Tables and cycles

Multiplication and division tables (B9)

B9 (mod 9) is the digital root: reduce any number to a single digit from 1 to 9 by adding its digits until one remains (e.g. 14 → 1+4 = 5). In decimal, results grow (2, 4, 6, 8…) and patterns are hidden; in B9, values stay between 1 and 9 and cycles become clear — the same sequence repeats every 9 terms.

Binary / Ternary

Binary sequences (×2 each row) and ternary (+3 each row). In decimal the numbers blow up; in B9 (digital root mod 9) they reduce to 1–9 and the cycles become obvious.

Binary

5 columns starting with 1, 3, 5, 7, 9 — each value doubles every row. B9 = digital root.

1	3	5	7	9
1	3	5	7	9
2	6	10	14	18
4	12	20	28	36
8	24	40	56	72
16	48	80	112	144
32	96	160	224	288
64	192	320	448	576
128	384	640	896	1152
256	768	1280	1792	2304
512	1536	2560	3584	4608
1024	3072	5120	7168	9216
2048	6144	10240	14336	18432
4096	12288	20480	28672	36864
8192	24576	40960	57344	73728
16384	49152	81920	114688	147456
32768	98304	163840	229376	294912
65536	196608	327680	458752	589824
131072	393216	655360	917504	1179648

Ternary

3 columns starting with 1, 2, 3 — add 3 each row. B9 = digital root.

1	2	3
1	2	3
4	5	6
7	8	9
10	11	12
13	14	15
16	17	18
19	20	21
22	23	24
25	26	27
28	29	30
31	32	33
34	35	36
37	38	39
40	41	42
43	44	45
46	47	48
49	50	51
52	53	54
55	56	57
58	59	60
61	62	63
64	65	66
67	68	69
70	71	72
73	74	75
76	77	78
79	80	81
82	83	84
85	86	87
88	89	90
91	92	93
94	95	96
97	98	99
100	101	102
103	104	105
106	107	108

Multiplication table

Full table: n = 1 to 36, all cycles ×1 to ×9. On the left, the decimal product (it grows with no visible loop); on the right in B9, the digital root mod 9: values 1 to 9 repeat and each column reveals a distinct cycle.

n	1	2	3	4	5	6	7	8	9
1	1	2	3	4	5	6	7	8	9
2	2	4	6	8	10	12	14	16	18
3	3	6	9	12	15	18	21	24	27
4	4	8	12	16	20	24	28	32	36
5	5	10	15	20	25	30	35	40	45
6	6	12	18	24	30	36	42	48	54
7	7	14	21	28	35	42	49	56	63
8	8	16	24	32	40	48	56	64	72
9	9	18	27	36	45	54	63	72	81
10	10	20	30	40	50	60	70	80	90
11	11	22	33	44	55	66	77	88	99
12	12	24	36	48	60	72	84	96	108
13	13	26	39	52	65	78	91	104	117
14	14	28	42	56	70	84	98	112	126
15	15	30	45	60	75	90	105	120	135
16	16	32	48	64	80	96	112	128	144
17	17	34	51	68	85	102	119	136	153
18	18	36	54	72	90	108	126	144	162
19	19	38	57	76	95	114	133	152	171
20	20	40	60	80	100	120	140	160	180
21	21	42	63	84	105	126	147	168	189
22	22	44	66	88	110	132	154	176	198
23	23	46	69	92	115	138	161	184	207
24	24	48	72	96	120	144	168	192	216
25	25	50	75	100	125	150	175	200	225
26	26	52	78	104	130	156	182	208	234
27	27	54	81	108	135	162	189	216	243
28	28	56	84	112	140	168	196	224	252
29	29	58	87	116	145	174	203	232	261
30	30	60	90	120	150	180	210	240	270
31	31	62	93	124	155	186	217	248	279
32	32	64	96	128	160	192	224	256	288
33	33	66	99	132	165	198	231	264	297
34	34	68	102	136	170	204	238	272	306
35	35	70	105	140	175	210	245	280	315
36	36	72	108	144	180	216	252	288	324

The matrix

In B9, each column (×1 to ×9) shows a cycle that repeats over 9 rows. In decimal you would only see growth; here the cycles are visible.

