Traditional Abacus and Bead ArithmeticSide 8 af 15
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 Suanfa1], 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
| 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 |
| 136 T 56789 9 | Column C: Apply rule 6/9>6+6 Change 6 in C into 6 and add 6 to D |
| 13 6 056789 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 |
| 1371 6 2789 9 | Column E: Apply rule 6/9>6+6 Change 6 in E into 6 and add 6 to F |
| 137173789 9 | Revise up |
| 13717 3 089 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 (Image: Jccsvq, CC BY-SA 4.0)
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 |
| 153 T 56789 8 | Column C: rule 3/8>3+6, change 3 in C into 3, add 6 to D |
| 15 3 056789 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 |
| 154320 8 49 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 |
| 1543209 7 3 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 |
Done! 123456789/9=15432098.625 |
Traditional division (帰除法) of 123456789 by 8 (Image: Jccsvq, CC BY-SA 4.0)
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 |
| 1763 5 1789 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 |
| 17636 5 289 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 |
| 1763668 2 5 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 (Image: Jccsvq, CC BY-SA 4.0)
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 |
| 205761 1 29 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 (Image: Jccsvq, CC BY-SA 4.0)
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 (Image: Jccsvq, CC BY-SA 4.0)
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 |
| 3086417 T 9 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 (Image: Jccsvq, CC BY-SA 4.0)
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 (Image: Jccsvq, CC BY-SA 4.0)
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 (Image: Jccsvq, CC BY-SA 4.0)
Multi-digit divisors (long division)
Division of 998001 by 999
| Abacus | Comment |
|---|---|
| ABCDEFGHIJKLM | Dividend in A-F, divisor 8 in K-M |
| 998001 999 | |
| 9 88001 999 | Chinese rule: 9/9>9+9 |
| -8 | Subtract 81 from BC |
| 9 T 8001 999 | |
| -1 | |
| 9 T 7001 999 | |
| -8 | Subtract 81 from CD |
| 999001 999 | |
| -1 | |
| 998901 999 | |
| 9 9 7901 999 | Chinese rule: 9/9>9+9 |
| -8 | Subtract 81 from CD |
| 999901 999 | |
| -1 | |
| 999801 999 | |
| -8 | Subtract 81 from DE |
| 998 T 01 999 | |
| -1 | |
| 998991 999 | |
| 99 8 791 999 | Chinese rule: 8/9>8+8 |
| -7 | Subtract 72 from DE |
| 998 T 91 999 | |
| -2 | |
| 998 T 71 999 | |
| -7 | Subtract 72 from EF |
| 9989 T 1 999 | |
| -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 998001 by 999
(Image: Jccsvq, CC BY-SA 4.0)
On a 5+2 abacus
(Image: Jccsvq, CC BY-SA 4.0)
On a 5+1 abacus
(Image: Jccsvq, CC BY-SA 4.0)
On a 5+3 abacus
Division of 888122 by 989
| Abacus | Comment |
|---|---|
| ABCDEFGHIJKLM | Dividend 888122 in A-F, divisor 989 in K-M |
| 888122 989 | |
| 8 68122 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 |
| 8 04122 989 | Subtract A×L=8×8=64 from BC |
| 896922 989 | Subtract A×M=8×9=72 from CD |
| 8 9 5922 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 |
| 89 7 612 989 | 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 |
| 89 7 052 989 | 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 (Image: Jccsvq, CC BY-SA 4.0)
Division of 888122 by 898
| Abacus | Comment |
|---|---|
| ABCDEFGHIJKLM | Dividend 888122in A-F, divisor 898 in K-M |
| 888122 898 | |
| 9 68122 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 |
| 9 8 5922 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 (Image: Jccsvq, CC BY-SA 4.0)
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 4598 1 12 | Column I: rule 1/8>1+2, change 1 in I into 1, add 2 to J |
| 896 4598 1 03 | subtract I×B=1×9=9 from JK |
| 896 4598 1 024 | 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 459821 3 44 | Column K: rule 3/8>3+6, change 3 in K into 3, add 6 to L |
| 896 459821 3 17 | subtract K×B=3×9=27 from LM |
| 896 459821 3 15 | 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
Traditional division (帰除法) of 412 by 896 (Image: Jccsvq, CC BY-SA 4.0)
References
- ↑ Xú Xīnlǔ (徐心魯) (1993) [1573]. Pánzhū Suànfǎ (盤珠算法) [Computational Methods with the Beads in a Tray] (in Chinese). Zhōngguó kēxué jìshù diǎnjí tōng huì (中國科學技術典籍通彙).
- ↑ Yoshida, Mitsuyoshi (吉田光由) (1634). Jinkoki (塵劫記) [Dust Record] (in Japanese).
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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