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| {{numeral systems}}
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| '''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 derives from this [[bijection]] (one-to-one correspondence) between the set of non-negative integers and the set of finite strings using a finite set of symbols (the "digits").
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| A '''bijective [[radix|base-''k'']] numeration''' is a bijective [[positional notation]]. 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''. Bijective base-''k'' numeration is also called ''k''-adic notation, not to be confused with the [[p-adic number|''p''-adic number system]].
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| 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. In particular, adding [[leading zero]]es does not change the value represented, so "1", "01" and "001" all represent the number [[1 (number)|one]]. Even though only the first is usual, the fact that the others are possible means that decimal is not bijective. However, [[unary numeral system|unary]], with only one digit, ''is'' bijective.
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| ==Definition==
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| The '''''k''-adic numeration system''' uses the digit-set {1, 2, ..., ''k''} (''k'' ≥ 1) to uniquely represent every non-negative integer, as follows:
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| * The integer zero is represented by the ''empty string''.
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| * The integer represented by the nonempty digit-string
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| ::''a''<sub>''n''</sub>''a''<sub>''n''−1</sub> ... ''a''<sub>1</sub>''a''<sub>0</sub>
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| :is
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| ::''a''<sub>''n''</sub> ''k''<sup>''n''</sup> + ''a''<sub>''n''−1</sub> ''k''<sup>''n''−1</sup> + ... + ''a''<sub>1</sub> ''k''<sup>1</sup> + ''a''<sub>0</sub> ''k''<sup>0</sup>.
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| * The digit-string representing the integer ''m'' > 0 is
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| ::''a''<sub>''n''</sub>''a''<sub>''n''−1</sub> ... ''a''<sub>1</sub>''a''<sub>0</sub>
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| :where
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| ::<math>
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| \begin{align}
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| a_0 & = & m - q_0 k , & & q_0 & = & f\left(\frac m k \right) & \\
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| a_1 & = & q_0 - q_1 k , & & q_1 & = & f\left(\frac {q_0} k \right) & \\
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| a_2 & = & q_1 - q_2 k , & & q_2 & = & f\left(\frac {q_1} k \right) & \\
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| & \vdots & & & & \vdots & & \\
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| a_n & = & q_{n-1} - 0 k , & & q_n & = & f\left(\frac {q_{n-1}} k \right) & = 0
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| \end{align}
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| </math>
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| :and
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| ::<math>f(x) = \lceil x \rceil - 1</math>,
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| :<math>\lceil x \rceil</math> being the least integer not less than ''x'' (the [[ceiling function]]).
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| ==Properties of bijective base-''k'' numerals==
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| For a given ''k'' ≥ 1,
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| * there are exactly ''k''<sup>''n''</sup> ''k''-adic numerals of length ''n'' ≥ 0;
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| * if ''k'' > 1, the number of digits in the ''k''-adic numeral representing a nonnegative integer ''n'' is [[Logarithm|<math>\lfloor \log_k (n+1) +\log_k(k - 1) \rfloor</math>]]{{cn|date=April 2013}}, in contrast to [[Floor and ceiling functions#Number of digits|<math>\lfloor \log_k n\rfloor +1 \ (n > 0)</math>]] for ordinary base-k numerals; if ''k'' = 1 (i.e., unary), then the number of digits is just ''n'';
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| * 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 binary and decimal representations are listed for comparison):
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| {| cellpadding="6"
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| |- align="right"
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| ! 1-adic:
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| | <tt> ε </tt>
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| | <tt> 1 </tt> || <tt> 11 </tt> || <tt> 111 </tt>
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| | <tt> 1111 </tt> || <tt> 11111 </tt> || <tt> ... </tt>
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| | colspan="11" align="left" |([[unary numeral system]])
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| |- align="right"
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| ! 2-adic:
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| | <tt> ε </tt>
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| | <tt> 1 </tt> || <tt> 2 </tt> || <tt> 11 </tt>
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| | <tt> 12 </tt>
