Traditional Abacus and Bead Arithmetic/Division/Traditional division examples

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 SuanfaXú Xīnlǔ (徐心魯) Pánzhū Suànfǎ (盤珠算法) Computational Methods with the Beads in a Tray 1573 1993 Chinese Zhōngguó kēxué jìshù diǎnjí tōng huì (中國科學技術典籍通彙), 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 JinkokiYoshida, Mitsuyoshi (吉田光由) Jinkoki (塵劫記) Dust Record 1634 Japanese https://dl.ndl.go.jp/info:ndljp/pid/3508170/7, 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 D136056789   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 alt=Traditional division (帰 除法) of 123456789 by 9|none|thumb|Traditional division (帰除法) of 123456789 by 9 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 thumb|Traditional division (帰除法) of 123456789 by 8|none 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 thumb|Traditional division (帰除法) of 123456789 by 7|none 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 thumb|Traditional division (帰除法) of 123456789 by 6|none 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 thumb|Traditional division (帰除法) of 123456789 by 5|none 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 thumb|Traditional division (帰除法) of 123456789 by 4|none 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 thumb|Traditional division (帰除法) of 123456789 by 3|none 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 thumb|Traditional division (帰除法) of 123456789 by 2|none 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! thumb|Traditional division (帰除法) of 888122 by 989|none 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! thumb|Traditional division (帰除法) of 888122 by 898|none 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 approximatedreference_name_A 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: reference_name_A aSee chapter: Abbreviated operations thumb|Traditional division (帰除法) of 412 by 896|none References 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). ../Multiplication Division by powers of two