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		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 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			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)