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The most familiar example of mixed-radix systems is in timekeeping and calendars. Western time radices include, both cardinally and ordinally, decimal years, decades, and centuries, septenary for days in a week, duodecimal months in a year, bases 28–31 for days within a month, as well as base 52 for weeks in a year.
General mixed radix systems were studied by Georg Cantor. [2] The term "factorial number system" is used by Knuth, [3] while the French equivalent "numération factorielle" was first used in 1888. [4] The term "factoradic", which is a portmanteau of factorial and mixed radix, appears to be of more recent date. [5]
"A base is a natural number B whose powers (B multiplied by itself some number of times) are specially designated within a numerical system." [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]
More general is using a mixed radix notation (here written little-endian) like for + +, etc. This is used in Punycode , one aspect of which is the representation of a sequence of non-negative integers of arbitrary size in the form of a sequence without delimiters, of "digits" from a collection of 36: a–z and 0–9, representing 0–25 and 26 ...
In contrast to decimal, or radix 10, which has a ones' place, tens' place, hundreds' place, and so on, radix b would have a ones' place, then a b 1 s' place, a b 2 s' place, etc. [2] For example, if b = 12, a string of digits such as 59A (where the letter "A" represents the value of ten) would represent the value 5 × 12 2 + 9 × 12 1 + 10 × ...
For example, 100 in decimal has three digits, so its cost of representation is 10×3 = 30, while its binary representation has seven digits (1100100 2), so the analogous calculation gives 2×7 = 14. Likewise, in base 3 its representation has five digits (10201 3 ), for a value of 3×5 = 15, and in base 36 (2S 36 ) one finds 36×2 = 72.
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The generalization to radix representations, for >, and to =, is a digit-reversal permutation, in which the base-digits of the index of each element are reversed to obtain the permuted index. The same idea can also been generalized to mixed radix number systems. In such cases, the digit-reversal permutation should simultaneously reverses the ...