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 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").

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.

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