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).

Read more about this topic:  Bijective Numeration

Famous quotes containing the word definition:

    “According to our social pyramid, all men who feel displaced racially, culturally, and/or because of economic hardships will turn on those whom they feel they can order and humiliate, usually women, children, and animals—just as they have been ordered and humiliated by those privileged few who are in power. However, this definition does not explain why there are privileged men who behave this way toward women.”
    —Ana Castillo (b. 1953)

    “The man who knows governments most completely is he who troubles himself least about a definition which shall give their essence. Enjoying an intimate acquaintance with all their particularities in turn, he would naturally regard an abstract conception in which these were unified as a thing more misleading than enlightening.”
    —William James (1842–1910)

    “... if, as women, we accept a philosophy of history that asserts that women are by definition assimilated into the male universal, that we can understand our past through a male lens—if we are unaware that women even have a history—we live our lives similarly unanchored, drifting in response to a veering wind of myth and bias.”
    —Adrienne Rich (b. 1929)