Septenary

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The septenary numeral system is the base-7 number system, and uses the digits 0-6.

1 Multiplication table
2 Fractions
3 Irrational Numbers
4 In fiction

Multiplication table [edit]

*	1	2	3	4	5	6	10 (7)	11	12
1	1	2	3	4	5	6	10	11	12
2	2	4	6	11	13	15	20	22	24
3	3	6	12	15	21	24	30	33	36
4	4	11	15	22	26	33	40	44	51
5	5	13	21	26	34	42	50	55	63
6	6	15	24	33	42	51	60	66	105
10 (7)	10	20	30	40	50	60	100	110	120
11	11	22	33	44	55	66	110	121	132
12	12	24	36	51	63	105	120	132	144

Fractions [edit]

Fractions expressed in septenary will repeat a sequence of digits unless the denominator is a power of seven. Few fractions can be expressed in a finite number of digits:

Decimal	Septimal (periodic part)
1/2	1/2 = 0.3
1/3	1/3 = 0.2
1/4	1/4 = 0.15
1/5	1/5 = 0.1254
1/6	1/6 = 0.1
1/7	1/10 = 0.1
1/8	1/11 = 0.06
1/9	1/12	1/10	1/13 = 0.0462
1/12	1/15 = 0.04
1/14	1/20 = 0.03
1/15	1/21 = 0.0316
1/16	1/22 = 0.03
1/18	1/24 = 0.025
1/19	1/25 = 0.024
1/20	1/26 = 0.0231
1/21	1/30 = 0.02
1/24	1/33 = 0.02
...	...
1/49	1/100 = 0.01

Irrational Numbers [edit]

Algebraic irrational number		In decimal	In septenary
√2	(the length of the diagonal of a unit square)	1.41421356237309... (≈ 1.414)	1.262034545211232611 ... (≈ 1.262)
√3	(the length of the diagonal of a unit cube, or twice the height of an equilateral triangle of unit side)	1.73205080756887... (≈ 1.732)	1.506044021410166456 ... (≈ 1.506 ≈ 1.5)
√5	(the length of the diagonal of a 1×2 rectangle)	2.2360679774997... (≈ 2.236)	2.143654106250351 ... (≈ 2.144)
φ	(phi, the golden ratio = (1+√5)⁄2	1.6180339887498... (≈ 1.618)	1.42166203646016... (≈ 1.422)
Transcendental irrational number		In decimal	In septenary
π	(pi, the ratio of circumference to diameter)	3.1415926535898... (≈ 3.1416)	3.06636514320361341... (≈ 3.1)
e	(the base of the natural logarithm)	2.718281828459045... (≈ 2.718)	2.50124106542265... (≈ 2.5)