Traditional Abacus and Bead Arithmetic/Division/Traditional division examples
Onedigit divisors (short division) edit
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
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 edit
Abacus  Comment 

ABCDEFGHIJKLM  Dividend in AI, 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 edit
Abacus  Comment 

ABCDEFGHIJKLM  Dividend in AI, 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 edit
Abacus  Comment 

ABCDEFGHIJKLM  Dividend in AI, 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 edit
Abacus  Comment 

ABCDEFGHIJKLM  Dividend in AI, 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 edit
Abacus  Comment 

ABCDEFGHIJKLM  Dividend in AI, 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 edit
Abacus  Comment 

ABCDEFGHIJKLM  Dividend in AI, 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 edit
Abacus  Comment 

ABCDEFGHIJKLM  Dividend in AI, 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 
Multidigit divisors (long division) edit
Division of 998001 by 999 edit
Abacus  Comment 

ABCDEFGHIJKLM  Dividend in AF, divisor 8 in KM 
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 

On a 5+2 abacus

On a 5+1 abacus

On a 5+3 abacus
Division of 888122 by 989 edit
Abacus  Comment 

ABCDEFGHIJKLM  Dividend 888122 in AF, divisor 989 in KM 
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 edit
Abacus  Comment 

ABCDEFGHIJKLM  Dividend 888122in AF, divisor 898 in KM 
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 edit
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 approximated^{a} 
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 edit
 ↑ 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).
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External resources edit
You can practice traditional division online with Soroban Trainer (see chapter: Introduction) using this file kijoho1digit.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).