# Traditional Abacus and Bead Arithmetic/Division/Special division tables
## Principle

Suppose we have to perform a large number of divisions by 36525, which could be the case if we do calendar calculations. Then we can simplify the task by creating a specialized division table for this divisor. Following what is stated in the chapter: Guide to traditional division, we will start by calculating the following three Euclidean divisions:
Creating a special division table for 36525 :
| 100000÷36525 | 100000÷36525 | 200000÷36525 | 200000÷36525 | 300000÷36525 | 300000÷36525 |
| Quotient | Remainder | Quotient | Remainder | Quotient | Remainder |
| 2 | 26950 | 5 | 17375 | 8 | 07800 |
Which can be summarized in the following specialized division table:
Division by 36525 table :
* 1/36525>2+26950
* 2/36525>5+17375
* 3/36525>8+07800
And now we can use this table to do divisions without touching the multiplication table. For example, how many Julian centuries of 36525 days can fit in 1 000 000 days?
1000000/36525 :
| Abacus | Comment |
| ABCDEFGHIJKLM | |
| 36525 1000000 | Use rule: 1/36525>2+26950 on column G |
| 36525 2000000 | change 1 in G into 2 |
| +26950 | add 26950 to H-L |
| 36525 2269500 | Use rule: 2/36525>5+17375 on column H |
| 36525 2569500 | change 2 in H into 5 |
| +17375 | add 17375 to I-M |
| 36525 2586875 | revise up |
| +1 | |
| -36525 | |
| 36525 2650350 | revise up |
| +1 | |
| -36525 | |
| 36525 2713825 | Done! 1000000÷36525=27, remainder 13825 |
And we have done a division by a five-digit divisor without using the multiplication table!

## Two-digit division tables

In the past, special division tables were used for divisors between 11 and 99.
Traditional division rules for 2-digit divisors :
| | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 |
| 1 | 9+01 | 8+04 | 7+09 | 7+02 | 6+10 | 6+04 | 5+15 | 5+10 | 5+05 |
| | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 |
| 1 | 4+16 | 4+12 | 4+08 | 4+04 | 4+00 | 3+22 | 3+19 | 3+16 | 3+13 |
| 2 | 9+11 | 9+02 | 8+16 | 8+08 | 8+00 | 7+18 | 7+11 | 7+04 | 6+26 |
| | 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 |
| 1 | 3+07 | 3+04 | 3+01 | 2+32 | 2+30 | 2+28 | 2+26 | 2+24 | 2+22 |
| 2 | 6+14 | 6+08 | 6+02 | 5+30 | 5+25 | 5+20 | 5+15 | 5+10 | 5+05 |
| 3 | 9+21 | 9+12 | 9+03 | 8+28 | 8+20 | 8+12 | 8+04 | 7+34 | 7+27 |
| | 41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 | 49 |
| 1 | 2+18 | 2+16 | 2+14 | 2+12 | 2+10 | 2+08 | 2+06 | 2+04 | 2+02 |
| 2 | 4+36 | 4+32 | 4+28 | 4+24 | 4+20 | 4+16 | 4+12 | 4+08 | 4+04 |
| 3 | 7+13 | 7+06 | 6+42 | 6+36 | 6+30 | 6+24 | 6+18 | 6+12 | 6+06 |
| 4 | 9+31 | 9+22 | 9+13 | 9+04 | 8+40 | 8+32 | 8+24 | 8+16 | 8+08 |
| | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59 |
| 1 | 1+49 | 1+48 | 1+47 | 1+46 | 1+45 | 1+44 | 1+43 | 1+42 | 1+41 |
| 2 | 3+47 | 3+44 | 3+41 | 3+38 | 3+35 | 3+32 | 3+29 | 3+26 | 3+23 |
| 3 | 5+45 | 5+40 | 5+35 | 5+30 | 5+25 | 5+20 | 5+15 | 5+10 | 5+05 |
| 4 | 7+43 | 7+36 | 7+29 | 7+22 | 7+15 | 7+08 | 7+01 | 6+52 | 6+46 |
| 5 | 9+41 | 9+32 | 9+23 | 9+14 | 9+05 | 8+52 | 8+44 | 8+36 | 8+28 |
| | 61 | 62 | 63 | 64 | 65 | 66 | 67 | 68 | 69 |
| 1 | 1+39 | 1+38 | 1+37 | 1+36 | 1+35 | 1+34 | 1+33 | 1+32 | 1+31 |
| 2 | 3+17 | 3+14 | 3+11 | 3+08 | 3+05 | 3+02 | 2+66 | 2+64 | 2+62 |
| 3 | 4+56 | 4+52 | 4+48 | 4+44 | 4+40 | 4+36 | 4+32 | 4+28 | 4+24 |
| 4 | 6+34 | 6+28 | 6+22 | 6+16 | 6+10 | 6+04 | 5+65 | 5+60 | 5+55 |
| 5 | 8+12 | 8+04 | 7+59 | 7+52 | 7+45 | 7+38 | 7+31 | 7+24 | 7+17 |
| 6 | 9+51 | 9+42 | 9+33 | 9+24 | 9+15 | 9+06 | 8+64 | 8+56 | 8+48 |
| | 71 | 72 | 73 | 74 | 75 | 76 | 77 | 78 | 79 |
| 1 | 1+29 | 1+28 | 1+27 | 1+26 | 1+25 | 1+24 | 1+23 | 1+22 | 1+21 |
| 2 | 2+58 | 2+56 | 2+54 | 2+52 | 2+50 | 2+48 | 2+46 | 2+44 | 2+42 |
| 3 | 4+16 | 4+12 | 4+08 | 4+04 | 4+00 | 3+72 | 3+69 | 3+66 | 3+63 |
| 4 | 5+45 | 5+40 | 5+35 | 5+30 | 5+25 | 5+20 | 5+15 | 5+10 | 5+05 |
| 5 | 7+03 | 6+68 | 6+62 | 6+56 | 6+50 | 6+44 | 6+38 | 6+32 | 6+26 |
| 6 | 8+32 | 8+24 | 8+16 | 8+08 | 8+00 | 7+68 | 7+61 | 7+54 | 7+47 |
| 7 | 9+61 | 9+52 | 9+43 | 9+34 | 9+25 | 9+16 | 9+07 | 8+76 | 8+68 |
| | 81 | 82 | 83 | 84 | 85 | 86 | 87 | 88 | 89 |
| 1 | 1+19 | 1+18 | 1+17 | 1+16 | 1+15 | 1+14 | 1+13 | 1+12 | 1+11 |
| 2 | 2+38 | 2+36 | 2+34 | 2+32 | 2+30 | 2+28 | 2+26 | 2+24 | 2+22 |
| 3 | 3+57 | 3+54 | 3+51 | 3+48 | 3+45 | 3+42 | 3+39 | 3+36 | 3+33 |
| 4 | 4+76 | 4+72 | 4+68 | 4+64 | 4+60 | 4+56 | 4+52 | 4+48 | 4+44 |
| 5 | 6+14 | 6+08 | 6+02 | 5+80 | 5+75 | 5+70 | 5+65 | 5+60 | 5+55 |
| 6 | 7+33 | 7+26 | 7+19 | 7+12 | 7+05 | 6+84 | 6+78 | 6+72 | 6+66 |
| 7 | 8+52 | 8+44 | 8+36 | 8+28 | 8+20 | 8+12 | 8+04 | 7+84 | 7+77 |
| 8 | 9+71 | 9+62 | 9+53 | 9+44 | 9+35 | 9+26 | 9+17 | 9+08 | 8+88 |
| | 91 | 92 | 93 | 94 | 95 | 96 | 97 | 98 | 99 |
| 1 | 1+09 | 1+08 | 1+07 | 1+06 | 1+05 | 1+04 | 1+03 | 1+02 | 1+01 |
| 2 | 2+18 | 2+16 | 2+14 | 2+12 | 2+10 | 2+08 | 2+06 | 2+04 | 2+02 |
| 3 | 3+27 | 3+24 | 3+21 | 3+18 | 3+15 | 3+12 | 3+09 | 3+06 | 3+03 |
| 4 | 4+36 | 4+32 | 4+28 | 4+24 | 4+20 | 4+16 | 4+12 | 4+08 | 4+04 |
| 5 | 5+45 | 5+40 | 5+35 | 5+30 | 5+25 | 5+20 | 5+15 | 5+10 | 5+05 |
| 6 | 6+54 | 6+48 | 6+42 | 6+36 | 6+30 | 6+24 | 6+18 | 6+12 | 6+06 |
| 7 | 7+63 | 7+56 | 7+49 | 7+42 | 7+35 | 7+28 | 7+21 | 7+14 | 7+07 |
| 8 | 8+72 | 8+64 | 8+56 | 8+48 | 8+40 | 8+32 | 8+24 | 8+16 | 8+08 |
| 9 | 9+81 | 9+72 | 9+63 | 9+54 | 9+45 | 9+36 | 9+27 | 9+18 | 9+09 |

