123456789/9=15432098.625

Traditional division (帰除法) of 123456789 by 8

123456789 divided by 7[edit | edit source]

123456789 divided by 7
Abacus Comment
ABCDEFGHIJKLM Dividend in A-I, divisor 8 at M
123456789   7
153456789   7 Column A: rule 1/7>1+3, change 1 in A into 1, add 3 to B
174456789   7 Column B: rule 5/7>7+1, change 5 in B into 7, add 1 to C
175956789   7 Column C: rule 4/7>5+5, change 4 in C into 5, add 5 to D
176256789   7 Revise up C, add 1 to C, subtract 7 from D
176256789   7 Column D: rule 2/7>2+6, change 2 in D into 2, add 6 to E
176346789   7 Revise up D, add 1 to D, subtract 7 from E
176351789   7 Column E: rule 4/7>5+5, change 4 in E into 5, add 5 to F
176364789   7 Revise up E, add 1 to E, subtract 7 from F
176365289   7 Column F: rule 4/7>5+5, change 4 in F into 5, add 5 to G
176366589   7 Revise up F, add 1 to F, subtract 7 from G
176366799   7 Column G: rule 5/7>7+1, change 5 in G into 7, add 1 to H
176366829   7 Revise up G, add 1 to G, subtract 7 from H
176366825   7 Column H: rule 2/7>2+6, change 2 in H into 2, add 6 to I
176366841   7 Revise up H twice, add 2 to H, subtract 14 from I.
Stop here! 123456789/9=17636684, remainder = 1
Traditional division (帰除法) of 123456789 by 7

123456789 divided by 6[edit | edit source]

123456789 divided by 6
Abacus Comment
ABCDEFGHIJKLM Dividend in A-I, divisor 8 at M
123456789   6
163456789   6 Column A: rule 1/6>1+4, change 1 in A into 1, add 4 to B
203456789   6 Revise up A, add 1 to A, subtract 6 from B
205456789   6 Column C: rule 3/6>5+0, change 3 in C into 5, add 0 to D
205696789   6 Column D: rule 4/6>6+4, change 4 in D into 6, add 4 to E
205736789   6 Revise up D, add 1 to D, subtract 6 from E
205756789   6 Column E: rule 3/6>5+0, change 3 in E into 5, add 0 to F
205760789   6 Revise up E, add 1 to E, subtract 6 from F
205761189   6 Revise up F, add 1 to F, subtract 6 from G
205761129   6 Column G: rule 1/6>1+4, change 1 in G into 1, add 4 to H
205761309   6 Revise up G twice, add 2 to G, subtract 12 from H
205761313   6 Revise up H, add 1 to H, subtract 6 from I
205761315   6 Column I: rule 3/6>5+0, change 3 in I into 5, add 0 to J.
Done! 123456789/6=20576131.5
Traditional division (帰除法) of 123456789 by 6

123456789 divided by 5[edit | edit source]

123456789 divided by 5
Abacus Comment
ABCDEFGHIJKLM Dividend in A-I, divisor 8 at M
123456789   5
223456789   5 Column A: Rule 1/5>2+0, change 1 in A into 2, add 0 to B
243456789   5 Column B: Rule 2/5>4+0, change 2 in B into 4, add 0 to C
246456789   5 Column C: Rule 3/5>6+0, change 3 in C into 6, add 0 to D
246856789   5 Column D: Rule 4/5>8+0, change 4 in D into 8, add 0 to E
246906789   5 Revise up D, add 1 to D, subtract 5 from E
246911789   5 Revise up E, add 1 to E, subtract 5 from F
246912789   5 Column F: Rule 1/5>2+0, change 1 in F into 2, add 0 to G
246913289   5 Revise up F, add 1 to F, subtract 5 from G
246913489   5 Column G: Rule 2/5>4+0, change 2 in G into 4, add 0 to H
246913539   5 Revise up G, Add 1 to G, subtract 5 from H
246913569   5 Column H: Rule 3/5>6+0, change 3 in H into 6, add 0 to I
246913574   5 Revise up H, add 1 to H, subtract 5 from I
246913578   5 Column I: Rule 4/5>8+0, change 4 in I into 8, add 0 to J.
Done! 123456789/5=24691357.8
Traditional division (帰除法) of 123456789 by 5

123456789 divided by 4[edit | edit source]