	×1	×2	×3	×4	×5	×6	×7	×8	×9
1	1	2	3	4	5	6	7	8	9
2	2	4	6	8	1	3	5	7	9
3	3	6	9	3	6	9	3	6	9
4	4	8	3	7	2	6	1	5	9
5	5	1	6	2	7	3	8	4	9
6	6	3	9	6	3	9	6	3	9
7	7	5	3	1	8	6	4	2	9
8	8	7	6	5	4	3	2	1	9
9	9	9	9	9	9	9	9	9	9

Division table

Decimal result (1/n, 2/n…) on the left; on the right in B9 (mod 9), the digital root of the inverse. In decimal the fractions have no readable cycle; in B9 each divisor (1, 2, 4, 5, 7, 8) has a cycle that repeats. Fill the B9 columns then check the matrix.

n	1/1	1/2	1/4	1/5	1/7	1/8
1	1	0.5	0.25	0.2	0.14	0.12
2	2	1	0.5	0.4	0.29	0.25
3	3	1.5	0.75	0.6	0.43	0.38
4	4	2	1	0.8	0.57	0.5
5	5	2.5	1.25	1	0.71	0.62
6	6	3	1.5	1.2	0.86	0.75
7	7	3.5	1.75	1.4	1	0.88
8	8	4	2	1.6	1.14	1
9	9	4.5	2.25	1.8	1.29	1.12
10	10	5	2.5	2	1.43	1.25
11	11	5.5	2.75	2.2	1.57	1.38
12	12	6	3	2.4	1.71	1.5
13	13	6.5	3.25	2.6	1.86	1.62
14	14	7	3.5	2.8	2	1.75
15	15	7.5	3.75	3	2.14	1.88
16	16	8	4	3.2	2.29	2
17	17	8.5	4.25	3.4	2.43	2.12
18	18	9	4.5	3.6	2.57	2.25
19	19	9.5	4.75	3.8	2.71	2.38
20	20	10	5	4	2.86	2.5
21	21	10.5	5.25	4.2	3	2.62
22	22	11	5.5	4.4	3.14	2.75
23	23	11.5	5.75	4.6	3.29	2.88
24	24	12	6	4.8	3.43	3
25	25	12.5	6.25	5	3.57	3.12
26	26	13	6.5	5.2	3.71	3.25
27	27	13.5	6.75	5.4	3.86	3.38
28	28	14	7	5.6	4	3.5
29	29	14.5	7.25	5.8	4.14	3.62
30	30	15	7.5	6	4.29	3.75
31	31	15.5	7.75	6.2	4.43	3.88
32	32	16	8	6.4	4.57	4
33	33	16.5	8.25	6.6	4.71	4.12
34	34	17	8.5	6.8	4.86	4.25
35	35	17.5	8.75	7	5	4.38
36	36	18	9	7.2	5.14	4.5

Division matrix (mod 9)

Same idea as multiplication: in B9 the cycle of each divisor (1, 2, 4, 5, 7, 8) appears. Columns 3, 6 and 9 are handled in mod 8 below.

	1	2	4	5	7	8
1	1	5	7	2	4	8
2	2	1	5	4	8	7
3	3	6	3	6	3	6
4	4	2	1	8	7	5
5	5	7	8	1	2	4
6	6	3	6	3	6	3
7	7	8	4	5	1	2
8	8	4	2	7	5	1
9	9	9	9	9	9	9

Division by 3, 6 and 9 (mod 8)

When we divide by 3, 6 or 9, we leave mod 9 arithmetic and enter mod 8: another reality, on the other side of the singularity. The 9 (and its divisors 3, 6) mark the passage into this 8-term cycle.

n	n/3	n	n/6	n	n/9
1	0.33	1	0.17	1	0.11
2	0.67	2	0.33	2	0.22
3	1	3	0.50	3	0.33
4	1.33	4	0.67	4	0.44
5	1.67	5	0.83	5	0.56
6	2	6	1	6	0.67
7	2.33	7	1.17	7	0.78
8	2.67	8	1.33	8	0.89
9	3	9	1.50	9	1
10	3.33	10	1.67	10	1.11
11	3.67	11	1.83	11	1.22
12	4	12	2	12	1.33
13	4.33	13	2.17	13	1.44
14	4.67	14	2.33	14	1.56
15	5	15	2.50	15	1.67
16	5.33	16	2.67	16	1.78
17	5.67	17	2.83	17	1.89
18	6	18	3	18	2
19	6.33	19	3.17	19	2.11
20	6.67	20	3.33	20	2.22
21	7	21	3.50	21	2.33
22	7.33	22	3.67	22	2.44
23	7.67	23	3.83	23	2.56
24	8	24	4	24	2.67
25	8.33	25	4.17	25	2.78
26	8.67	26	4.33	26	2.89
27	9	27	4.50	27	3
28	9.33	28	4.67	28	3.11
29	9.67	29	4.83	29	3.22
30	10	30	5	30	3.33
31	10.33	31	5.17	31	3.44
32	10.67	32	5.33	32	3.56

Division matrix mod 8 (3, 6, 9)

Mod 8 values: 1/3 and 1/9 cycle of 8; 1/6 cycle of 16 (n = 1 to 32).

