 Mixed radix

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UrnfieldList of numeral system topics Positional systems by base Decimal (10) 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 20, 24, 30, 36, 60, 64 List of numeral systems Mixed radix numeral systems are nonstandard positional numeral systems in which the numerical base varies from position to position. Such numerical representation applies when a quantity is expressed using a sequence of units that are each a multiple of the next smaller one, but not by the same factor. Such units are common for instance in measuring time; a time of 32 weeks, 5 days, 7 hours, 45 minutes, 15 seconds, and 500 milliseconds might be expressed as a number of minutes in mixedradix notation as:
... 32, 5, 7, 45; 15, 500 ... ∞, 7, 24, 60; 60, 1000
or as
 32_{∞}5_{7}7_{24}45_{60}.15_{60}500_{1000}
In the tabular format, the digits are written above their base, and a semicolon is used to indicate the radix point. In numeral format, each digit has its associated base attached as a subscript, and the position of the radix point is indicated by a full stop. The base for each digit is the number of corresponding units that make up the next larger unit. As a consequence there is no base (written as ∞) for the first (most significant) digit, since here the "next larger unit" does not exist (and note that one could not add a larger unit of "month" or "year" to the sequence of units, as they are not integer multiples of "week").
Contents
Examples
A mixed radix numeral system can often benefit from a tabular summary. The familiar system for describing the 604800 seconds of a week starting from midnight on Sunday runs as follows:
Radix: 7 2 12 60 60 Denomination: day halfday hour minute second Place value (seconds): 86400 43200 3600 60 1 Digit translations … day: 0=Sunday, 1=Monday, 2=Tuesday, 3=Wednesday, 4=Thursday, 5=Friday, 6=Saturday halfday: 0=AM, 1=PM hour: 0 is written as "12" (!) In this numeral system, the mixed radix numeral 3_{7}1_{2}5_{12}51_{60}57_{60} seconds would be interpreted as 05:51:57 p.m. on Wednesday, and 0_{7}0_{2}0_{12}02_{60}24_{60} would be 12:02 :24 a.m. on Sunday. Ad hoc notations for mixed radix numeral systems are commonplace.
A second example of a mixed radix numeral system in current use is in the design and use of currency, where a limited set of denominations are printed or minted with the objective of being able to represent any monetary quantity; the amount of money is then represented by the number of coins or banknotes of each denomination. When deciding which denominations to create (and hence which radices to mix), a compromise is aimed for between a minimal number of different denominations, and a minimal number of individual pieces of coinage required to represent typical quantities. So, for example, in the UK, banknotes are printed for £50, £20, £10 and £5, and coins are minted for £2, £1, 50p, 20p, 10p, 5p, 2p and 1p—these follow the 125 series of preferred values. In theory, balanced ternary minimizes the number of pieces of coinage required to represent any quantity.
A historical example of a mixed radix numeral system is the system of Mayan numerals, which generally used base20, except for the second place (the "10s" in decimal) which was base18, so that the first two places counted up to 360 (an approximation to the number of days in the year).
Mixedradix representation is also relevant to mixedradix versions of the CooleyTukey FFT algorithm, in which the indices of the input values are expanded in a mixedradix representation, the indices of the output values are expanded in a corresponding mixedradix representation with the order of the bases and digits reversed, and each subtransform can be regarded as a Fourier transform in one digit for all values of the remaining digits.
Manipulation
Mixedradix numbers of the same base can be manipulated using a generalization of manual arithmetic algorithms. Conversion of values from one mixed base to another is easily accomplished by first converting the place values of the one system into the other, and then applying the digits from the one system against these.
APL includes operators to convert to and from mixedradix systems.
Factorial number system
Main article: Factorial number systemAnother proposal is the socalled factorial number system:
Radix 8 7 6 5 4 3 2 1 Place value 7! 6! 5! 4! 3! 2! 1! 0! Place value in decimal 5040 720 120 24 6 2 1 0 Highest digit allowed 7 6 5 4 3 2 1 0 For example, the biggest number that could be represented with six digits would be 543210 which equals 719 in decimal: 5×5! + 4×4! + 3×3! + 2×2! + 1×1! It might not be clear at first sight but the factorial based numbering system is unambiguous and complete. Every number can be represented in one and only one way because the sum of respective factorials multiplied by the index is always the next factorial minus one:
There is a natural mapping between the integers 0, ..., n! − 1 and permutations of n elements in lexicographic order, which uses the factorial representation of the integer, followed by an interpretation as a Lehmer code.
The above equation is a particular case of the following general rule for any radix (either standard or mixed) base representation which expresses the fact that any radix (either standard or mixed) base representation is unambiguous and complete. Every number can be represented in one and only one way because the sum of respective weights multiplied by the index is always the next weight minus one:
 , where ,
which can be easily proved with mathematical induction.
Prime radix number system
Another interesting proposal is the number system with successive prime numbers as radix, whose place values are primorial numbers:
radix: 17 13 11 7 5 3 2 place value: (p_{6}=13)# (p_{5}=11)# (p_{4}=7)# (p_{3}=5)# (p_{2}=3)# (p_{1}=2)# (p_{0}=1)# decimal: 30030 2310 210 30 6 2 1  where , and p_{j} = j^{th} prime, p_{0}# = p_{0} = 1.
References
 Donald Knuth. The Art of Computer Programming, Volume 2: Seminumerical Algorithms, Third Edition. AddisonWesley, 1997. ISBN 0201896842. Pages 65–66, 208–209, and 290.
 Georg Cantor. Über einfache Zahlensysteme, Zeitschrift für Math. und Physik 14(1869), 121–128.
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