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Mixed radix numeral systems are unique 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, 2 weeks, 5 days, 7 hours, 45 minutes, 15 seconds, and 500 milliseconds can be rendered in mixed-radix notation as:
<math>\begin{bmatrix} 2, 5, 7, 45, 15, 500 \\
\quad 7, 24, 60, 60; 1000
\end{bmatrix} \mbox{seconds}
<math>
The digits are written above their base, and a semicolon is used to indicate the radix point.
Mixed-radix numbers of the same base can be manipulated using a generalization of manual arithmetic algorithms.
Factorial based radix
An interesting proposal is a factorial based radix:
radix: 5! 4! 3! 2! 1!
decimal: 120 24 6 2 1
For example, the biggest number that could be represented with five digits would be 54321 which equals 719 in decimal: 5*5!+4*4!+3*3!+2*2!+1*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=1}^{n} {i}{i!} = {(n+1)!} - 1. <math>
This can be easily proved with mathematical induction.
The factorial based radix is also called the factoradic. There is a natural mapping between the integers 1 .. n! 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
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