123456789 divided by 4
Abacus Comment
ABCDEFGHIJKLM Dividend in A-I, divisor 8 at M
123456789   4
243456789   4 Column A: rule 1/4>2+2, change 1 in A into 2, add 2 to B
303456789   4 Revise up A, add 1 to A, subtract 4 from B
307656789   4 Column C: rule 3/4>7+2, change 3 in C into 7, add 2 to D
308256789   4 Revise up C, add 1 to C, subtract 4 from D
308556789   4 Column D: rule 2/4>5+0, change 2 in D into 5, add 0 to E
308616789   4 Revise up D, add 1 to D, subtract 4 from E
308628789   4 Column E: rule 1/4>2+2, change 1 in E into 2, add 2 to F
308640789   4 Revise up E twice, add 2 to E, subtract 8 from F
308641389   4 Revise up F, add 1 to F, subtract 4 from G
3086417T9   4 Column G: rule 3/4>7+2, change 3 in G into 7, add 2 to H
308641929   4 Revise up G twice, add 2 to G, subtract 8 from H
308641959   4 Column H: rule 2/4>5+0, change 2 in H into 5, add 0 to I
308641971   4 Revise up H twice, add 2 to H, subtract 8 from I
3086419722  4 Column I: rule 1/4>2+2, change 1 in I into 2, add 2 to J
3086419725  4 Column J: rule 2/4>5+0, change 2 in J into 5, add 0 to K.
Done! 123456789/4=30864197.25
Traditional division (帰除法) of 123456789 by 4

123456789 divided by 3[edit | edit source]

123456789 divided by 3
Abacus Comment
ABCDEFGHIJKLM Dividend in A-I, divisor 8 at M
123456789   3
333456789   3 Column A: rule 1/3>3+1, change 1 in A into 3, add 1 to B
403456789   3 Revise up A, add 1 to A, subtract 3 from B
410456789   3 Revise up B, add 1 to B, subtract 3 from C
411156789   3 Revise up C, add 1 to C, subtract 3 from D
411366789   3 Column D: rule 1/3>3+1, change 1 in D into 3, add 1 to E
411506789   3 Revise up D twice, add 2 to D, subtract 6 from E
411520789   3 Revise up E twice, add 2 to E, subtract 6 from F
411522189   3 Revise up F twice, add 2 to F, subtract 6 from G
411522399   3 Column G: rule 1/3>3+1, change 1 in G into 3, add 1 to H
411522609   3 Revise up G three times, add 3 to G, subtract 9 from H
411522630   3 Revise up H three times, add 3 to H, subtract 9 from I.
Done! 123456789/3=41152263
Traditional division (帰除法) of 123456789 by 3

123456789 divided by 2[edit | edit source]

123456789 divided by 2
Abacus Comment
ABCDEFGHIJKLM Dividend in A-I, divisor 8 at M
123456789   2
523456789   2 Column A: rule 1/2>5+0, change 1 in A into 5, add 0 to B
603456789   2 Revise up A, add 1 to A, subtract 2 from B
611456789   2 Revise up B, add 1 to B, subtract 2 from C
615456789   2 Column C: rule 1/2>5+0, change 1 in C into 5, add 0 to D
617056789   2 Revise up C twice, add 2 to C, subtract 4 from D
617216789   2 Revise up D twice, add 2 to D, subtract 4 from E
617256789   2 Column E: rule 1/2>5+0, change 1 in E into 5, add 0 to F
617280789   2 Revise up E three times, add 3 to E, subtract 6 from F
617283189   2 Revise up F three times, add 3 to F, subtract 6 from G
617283589   2 Column G: rule 1/2>5+0, change 1 in G into 5, add 0 to H
617283909   2 Revise up G four times, add 4 to G, subtract 8 from H
617283941   2 Revise up H four times, add 4 to H, subtract 8 from I
617283945   2 Column I: rule 1/2>5+0, change 1 in I into 5, add 0 to J.
Done! 123456789/2=61728394.5
Traditional division (帰除法) of 123456789 by 2

Multi-digit divisors (long division)[edit | edit source]

Division of 998001 by 999[edit | edit source]

Division of 998001 by 999
Abacus Comment
ABCDEFGHIJKLM Dividend in A-F, divisor 8 in K-M
998001    999
988001    999 Chinese rule: 9/9>9+9
-8 Subtract 81 from BC
9T8001    999
 -1
9T7001    999
 -8 Subtract 81 from CD
999001    999
  -1
998901    999
997901    999 Chinese rule: 9/9>9+9
 -8 Subtract 81 from CD
999901    999
  -1
999801    999
  -8 Subtract 81 from DE
998T01    999
   -1
998991    999
998791    999 Chinese rule: 8/9>8+8
  -7 Subtract 72 from DE
998T91    999
   -2
998T71    999
   -7 Subtract 72 from EF
9989T1    999
    -2
998999    999
    -9 Revising up (from right to left to save a hand displacement)
998990    999
   -9
998900    999
  -9
998000    999
 +1
999000    999 Done! 998001/999 = 999

Division of 888122 by 989[edit | edit source]