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| | <tt> 21 </tt> || <tt> 22 </tt> || <tt> 111 </tt>
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| | <tt> 112 </tt> || <tt> 121 </tt>
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| | <tt> 122 </tt> || <tt> 211 </tt> || <tt> 212 </tt>
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| | <tt> 221 </tt>
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| | <tt> 222 </tt> || <tt> 1111 </tt> || <tt> 1112 </tt>
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| | <tt> ... </tt>
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| |- align="right"
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| ! binary:
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| | <tt> 0 </tt>
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| | <tt> 1 </tt> || <tt> 10 </tt> || <tt> 11 </tt>
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| | <tt> 100 </tt>
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| | <tt> 101 </tt> || <tt> 110 </tt> || <tt> 111 </tt>
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| | <tt> 1000 </tt> || <tt> 1001 </tt>
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| | <tt> 1010 </tt> || <tt> 1011 </tt> || <tt> 1100 </tt>
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| | <tt> 1101 </tt>
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| | <tt> 1110 </tt> || <tt> 1111 </tt> || <tt> 10000 </tt>
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| | <tt> ... </tt>
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| |- align="right"
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| ! 3-adic:
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| | <tt> ε </tt>
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| | <tt> 1 </tt> || <tt> 2 </tt> || <tt> 3 </tt>
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| | <tt> 11 </tt>
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| | <tt> 12 </tt> || <tt> 13 </tt> || <tt> 21 </tt>
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| | <tt> 22 </tt> || <tt> 23 </tt>
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| | <tt> 31 </tt> || <tt> 32 </tt> || <tt> 33 </tt>
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| | <tt> 111 </tt>
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| | <tt> 112 </tt> || <tt> 113 </tt> || <tt> 121 </tt>
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| | <tt> ... </tt>
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| |- align="right"
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| ! 10-adic:
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| | <tt> ε </tt>
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| | <tt> 1 </tt> || <tt> 2 </tt> || <tt> 3 </tt>
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| | <tt> 4 </tt>
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| | <tt> 5 </tt> || <tt> 6 </tt> || <tt> 7 </tt>
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| | <tt> 8 </tt> || <tt> 9 </tt>
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| | <tt> A </tt> || <tt> 11 </tt> || <tt> 12 </tt>
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| | <tt> 13 </tt>
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| | <tt> 14 </tt> || <tt> 15 </tt> || <tt> 16 </tt>
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| | <tt> ... </tt>
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| |- align="right"
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| ! decimal:
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| | <tt> 0 </tt> || <tt> 1 </tt>
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| | <tt> 2 </tt> || <tt> 3 </tt>
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| | <tt> 4 </tt> || <tt> 5 </tt> || <tt> 6 </tt>
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| | <tt> 7 </tt>
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| | <tt> 8 </tt> || <tt> 9 </tt> || <tt> 10 </tt>
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| | <tt> 11 </tt> || <tt> 12 </tt>
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| | <tt> 13 </tt> || <tt> 14 </tt> || <tt> 15 </tt>
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| | <tt> 16 </tt> || <tt> ... </tt>
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| |}
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| ==Examples==
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| :(34152)<sub>five-adic</sub> = 3×5<sup>4</sup> + 4×5<sup>3</sup> + 1×5<sup>2</sup> + 5×5<sup>1</sup> + 2×5<sup>0</sup> = (2427)<sub>decimal</sub>.
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| :(119A)<sub>ten-adic</sub> = 1×10<sup>3</sup> + 1×10<sup>2</sup> + 9×10<sup>1</sup> + 10×10<sup>0</sup> = (1200)<sub>decimal</sub>.
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| 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}.
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| ==The bijective base-10 system==
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| The bijective base-10 system is also known as '''decimal without a zero'''. It is a base [[10 (number)|ten]] positional [[numeral system]] that does not use a digit to represent [[0 (number)|zero]]. It instead has a digit to represent ten, such as ''A''.
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| 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 integer]]s which 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.
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| [[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.
