Bijective numeration is any numeral system that establishes a bijection between the set of non-negative integers and the set of finite strings over a finite set of digits. A numeral is a symbol or group of symbols that represents a number. ... In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one-to-one and onto. ... The integers consist of the positive natural numbers (1, 2, 3, …) the negative natural numbers (−1, −2, −3, ...) and the number zero. ...

Bijective base-k numeration uses the specific digit-set {1, 2, ..., k}(k ≥ 1), and is as follows:

• The integer zero is represented by the empty string.

• The integer represented by the nonempty digit-string

a_na_n-1...a₁a₀

is

a_n*kⁿ + a_n-1*k^n-1 + ... + a₁*k¹ + a₀*k⁰.

• The digit-string representing the integer m > 0 is

a_na_n-1...a₁a₀

where

a₀ = m - q₀*k,   q₀ = f(m/k);

a₁ = q₀ - q₁*k,  q₁ = f(q₀/k);

a₂ = q₁ - q₂*k,  q₂ = f(q₁/k);

...

a_n = q_n-1 - 0*k, q_n = f(q_n-1/k) = 0;

and

f(x) = ceil(x) - 1,

ceil(x) being the least integer not less than x (the ceiling function).

Bijective base-k numeration is also called k-adic notation, not to be confused with the p-adic number system. Bijective base-1 is also called unary. In mathematics, the floor function is the function defined as follows: for a real number x, floor(x) is the largest integer less than or equal to x. ... The p-adic number systems were first described by Kurt Hensel in 1897. ... The unary numeral system is the simplest numeral system to represent natural numbers: in order to represent a number N, an arbitrarily chosen symbol is repeated N times. ...

Properties of bijective base-k numerals

For a given k ≥ 1,

• there are exactly kⁿ bijective base-k numerals of length n ≥ 0;

• a list of bijective base-k 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 bijective base-1, base-2 and base-3 numerals are as follows (with the ordinary decimal representations listed for comparison):

The unary numeral system is the simplest numeral system to represent natural numbers: in order to represent a number N, an arbitrarily chosen symbol is repeated N times. ...

Examples

34152(five-adic) = 3*5^4 + 4*5^3 + 1*5^2 + 5*5^1 + 2*5^0 = 2427(decimal).

119A(ten-adic) = 1*10^3 + 1*10^2 + 9*10^1 + 10*10^0 = 1200(decimal).

In the last example, the digit 'A' represents the integer ten.