	3	6	9
1	3	2	1
2	6	3	2
3	1	5	3
4	4	6	4
5	7	8	5
6	2	1	6
7	5	3	7
8	8	4	8
9	3	6	1
10	6	7	2
11	1	1	3
12	4	2	4
13	7	4	5
14	2	5	6
15	5	7	7
16	8	8	8
17	3	2	1
18	6	3	2
19	1	5	3
20	4	6	4
21	7	8	5
22	2	1	6
23	5	3	7
24	8	4	8
25	3	6	1
26	6	7	2
27	1	1	3
28	4	2	4
29	7	4	5
30	2	5	6
31	5	7	7
32	8	8	8

1	2	3
1	2	3
4	5	6
7	8	9
10	11	12
13	14	15
16	17	18
19	20	21
22	23	24
25	26	27
28	29	30
31	32	33
34	35	36
37	38	39
40	41	42
43	44	45
46	47	48
49	50	51
52	53	54
55	56	57
58	59	60
61	62	63
64	65	66
67	68	69
70	71	72
73	74	75
76	77	78
79	80	81
82	83	84
85	86	87
88	89	90
91	92	93
94	95	96
97	98	99
100	101	102
103	104	105
106	107	108

n	1	2	3	4	5	6	7	8	9
1	1	2	3	4	5	6	7	8	9
2	2	4	6	8	10	12	14	16	18
3	3	6	9	12	15	18	21	24	27
4	4	8	12	16	20	24	28	32	36
5	5	10	15	20	25	30	35	40	45
6	6	12	18	24	30	36	42	48	54
7	7	14	21	28	35	42	49	56	63
8	8	16	24	32	40	48	56	64	72
9	9	18	27	36	45	54	63	72	81
10	10	20	30	40	50	60	70	80	90
11	11	22	33	44	55	66	77	88	99
12	12	24	36	48	60	72	84	96	108
13	13	26	39	52	65	78	91	104	117
14	14	28	42	56	70	84	98	112	126
15	15	30	45	60	75	90	105	120	135
16	16	32	48	64	80	96	112	128	144
17	17	34	51	68	85	102	119	136	153
18	18	36	54	72	90	108	126	144	162
19	19	38	57	76	95	114	133	152	171
20	20	40	60	80	100	120	140	160	180
21	21	42	63	84	105	126	147	168	189
22	22	44	66	88	110	132	154	176	198
23	23	46	69	92	115	138	161	184	207
24	24	48	72	96	120	144	168	192	216
25	25	50	75	100	125	150	175	200	225
26	26	52	78	104	130	156	182	208	234
27	27	54	81	108	135	162	189	216	243
28	28	56	84	112	140	168	196	224	252
29	29	58	87	116	145	174	203	232	261
30	30	60	90	120	150	180	210	240	270
31	31	62	93	124	155	186	217	248	279
32	32	64	96	128	160	192	224	256	288
33	33	66	99	132	165	198	231	264	297
34	34	68	102	136	170	204	238	272	306
35	35	70	105	140	175	210	245	280	315
36	36	72	108	144	180	216	252	288	324

	×1	×2	×3	×4	×5	×6	×7	×8	×9
1	1	2	3	4	5	6	7	8	9
2	2	4	6	8	1	3	5	7	9
3	3	6	9	3	6	9	3	6	9
4	4	8	3	7	2	6	1	5	9
5	5	1	6	2	7	3	8	4	9
6	6	3	9	6	3	9	6	3	9
7	7	5	3	1	8	6	4	2	9
8	8	7	6	5	4	3	2	1	9
9	9	9	9	9	9	9	9	9	9

