Definitions

# Bijective numeration

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. In particular, bijective base-k numeration represents a non-negative integer by using a string of digits from the set {1, 2, ..., k} (k ≥ 1) to encode the integer's expansion in powers of k. (Although slightly misleading, this is the terminology in the literature. Ordinary base-k numeration also establishes a bijection, but not in the required sense, due to the absence of leading zeros; for example, there are only 90 two-digit decimal numerals, rather than the required 102.) 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.

## 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

a0 = mq0 k,   q0 = f(m/k);

a1 = q0q1 k,  q1 = f(q0/k);

a2 = q1q2 k,  q2 = f(q1/k);

...

an = qn−1 − 0 kqn = f(qn−1/k) = 0;

and

f(x) = ceil(x) − 1,

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

## Properties of bijective base-k numerals

For a given k ≥ 1,

• there are exactly kn k-adic numerals of length n ≥ 0;
• 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 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 ...
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 ...

## Examples

(34152)five-adic = 3×54 + 4×53 + 1×52 + 5×51 + 2×50 = (2427)decimal.

(119A)ten-adic = 1×103 + 1×102 + 9×101 + 10×100 = (1200)decimal.

In the last example, the digit "A" represents the integer ten. For 10 ≤ k ≤ 35, it is common to use successive letters of a common alphabet for the digits after 9; e.g., bijective hexadecimal would use the sixteen digits {1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, G}.

## The bijective base-10 system

The bijective base-10 system is also known as decimal without a zero. It is a base ten positional numeral system which does not use a digit to represent zero; it instead has a digit to represent ten, such as "A".

As with conventional decimal, each digit position represents a power of ten, so for example 123 is "one hundred, plus two tens, plus three units". All positive integers which are represented solely with non-zero digits in conventional decimal (such as 123) have the same representation in decimal without a zero. Those which use a zero need to be rewritten, so for example 10 becomes A, conventional 20 becomes 1A, conventional 100 becomes 9A, conventional 101 becomes A1, conventional 302 becomes 2A2, conventional 1000 becomes 99A, conventional 1110 becomes AAA, conventional 2010 becomes 19AA, and so on.

Addition and multiplication in decimal without a zero are essentially the same as with conventional decimal, except that carries occur when a position exceeds ten, rather than when it exceeds nine. So to calculate 643 + 759, there are twelve units (write 2 at the right and carry 1 to the tens), ten tens (write A with no need to carry to the hundreds), thirteen hundreds (write 3 and carry 1 to the thousands), and one thousand (write 1), to give the result 13A2 rather than the conventional 1402.

## Systematic naming using the alphabet

Many spreadsheets including Microsoft Excel use the 26-adic counting system with the "digits" A-Z to label the columns of a spreadsheet, starting A, B, C... Z, AA, AB... AZ, BA... ZZ, AAA, etc. The numbering starts at 1 or A, so the empty string is not used. A variant of this system is used to name variable stars, it can be applied to any problem where a systematic naming using letters is desired, while using the shortest possible strings.

## Historical notes

The fact that every non-negative integer has a unique representation in bijective base-k (k ≥ 1), is a "folk theorem" that has been rediscovered many times. Early instances are Smullyan (1961) for the case k = 2, and Böhm (1964) for all k ≥ 1 (the latter using these representations to perform computations in the programming language P′′). Knuth (1969) mentions the special case of k = 10, and Salomaa (1973) discusses the cases k ≥ 2. Forslund (1995) considers that if ancient numeration systems used bijective base-k, they might not be recognised as such in archaeological documents, due to general unfamiliarity with this system. (The latter article is notable in that it does not cite existing literature on this system, but appears to unwittingly reinvent it.)

## References

• Böhm, C. "On a family of Turing machines and the related programming language", ICC Bulletin 3, p. 191, July 1964.
• Knuth, D. E. The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 1st ed., Addison-Wesley, 1969. (Solution to Exercise 4.1-24, p. 495., discusses bijective base-10.)
• Salomaa, A. Formal Languages, Academic Press, 1973. (Note 9.1, pp. 90-91, discusses bijective base-k for all k ≥ 2.)
• Smullyan, R. "Theory of Formal Systems", Annals of Mathematics Studies, Number 47, Princeton, 1961.