Mixed radix

Mixed radix numeral systems are more general than the usual ones in that the numerical base may vary from position to position. Such numerical representation is advantageous when representing units that are equivalent to each other, but not by the same ratio. For example, 32 weeks, 5 days, 7 hours, 45 minutes, 15 seconds, and 500 milliseconds might be rendered relative to minutes in mixed-radix notation as:

...  3,  2, 5,  7, 45; 15,  500 , or as
... 10, 10, 7, 24, 60; 60, 1000
 
     310210577244560.15605001000

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 radix point's position is indicated by a full stop.

An MRN system can often benefit from a tabular summary. The familiar system for describing the 604800 seconds of a week starting from Sunday Midnight runs as follows:

                Radix:   7       2     12    60     60
         Denomination:  day  half-day 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
  half-day: 0–AM 1–PM
      hour: 0 is written as "12" (!)

So the MRN 3₇1₂5₁₂51₆₀57₆₀ seconds (from Midnight Sunday) is interpreted as 05:51:57 PM Wednesday, 0₇0₂0₁₂02₆₀24₆₀ as 12:02:24 AM Sunday. Ad-hoc notations for MRN 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 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 mint (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.

An example of a mixed radix numeral system in history is the system of Mayan numerals, which generally used base-20, except for the second place (the "10s" in decimal) which was base-18, so that the first two places counted up to 360 (an approximation to the number of days in the year).

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.

Factorial based radix

Main article: Factoradic

An interesting proposal is a factorial based radix, also known as factoradic:

radix:           6   5   4   3   2   1
place value:     5!  4!  3!  2!  1!  0!
decimal:       120  24   6   2   1   1

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! + 0×1!. It might not be clear at first sight but factorial based numbering system is also unambiguous. No number can be represented by more than one way because the sum of respective factorials multiplied by the index is always the next factorial minus one:

<math> \sum_{i=0}^{n} i\cdot i! = {(n+1)!} - 1. <math>

This can be easily proved with mathematical induction.

There is a natural mapping between the integers 0, ..., n! − 1 and permutations of n elements in lexicographic order, when the integers are expressed in factoradic form.

External resources

Donald E. Knuth. The Art of Computer Programming, Volume 2. pp209

External links

• "Comments on “An Arithmetic Free Parallel Mixed-Radix Conversion Algorithm”". Submitted to IEEE Trans. Circuits and Systems. By Antonio García,Student Member,IEEE, and Graham A. Jullien, Senior Member, IEEE. (PDF) (http://www.atips.ca/research/documents/ca/rns/1999_Comments-RNS.pdf) Contains additional references to MRN conversion.