Bijective numeration

Definition

The base-k bijective 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

is

The digit-string representing the integer m > 0 is

whereand ⌈ x ⌉ being the least integer not less than x (the ceiling function).

Properties of bijective base-k numerals

For a given base k ≥ 1,

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

if k > 1, the number of digits in the bijective base-k numeral representing a nonnegative integer n is ⌊ log k ⁡ ( n + 1 ) + log k ⁡ ( k − 1 ) ⌋ , in contrast to ⌊ log k ⁡ n ⌋ + 1   ( n > 0 ) for ordinary base-k numerals; if k = 1 (i.e., unary), then the number of digits is just n;

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 0 to denote the empty string, the base 1, 2, 3, 8, 10, 12, and 16 numerals are as follows (where the ordinary representations are listed for comparison):

Examples

34152 (in bijective base-5) = 3×5⁴ + 4×5³ + 1×5² + 5×5¹ + 2×1 = 2427 (in decimal).119A (in bijective base-10, with "A" representing the digit value ten) = 1×10³ + 1×10² + 9×10¹ + 10×1 = 1200 (in decimal).

The bijective base-10 system

The bijective base-10 system is a base ten positional numeral system that 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 that are represented solely with non-zero digits in conventional decimal (such as 123) have the same representation in decimal without a zero. Those that use a zero must 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.

The bijective base-26 system

In the bijective base-26 system one may use the Latin alphabet letters "A" to "Z" to represent the 26 digit values one to twenty-six.

With this choice of notation, the number sequence (starting from 1) begins A, B, C, ..., X, Y, Z, AA, AB, AC, ..., AX, AY, AZ, BA, BB, BC, ...

Each digit position represents a power of twenty-six, so for example, the numeral ABC represents the value 1 × 26² + 2 × 26¹ + 3 × 26⁰.

Many spreadsheets including Microsoft Excel use this system to assign labels to the columns of a spreadsheet, starting A, B, C, ..., Z, AA, AB, ..., AZ, BA, ..., ZZ, AAA, etc. For instance, in Excel 2013, there can be up to 16384 columns, labeled from A to XFD. 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.