## Some examples

Dividing by numbers that start with 1 is awkward, the following table may be used to divide by 19.
Division by 19 table :
| 19 ||
| 1 | 5+05 |
Division by 99 table :
| 99 ||
| 1 | 1+01 |
| 2 | 2+02 |
| 3 | 3+03 |
| 4 | 4+04 |
| 5 | 5+05 |
| 6 | 6+06 |
| 7 | 7+07 |
| 8 | 8+08 |
| 9 | 9+09 |
9801÷99 :
| Abacus | Comment |
| ABCDEFGHI | |
| 9801 99 | Dividend AD, divisor HI |
| 9891 99 | A: Rule 9/99>9+09 |
| 9899 99 | B: Rule 8/99>8+08 |
| +1 | revising up |
| -99 | |
| 99 99 | Done! No remainder, quotient: 99 |
Dividing by 𝝅 is common in applications, here are the tables for two approximations of this irrational number.
Division by 𝝅 tables :
| 314 | 314 | 31416 | 31416 |
| 1 | 3+058 | 1 | 3+05752 |
| 2 | 6+116 | 2 | 6+11504 |
| 3 | 9+174 | 3 | 9+17256 |
Finally, the division by 666 table.
Division by 666 table :
| 666 ||
| 1 | 1+334 |
| 2 | 3+002 |
| 3 | 4+336 |
| 4 | 6+004 |
| 5 | 7+338 |
| 6 | 9+006 |
However, It is not advisable to divide by this number; results can be unpredictable… and uncontrollable! In any case, remember the advice:
   I say to you againe, doe not call up Any that you can not put downe; by the Which I meane, Any that can in Turne call up somewhat against you, whereby your Powerfullest Devices may not be of use.
   H. P. Lovecraft - The Case of Charles Dexter Ward (1941)

## Further reading

* Murakami, Masaaki (2020). "Specially Crafted Division Tables" (PDF). 算盤 Abacus: Mystery of the Bead. Archived from the original (PDF) on August 1, 2021. {{cite web}}: Unknown parameter |accesdate= ignored (|access-date= suggested) (help)
* ↑ Martzloff, Jean-Claude (2006), A history of chinese mathematics, Springer, p. 221, ISBN 978-3-540-33782-9
* ↑ Cabrera, Jesús (2021), "Tide Abacus", jccAbacus, retrieved 4 August 2021