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Mixed radix numeral systems are non-standard 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 mixed-radix notation as: ... 32, 5, 7, 45; 15, 500 ... ∞, 7, 24, 60; 60, 1000 or as :32∞577244560.15605001000 In the tabular format, the digits are written above their base, and a semicolon indicates the radix point. In numeral format, each digit has its associated base attached as a subscript, and the radix point is marked by a full stop or period. 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"). ==Examples== The most familiar example of mixed radix systems is in timekeeping and calendars. Western time radices include decimal centuries, decades and years as well as duodecimal months, trigesimal (and untrigesimal) days, overlapped with base 52 weeks and septenary days. One variant uses tridecimal months, quaternary weeks, and septenary days. Time is further divided by quadrivigesimal hours, sexagesimal minutes and seconds, then decimal fractions thereof. A mixed radix numeral system can often benefit from a tabular summary. The system for describing the 604800 seconds of a week starting from midnight on Sunday runs as follows: In this numeral system, the mixed radix numeral 371251251605760 seconds would be interpreted as 5:51:57 p.m. on Wednesday, and 070201202602460 would be 12:02:24 a.m. on Sunday. ''Ad hoc'' notations for mixed radix numeral systems are commonplace. The Maya calendar consists of several overlapping cycles of different radices. A short count ''tzolk'in'' overlaps vigesimal named days with tridecimal numbered days. A ''haab''' consists of vigesimal days, octodecimal ''months'', and base-52 years forming a ''round''. In addition, a ''long count'' of vigesimal days, octodecimal ''winal'', then vigesimal ''tun'', ''k'atun'', ''b'ak'tun'', etc. tracks historical dates. 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 1-2-5 series of preferred values. Mixed-radix representation is also relevant to mixed-radix versions of the Cooley-Tukey FFT algorithm, in which the indices of the input values are expanded in a mixed-radix representation, the indices of the output values are expanded in a corresponding mixed-radix 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. Primary notation uses increasing consecutive prime numbers for each radix. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Mixed radix」の詳細全文を読む スポンサード リンク
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