Bijective numeration is any numeral system in which every non-negative integer can be represented in exactly one way using a finite string of digits.
The name refers to the bijection (i.e. one-to-one correspondence) that exists in this case between the set of non-negative integers and the set of finite strings using a finite set of symbols (the "digits").
Most ordinary numeral systems, such as the common decimal system, are not bijective because more than one string of digits can represent the same positive integer.
Even though only the first is usual, the fact that the others are possible means that the decimal system is not bijective.
However, the unary numeral system, with only one digit, is bijective.
It uses a string of digits from the set {1, 2, ..., k} (where k ≥ 1) to encode each positive integer; a digit's position in the string defines its value as a multiple of a power of k. Smullyan (1961) calls this notation k-adic, but it should not be confused with the p-adic numbers: bijective numerals are a system for representing ordinary integers by finite strings of nonzero digits, whereas the p-adic numbers are a system of mathematical values that contain the integers as a subset and may need infinite sequences of digits in any numerical representation.
The base-k bijective numeration system uses the digit-set {1, 2, ..., k} (k ≥ 1) to uniquely represent every non-negative integer, as follows: In contrast, standard positional notation can be defined with a similar recursive algorithm where For base
numeration system could be extended to negative integers in the same way as the standard base-
numeral system by use of an infinite number of the digit
This is because the Euler summation meaning that and for every positive number
This representation is no longer bijective, as the entire set of left-infinite sequences of digits is used to represent the
, The bijective base-10 system is a base ten positional numeral system that does not use a digit to represent zero.
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 the bijective base-10 system.
Addition and multiplication in this system 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.
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.
(A=1, B=2, C=3, ..., Z=26) 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 WI represents the value 23 × 261 + 9 × 260 = 607 in base 10.
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 (214 in binary code), labeled from A to XFD.
A variant of this system is used to name variable stars.
[4] It can be applied to any problem where a systematic naming using letters is desired, while using the shortest possible strings.
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 Foster (1947) for the case k = 10, and Smullyan (1961) and Böhm (1964) for all k ≥ 1.
Smullyan uses this system to provide a Gödel numbering of the strings of symbols in a logical system; Böhm uses these representations to perform computations in the programming language P′′.
Forslund (1995) appears to be another rediscovery, and hypothesizes that if ancient numeration systems used bijective base-k, they might not be recognized as such in archaeological documents, due to general unfamiliarity with this system.