Septenary

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

The first 20 counting numbers in base 7

Base 10	-	1	-	2	-	3	-	4	-	5	-	6	-	7	-	8	-	9	-	10	-	11	-	12	-	13	-	14	-	15	-	16	-	17	-	18	-	19	-	20	} 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+√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 The Tau of Sci-fi Table-top battle game Warhammer 40,000 use a base-7 counting system. In the New Series Adventures of 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. In the Halo videogame series, the Forerunners use a base-7 number system. In the online RPG Kingdom of Loathing, the Dwarven miners use a base-7 number system. References External links The first 1000 counting numbers in base 7-scientific-library.com