Traditional Abacus and Bead Arithmetic/Division/Special division tables

PrincipleEdit

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 200000÷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 tablesEdit

In the past, special division tables were used for divisors between 11 and 99[1].

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 examplesEdit

Dividing by numbers that start with 1 is awkward, the following table may be used to divide by 19[2].

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 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 readingEdit


  1. Martzloff, Jean-Claude (2006), A history of chinese mathematics, Springer, p. 221, ISBN 978-3-540-33782-9 
  2. Cabrera, Jesús (2021), "Tide Abacus", jccAbacus, https://sites.google.com/view/jccabacus/abacus-articles/advanced-abacus-algorithms/tide-abacus?authuser=0, retrieved 4 August 2021