Factorial Number System
Another proposal is the so-called factorial number system:
| Radix | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 |
| Place value | 7! | 6! | 5! | 4! | 3! | 2! | 1! | 0! |
| Place value in decimal | 5040 | 720 | 120 | 24 | 6 | 2 | 1 | 1 |
| Highest digit allowed | 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 |
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! It might not be clear at first sight but the factorial based numbering system is unambiguous and complete. Every number can be represented in one and only one way because the sum of respective factorials multiplied by the index is always the next factorial minus one:
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:
- , where ,
which can be easily proved with mathematical induction.
Read more about this topic: Mixed Radix
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