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| The [[Greek numerals|system of common numerals]] used in [[ancient Greece]] prior to the [[Hellenistic Age]] was a bijective base-10 number system in which letters of the [[Greek alphabet]] were assigned values between 1 and 900. This was the system used to reckon the year based on the four-year [[Olympiads]], so for instance 480 BCE (the date of the [[Battle of Thermopylae]]) would be written ἔτει αʹ Ὀλυμπιάδος οδʹ, that is, the 1st year of the 74th Olympiad. These numbers are still commonly used in Greece for [[Ordinal number (linguistics)|ordinals]].
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| ==The bijective base-26 system==
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| {{See also|Hexavigesimal}}
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| The bijective base-26 system is also known as '''base 26 without a zero'''. It is a base [[26 (number)|twenty-six]] positional [[numeral system]] that does not use a digit to represent [[0 (number)|zero]]. It uses digits "A" to "Z" to represent [[1 (number)|one]] to [[26 (number)|twenty-six]].
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| The number sequence goes A, B, C, ..., X, Y, Z, AA, AB, AC, ..., AX, AY, AZ, BA, BB, BC, ...
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| Each digit position represents a power of twenty-six, so for example, ABC is "one 26<sup>2</sup>, plus two 26<sup>1</sup>, plus three 26<sup>0</sup>" since A is worth one, B is worth two, and C is worth three.
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| In this representation the number WIKIPEDIA is:
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| :23×26<sup>8</sup> + 9×26<sup>7</sup> + 11×26<sup>6</sup> + 9×26<sup>5</sup> + 16×26<sup>4</sup> + 5×26<sup>3</sup> + 4×26<sup>2</sup> + 9×26<sup>1</sup> + 1×26<sup>0</sup> = 4,878,821,185,187
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| ===Systematic naming using the alphabet===
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| Many [[spreadsheet]]s 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 star designation|variable stars]], it can be applied to any problem where a systematic naming using letters is desired, while using the shortest possible strings.
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| ==Historical notes==
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| The fact that every non-negative integer has a unique representation in bijective base-''k'' (''k'' ≥ 1), is a "[[mathematical folklore|folk theorem]]" that has been rediscovered many times. Early instances are [[James Foster|Foster, J. E.]] (1947), [[Raymond Smullyan|Smullyan]] (1961) for the case ''k'' = 2, and [[Corrado Böhm|Böhm]] (1964) for all ''k'' ≥ 1 (the latter using these representations to perform computations in the programming language [[P prime prime|P′′]]). [[Donald Knuth|Knuth]] (1969) mentions the special case of ''k'' = 10, and [[Arto Salomaa|Salomaa]] (1973) discusses the cases ''k'' ≥ 2. Forslund (1995) considers 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. (The latter article is notable in that it does not cite existing literature on this system, but appears to unwittingly reinvent it.)
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| ==References==
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| * [[Corrado Böhm|Böhm, C.]], "On a family of Turing machines and the related programming language", ''ICC Bulletin 3'', p. 191, July 1964.
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| * [[James Foster|Foster, J. E.]], "A Number System Without a Zero Symbol", ''Mathematics Magazine'', Vol.21, No.1 pp. 39-41, 1947.
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| * [[Donald Knuth|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.)
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| * [[Arto Salomaa|Salomaa, A.]], ''Formal Languages'', Academic Press, 1973. (Note 9.1, pp. 90-91, discusses bijective base-''k'' for all ''k'' ≥ 2.)
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| * [[Raymond Smullyan|Smullyan, R.]], "Theory of Formal Systems", ''Annals of Mathematics Studies'', Number 47, Princeton, 1961.
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| ==External links==
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| *[http://www.maths.soton.ac.uk/EMIS/journals/SWJPAM/vol1-95.html Forslund, R. R.: "A logical alternative to the existing positional number system", ''Southwest Journal of Pure and Applied Mathematics'', Volume 1 (September 1995), pp. 27-29.]
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| [[Category:Numeral systems]]
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| [[Category:Non-standard positional numeral systems]]
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