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Mixed radix
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Mixed radix
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:
or as
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 one could not add a larger unit of "month" or "year" to the sequence of units, as they are not integer multiples of "week").
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. 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 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):
In this numeral system, the mixed radix numeral 37172451605760 seconds would be interpreted as 17:51:57 on Wednesday, and 0702402602460 would be 00:02:24 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 base 20 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 base 24 tun, k'atun, b'ak'tun, etc., tracks historical dates.
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Mixed radix
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:
or as
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 one could not add a larger unit of "month" or "year" to the sequence of units, as they are not integer multiples of "week").
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. 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 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):
In this numeral system, the mixed radix numeral 37172451605760 seconds would be interpreted as 17:51:57 on Wednesday, and 0702402602460 would be 00:02:24 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 base 20 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 base 24 tun, k'atun, b'ak'tun, etc., tracks historical dates.