# 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 Suanfa, 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, 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 998001 by 999
* On a 5+2 abacus
* On a 5+1 abacus
* On a 5+3 abacus

### 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 approximated |
| 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

* ↑ 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)
* ↑ 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).