Bijective Numeration - Properties of Bijective Base-k Numerals

Properties of Bijective Base-k Numerals

For a given k ≥ 1,

  • there are exactly kn k-adic numerals of length n ≥ 0;
  • if k > 1, the number of digits in the k-adic numeral representing a nonnegative integer n is, in contrast to for ordinary base-k numerals; if k = 1 (i.e., unary), then the number of digits is just n;
  • a list of k-adic numerals, in natural order of the integers represented, is automatically in shortlex order (shortest first, lexicographical within each length). Thus, using ε to denote the empty string, the 1-, 2-, 3-, and 10-adic numerals are as follows (where the ordinary binary and decimal representations are listed for comparison):
1-adic: ε 1 11 111 1111 11111 ... (unary numeral system)
2-adic: ε 1 2 11 12 21 22 111 112 121 122 211 212 221 222 1111 1112 ...
binary: 0 1 10 11 100 101 110 111 1000 1001 1010 1011 1100 1101 1110 1111 10000 ...
3-adic: ε 1 2 3 11 12 13 21 22 23 31 32 33 111 112 113 121 ...
10-adic: ε 1 2 3 4 5 6 7 8 9 A 11 12 13 14 15 16 ...
decimal: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 ...

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