	1	2	4	5	7	8
1	1	5	7	2	4	8
2	2	1	5	4	8	7
3	3	6	3	6	3	6
4	4	2	1	8	7	5
5	5	7	8	1	2	4
6	6	3	6	3	6	3
7	7	8	4	5	1	2
8	8	4	2	7	5	1
9	9	9	9	9	9	9

	3	6	9
1	3	2	1
2	6	3	2
3	1	5	3
4	4	6	4
5	7	8	5
6	2	1	6
7	5	3	7
8	8	4	8
9	3	6	1
10	6	7	2
11	1	1	3
12	4	2	4
13	7	4	5
14	2	5	6
15	5	7	7
16	8	8	8
17	3	2	1
18	6	3	2
19	1	5	3
20	4	6	4
21	7	8	5
22	2	1	6
23	5	3	7
24	8	4	8
25	3	6	1
26	6	7	2
27	1	1	3
28	4	2	4
29	7	4	5
30	2	5	6
31	5	7	7
32	8	8	8

1	2	3
1	2	3
4	5	6
7	8	9
10	11	12
13	14	15
16	17	18
19	20	21
22	23	24
25	26	27
28	29	30
31	32	33
34	35	36
37	38	39
40	41	42
43	44	45
46	47	48
49	50	51
52	53	54
55	56	57
58	59	60
61	62	63
64	65	66
67	68	69
70	71	72
73	74	75
76	77	78
79	80	81
82	83	84
85	86	87
88	89	90
91	92	93
94	95	96
97	98	99
100	101	102
103	104	105
106	107	108

n	1	2	3	4	5	6	7	8	9
1	1	2	3	4	5	6	7	8	9
2	2	4	6	8	10	12	14	16	18
3	3	6	9	12	15	18	21	24	27
4	4	8	12	16	20	24	28	32	36
5	5	10	15	20	25	30	35	40	45
6	6	12	18	24	30	36	42	48	54
7	7	14	21	28	35	42	49	56	63
8	8	16	24	32	40	48	56	64	72
9	9	18	27	36	45	54	63	72	81
10	10	20	30	40	50	60	70	80	90
11	11	22	33	44	55	66	77	88	99
12	12	24	36	48	60	72	84	96	108
13	13	26	39	52	65	78	91	104	117
14	14	28	42	56	70	84	98	112	126
15	15	30	45	60	75	90	105	120	135
16	16	32	48	64	80	96	112	128	144
17	17	34	51	68	85	102	119	136	153
18	18	36	54	72	90	108	126	144	162
19	19	38	57	76	95	114	133	152	171
20	20	40	60	80	100	120	140	160	180
21	21	42	63	84	105	126	147	168	189
22	22	44	66	88	110	132	154	176	198
23	23	46	69	92	115	138	161	184	207
24	24	48	72	96	120	144	168	192	216
25	25	50	75	100	125	150	175	200	225
26	26	52	78	104	130	156	182	208	234
27	27	54	81	108	135	162	189	216	243
28	28	56	84	112	140	168	196	224	252
29	29	58	87	116	145	174	203	232	261
30	30	60	90	120	150	180	210	240	270
31	31	62	93	124	155	186	217	248	279
32	32	64	96	128	160	192	224	256	288
33	33	66	99	132	165	198	231	264	297
34	34	68	102	136	170	204	238	272	306
35	35	70	105	140	175	210	245	280	315
36	36	72	108	144	180	216	252	288	324

	×1	×2	×3	×4	×5	×6	×7	×8	×9
1	1	2	3	4	5	6	7	8	9
2	2	4	6	8	1	3	5	7	9
3	3	6	9	3	6	9	3	6	9
4	4	8	3	7	2	6	1	5	9
5	5	1	6	2	7	3	8	4	9
6	6	3	9	6	3	9	6	3	9
7	7	5	3	1	8	6	4	2	9
8	8	7	6	5	4	3	2	1	9
9	9	9	9	9	9	9	9	9	9

	1	2	4	5	7	8
1	1	5	7	2	4	8
2	2	1	5	4	8	7
3	3	6	3	6	3	6
4	4	2	1	8	7	5
5	5	7	8	1	2	4
6	6	3	6	3	6	3
7	7	8	4	5	1	2
8	8	4	2	7	5	1
9	9	9	9	9	9	9