Division of 888122 by 989
Abacus Comment
ABCDEFGHIJKLM Dividend 888122 in A-F, divisor 989 in K-M
888122    989
868122    989 Focus on A and use rule: 8/9>8+8 i.e. change 8 in A to 8 (nothing to do) and add 8 to B
804122    989 Subtract A×L=8×8=64 from BC
896922    989 Subtract A×M=8×9=72 from CD
895922    989 Focus on B and use rule: 9/9>9+9 i.e. change 9 in B to 9 (nothing to do) and add 9 to C
898722    989 Subtract B×L=9×8=72 from CD
897912    989 Subtract B×M=9×9=81 from DE
897612    989 Focus on C and use rule: 7/9>7+7 i.e. change 7 in B to 7 (nothing to do) and add 7 to D
897052    989 Subtract C×L=7×8=56 from DE
897989    989 Subtract C×M=7×9=63 from EF
898000    989 Revise up: add 1 to C and subtract 989 from DEF. Remainder in DEF is zero, so that 888122/989 = 898. Done!
Traditional division (帰除法) of 888122 by 989

Division of 888122 by 898[edit | edit source]

Division of 888122 by 898
Abacus Comment
ABCDEFGHIJKLM Dividend 888122in A-F, divisor 898 in K-M
888122    898
968122    898 Focus on A and use rule: 8/8>9+8, i.e. change 8 in A to 9 and add 8 to B
987122    898 Subtract A×L=9×9=81 from BC
979922    898 Subtract A×M=9×8=72 from CD
985922    898 Focus on B and use rule: 7/8>8+6, i.e. change 7 in B to 8 and add 6 to C
988722    898 Subtract B×L=8×9=72 from CD
988082    898 Subtract B×M=8×8=64 from DE
989882    898 Focus on C and use rule: 8/8>9+8, i.e. change 8 in C to 9 and add 8 to D
989072    898 Subtract C×L=9×9=81 from DE
989000    898 Subtract C×M=9×8=72 from EF. Remainder in DEF is zero, so that 888122/898 = 989. Done!
Traditional division (帰除法) of 888122 by 898

Division of 412 by 896[edit | edit source]

Division of 412 by 896
Abacus Comment
ABCDEFGHIJKLM
896 412 This time the divisor goes to the left and the dividend to the right
896 512 Column E: rule 4/8>5+0, change 4 in E into 5, add 0 to F
896 492 cannot subtract E×B=5×9=45 from FG, revise down E: subtract 1 from E, add 8 to F
896 456 subtract E×B=4×9=36 from FG
896 4536 subtract E×C=4×6=24 from GH
896 4656 Column F: rule 5/8>6+2, change 5 in F into 6, add 2 to G
896 4602 subtract F×B=6×9=54 from GH
896 4582 cannot subtract F×C=6×6=36 from HI, revise down F: subtract 1 from F, add 8 to G
896 4591 and add 9 to H to return the excess 89 subtracted from GH
896 4588 Continue normally and subtract F×C=3×6=30 from HI
896 45916 Column G: rule 8/8>9+8, change 8 in G into 9, add 8 to H
896 45979 subtract G×B=9×9=81 from HI
896 459736 subtract G×C=9×6=54 from IJ
896 459896 Column H: rule 7/8>8+6, Change 7 in H into 8, add 6 to I
896 459824 subtract H×B=8×9=72 from IJ
896 4598192 subtract H×C=8×6=48 from JK
896 4598112 Column I: rule 1/8>1+2, change 1 in I into 1, add 2 to J
896 4598103 subtract I×B=1×9=9 from JK
896 45981024 subtract I×C=1×6=6 from KL
896 45982128 revise up I: add 1 to I, subtract 896 from JKL
896 45982148 Column J: rule 1/8>1+2, Change 1 in J into 1, add 2 to K
896 45982139 subtract J×B=1×9=9 from KL
896 459821384 subtract J×C=1×6=6 from LM
896 459821344 Column K: rule 3/8>3+6, change 3 in K into 3, add 6 to L
896 459821317 subtract K×B=3×9=27 from LM
896 459821315 subtract K×C=3×6=18 from M … from now it is approximateda
896 459821425 revise up K: add 1 to K, subtract 896 from LM…
896 459821429 Column L: rule 2/8>2+4, Change 2 in L into 2, add 4 to M
896 459821427 subtract L×B=2×9=18 from M…
896 459821428 Column M: rule 7/8>8+6, Change 7 in M into 8, add 4 to … Done! 412/896=0.459821428

Note: ^a See chapter: Abbreviated operations

Traditional division (帰除法) of 412 by 896

References[edit | edit source]

  1. Xú Xīnlǔ (徐心魯) (1993) [1573] (in Chinese). Pánzhū Suànfǎ (盤珠算法) [Computational Methods with the Beads in a Tray]. Zhōngguó kēxué jìshù diǎnjí tōng huì (中國科學技術典籍通彙). 
  2. Yoshida, Mitsuyoshi (吉田光由) (1634) (in Japanese). Jinkoki (塵劫記) [Dust Record]. https://dl.ndl.go.jp/info:ndljp/pid/3508170/7. 

External resources[edit | edit source]

You can practice traditional division online with Soroban Trainer (see chapter: Introduction) using this file kijoho-1digit.sbk that you should download to your computer and then submit it to Soroban Trainer (It is a text file that you can inspect with any text editor and that you can safely download to your computer).