Septenary

Septenary

[sep-tuh-ner-ee]
The septenary numeral system is the base-7 number system, and uses the digits 0-6.

Multiplication table

1 2 3 4 5 6 10
2 4 6 11 13 15 20
3 6 12 15 21 24 30
4 11 15 22 26 33 40
5 13 21 26 34 42 50
6 15 24 33 42 51 60
10 20 30 40 50 60 100

Fractions

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 = 0.053...
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

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 = ) 1.6180339887498... (≈ 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)

One advantage of this system is that 3.1 (22/7) is a reasonably good approximation for Pi.

In Fiction

  • The Tau of Sci-fi Table-top battle game Warhammer 40,000 use a base-7 counting system.
  • In the TV series Doctor Who the time lords employ a number system using base 7.
  • The Halo 3 Alternate Reality Game "IRIS" used a countdown clock in base-7.
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