	3	6	9
1	3	2	1
2	6	3	2
3	1	5	3
4	4	6	4
5	7	8	5
6	2	1	6
7	5	3	7
8	8	4	8
9	3	6	1
10	6	7	2
11	1	1	3
12	4	2	4
13	7	4	5
14	2	5	6
15	5	7	7
16	8	8	8
17	3	2	1
18	6	3	2
19	1	5	3
20	4	6	4
21	7	8	5
22	2	1	6
23	5	3	7
24	8	4	8
25	3	6	1
26	6	7	2
27	1	1	3
28	4	2	4
29	7	4	5
30	2	5	6
31	5	7	7
32	8	8	8

1	2	3
1	2	3
4	5	6
7	8	9
10	11	12
13	14	15
16	17	18
19	20	21
22	23	24
25	26	27
28	29	30
31	32	33
34	35	36
37	38	39
40	41	42
43	44	45
46	47	48
49	50	51
52	53	54
55	56	57
58	59	60
61	62	63
64	65	66
67	68	69
70	71	72
73	74	75
76	77	78
79	80	81
82	83	84
85	86	87
88	89	90
91	92	93
94	95	96
97	98	99
100	101	102
103	104	105
106	107	108

n	1	2	3	4	5	6	7	8	9
1	1	2	3	4	5	6	7	8	9
2	2	4	6	8	10	12	14	16	18
3	3	6	9	12	15	18	21	24	27
4	4	8	12	16	20	24	28	32	36
5	5	10	15	20	25	30	35	40	45
6	6	12	18	24	30	36	42	48	54
7	7	14	21	28	35	42	49	56	63
8	8	16	24	32	40	48	56	64	72
9	9	18	27	36	45	54	63	72	81
10	10	20	30	40	50	60	70	80	90
11	11	22	33	44	55	66	77	88	99
12	12	24	36	48	60	72	84	96	108
13	13	26	39	52	65	78	91	104	117
14	14	28	42	56	70	84	98	112	126
15	15	30	45	60	75	90	105	120	135
16	16	32	48	64	80	96	112	128	144
17	17	34	51	68	85	102	119	136	153
18	18	36	54	72	90	108	126	144	162
19	19	38	57	76	95	114	133	152	171
20	20	40	60	80	100	120	140	160	180
21	21	42	63	84	105	126	147	168	189
22	22	44	66	88	110	132	154	176	198
23	23	46	69	92	115	138	161	184	207
24	24	48	72	96	120	144	168	192	216
25	25	50	75	100	125	150	175	200	225
26	26	52	78	104	130	156	182	208	234
27	27	54	81	108	135	162	189	216	243
28	28	56	84	112	140	168	196	224	252
29	29	58	87	116	145	174	203	232	261
30	30	60	90	120	150	180	210	240	270
31	31	62	93	124	155	186	217	248	279
32	32	64	96	128	160	192	224	256	288
33	33	66	99	132	165	198	231	264	297
34	34	68	102	136	170	204	238	272	306
35	35	70	105	140	175	210	245	280	315
36	36	72	108	144	180	216	252	288	324

	×1	×2	×3	×4	×5	×6	×7	×8	×9
1	1	2	3	4	5	6	7	8	9
2	2	4	6	8	1	3	5	7	9
3	3	6	9	3	6	9	3	6	9
4	4	8	3	7	2	6	1	5	9
5	5	1	6	2	7	3	8	4	9
6	6	3	9	6	3	9	6	3	9
7	7	5	3	1	8	6	4	2	9
8	8	7	6	5	4	3	2	1	9
9	9	9	9	9	9	9	9	9	9

	1	2	4	5	7	8
1	1	5	7	2	4	8
2	2	1	5	4	8	7
3	3	6	3	6	3	6
4	4	2	1	8	7	5
5	5	7	8	1	2	4
6	6	3	6	3	6	3
7	7	8	4	5	1	2
8	8	4	2	7	5	1
9	9	9	9	9	9	9

	3	6	9
1	3	2	1
2	6	3	2
3	1	5	3
4	4	6	4
5	7	8	5
6	2	1	6
7	5	3	7
8	8	4	8
9	3	6	1
10	6	7	2
11	1	1	3
12	4	2	4
13	7	4	5
14	2	5	6
15	5	7	7
16	8	8	8
17	3	2	1
18	6	3	2
19	1	5	3
20	4	6	4
21	7	8	5
22	2	1	6
23	5	3	7
24	8	4	8
25	3	6	1
26	6	7	2
27	1	1	3
28	4	2	4
29	7	4	5
30	2	5	6
31	5	7	7
32	8	8	8