Septenary

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The septenary (or septal, heptimal) numeral system is the base-7 number system, and uses the digits 0-6.

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 Septenary Decimal Septenary
1/2 1/2 = 0.3 1/26 1/35 = 0.016122650544
1/3 1/3 = 0.2 1/27 1/36 = 0.01546324
1/4 1/4 = 0.15 1/28 1/40 = 0.015
1/5 1/5 = 0.1254 1/29 1/41 = 0.0145536
1/6 1/6 = 0.1 1/30 1/42 = 0.0143
1/7 1/10 = 0.1 1/31 1/43 = 0.014031062154342
1/8 1/11 = 0.06 1/32 1/44 = 0.0135
1/9 1/12 = 0.053 1/33 1/45 = 0.0132520564
1/10 1/13 = 0.0462 1/34 1/46 = 0.0130421565362451
1/11 1/14 = 0.0431162355 1/35 1/50 = 0.01254
1/12 1/15 = 0.04 1/36 1/51 = 0.012346
1/13 1/16 = 0.035245631421 1/37 1/52 = 0.012161463
1/14 1/20 = 0.03 1/38 1/53 = 0.012
1/15 1/21 = 0.0316 1/39 1/54 = 0.011536432605
1/16 1/22 = 0.03 1/40 1/55 = 0.0114
1/17 1/23 = 0.0261143464055232 1/41 1/56 = 0.0112363262135202250565543034045314644161
1/18 1/24 = 0.025 1/42 1/60 = 0.01
1/19 1/25 = 0.024 1/43 1/61 = 0.010656
1/20 1/26 = 0.0231 1/44 1/62 = 0.0105365613
1/21 1/30 = 0.02 1/45 1/63 = 0.010423264552
1/22 1/31 = 0.0214064526 1/46 1/64 = 0.0103124052165635426145
1/23 1/32 = 0.0206251134364604155323 1/47 1/65 = 0.01020411224523503061534
1/24 1/33 = 0.02 1/48 1/66 = 0.01
1/25 1/34 = 0.0165 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)

Note: One feature of this system is that 3.1 (= 22/7) approximates π with a relative error of 0.04%.

In fiction[edit]

  • The Halo 3 Alternate Reality Game "IRIS" used a countdown clock in base-7.
  • In the Halo videogame series, the Forerunners use a base-7 number system.