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mixed radix
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- the content below is remote from Wikipedia
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{{Short description|Type of numeral systems}}{{numeral systems}}{{No footnotes|date=July 2021}}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:
- the content below is remote from Wikipedia
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... 32, 5, 07, 45; 15, 500
... â, 7, 24, 60; 60, 1000
or as
... â, 7, 24, 60; 60, 1000
32â5707244560.15605001000
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 â) for the first (most significant) digit, since here the "next larger unit" does not exist (and 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, both cardinally and ordinally, decimal years, decades, and centuries, septenary for days in a week, duodecimal months in a year, bases 28â31 for days within a month, as well as base 52 for weeks in a year. Time is further divided into hours counted in base 24 hours, sexagesimal minutes within an hour and seconds within a minute, with decimal fractions of the latter. A standard form for dates is {{samp|2021-04-10 16:31:15}}, which would be a mixed radix number by this definition, with the consideration that the quantities of days vary both per month, and with leap years. One proposed calendar instead uses base 13 months, quaternary weeks, and septenary days.A mixed radix numeral system is often best expressed with a table. A table describing what can be understood as the 604800 seconds of a week is as follows, with the week beginning on hour 0 of day 0 (midnight on Sunday):{| class="wikitable" style="text-align:right;"! {{rh}} | Radix| 60 | ||
| second | ||
| 1 |
Manipulation
Mixed-radix 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 and J include operators to convert to and from mixed-radix systems.Factorial number system
Another proposal is the so-called factorial number system:{| class="wikitable" style="text-align:right;"
sum_{i=0}^{n} (([i+1]+1)-1) cdot ([i]+1)! = ([n+1]+1)! - 1
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:
sum_{i=0}^{n} (m_{i+1} - 1) cdot M_i = M_{n+1} - 1 , where M_i = prod_{j=1}^{i} m_j, m_j > 1, M_0 = 1 ,
which can be easily proved with mathematical induction.References
- Donald Knuth. The Art of Computer Programming, Volume 2: Seminumerical Algorithms, Third Edition. Addison-Wesley, 1997. {{ISBN|0-201-89684-2}}. Pages 65–66, 208–209, and 290.
- Georg Cantor. Ãber einfache Zahlensysteme, Zeitschrift für Math. und Physik 14(1869), 121–128.
External links
- Mixed Radix Calculator â Mixed Radix Calculator in C
- content above as imported from Wikipedia
- "mixed radix" does not exist on GetWiki (yet)
- time: 9:21am EDT - Sat, May 18 2024
- "mixed radix" does not exist on GetWiki (yet)
- time: 9:21am EDT - Sat, May 18 2024
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