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It is the repeating part in the decimal expansion of the rational number 1 / 7 = 0. 142857. Thus, multiples of 1 / 7 are simply repeated copies of the corresponding multiples of 142857:
For example, in duodecimal, 1 / 2 = 0.6, 1 / 3 = 0.4, 1 / 4 = 0.3 and 1 / 6 = 0.2 all terminate; 1 / 5 = 0. 2497 repeats with period length 4, in contrast with the equivalent decimal expansion of 0.2; 1 / 7 = 0. 186A35 has period 6 in duodecimal, just as it does in decimal.
[1]: 38 The term is not equivalent to radix, as it applies to all numerical notation systems (not just positional ones with a radix) and most systems of spoken numbers. [1] Some systems have two bases, a smaller (subbase) and a larger (base); an example is Roman numerals, which are organized by fives (V=5, L=50, D=500, the subbase) and tens (X ...
For example, decimal 365 (10) or senary 1 405 (6) corresponds to binary 1 0110 1101 (2) (nine bits) and to ternary 111 112 (3) (six digits). However, they are still far less compact than the corresponding representations in bases such as decimal – see below for a compact way to codify ternary using nonary (base 9) and septemvigesimal (base 27).
For example, in the decimal system (base 10), the numeral 4327 means (4×10 3) + (3×10 2) + (2×10 1) + (7×10 0), noting that 10 0 = 1. In general, if b is the base, one writes a number in the numeral system of base b by expressing it in the form a n b n + a n − 1 b n − 1 + a n − 2 b n − 2 + ... + a 0 b 0 and writing the enumerated ...
Of particular interest are the quater-imaginary base (base 2i) and the base −1 ± i systems discussed below, both of which can be used to finitely represent the Gaussian integers without sign. Base −1 ± i, using digits 0 and 1, was proposed by S. Khmelnik in 1964 [3] and Walter F. Penney in 1965. [4] [6]
Balanced ternary is a ternary numeral system (i.e. base 3 with three digits) that uses a balanced signed-digit representation of the integers in which the digits have the values −1, 0, and 1.
If an unknown weight W is balanced with 3 (3 1) on its pan and 1 and 27 (3 0 and 3 3) on the other, then its weight in decimal is 25 or 10 1 1 in balanced base-3. 10 1 1 3 = 1 × 3 3 + 0 × 3 2 − 1 × 3 1 + 1 × 3 0 = 25.