Bijective Numeration - Definition

Definition

The k-adic numeration system uses the digit-set {1, 2, ..., k} (k ≥ 1) to uniquely represent every non-negative integer, as follows:

  • The integer zero is represented by the empty string.
  • The integer represented by the nonempty digit-string
anan−1 ... a1a0
is
an kn + an−1 kn−1 + ... + a1 k1 + a0 k0.
  • The digit-string representing the integer m > 0 is
anan−1 ... a1a0
where
\begin{align} a_0 & = & m - q_0 k, & & q_0 & = & f\left(\frac m k \right) & \\ a_1 & = & q_0 - q_1 k, & & q_1 & = & f\left(\frac {q_0} k \right) & \\ a_2 & = & q_1 - q_2 k, & & q_2 & = & f\left(\frac {q_1} k \right) & \\ & \vdots & & & & \vdots & & \\ a_n & = & q_{n-1} - 0 k, & & q_n & = & f\left(\frac {q_{n-1}} k \right) & = 0 \end{align}
and
,
being the least integer not less than x (the ceiling function).

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