## One-digit divisors (short division)

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

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

### 123456789 divided by 8

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

### 123456789 divided by 7

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

### 123456789 divided by 6

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

### 123456789 divided by 5

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

### 123456789 divided by 4

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

### 123456789 divided by 3

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

### 123456789 divided by 2

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

## Multi-digit divisors (long division)

### Division of 998001 by 999

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

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!

### Division of 888122 by 898

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!

### Division of 412 by 896

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

## References

1. Xú Xīnlǔ (徐心魯) (1993) [1573]. Pánzhū Suànfǎ (盤珠算法) (in Chinese). Zhōngguó kēxué jìshù diǎnjí tōng huì (中國科學技術典籍通彙). `{{cite book}}`: Unknown parameter `|trans_title=` ignored (`|trans-title=` suggested) (help)
2. Yoshida, Mitsuyoshi (吉田光由) (1634). Jinkoki (塵劫記) (in Japanese). `{{cite book}}`: Unknown parameter `|trans_title=` ignored (`|trans-title=` suggested) (help)

## External resources

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).

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