【双射记数】
双射记数系统（Bijective numeration）是一种表示数字的位置数值系统，每个非负整数都可使用有限数字串表示。该名称「双射」指的是非负整数集和用有限符号集的有限字符串集间存在双射（即一一对应）。
大多数数字系统，例如十进制，都不是双射的；因为不止一串数字可以表示同一个正整数：添加前导零不会改变表示的值，例如「1」、「01」、「001」都表示数字「1」。而一进制因为只有一个数字「1」所以必然「是」双射的。
双射进制-k记数系统是一个双射进位制。使用集合{1, 2, ..., k}（其中k≥1）编码正整数；值的位置定义为「k」的幂倍数。1961称此为k-adic：用有限非零数字串表示普通整数的系统，而p-adic数是包含整数作为子集的一个数学值系统，并且可能需要无限数字序列表示。
== 定义 ==
双射进位制使用集合{1, 2, ..., k}（其中k≥ 1）来唯一编码每个非负整数：
* 零由空字符串表示。
* 由非空数字串表示的整数
:anan−1 ... a1a0 = an kn + an−1 kn−1 + ... + a1 k1 + a0 k0.
* 表示整数的数字串m>0是anan−1 ... a1a0
当
:
且
:
是不小于的最小整数x（上取整函数）
相反，标准进位制可用类似递归算法定义当
:
=== 扩展到整数 ===
在进制, the bijective 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 , where , represented as a left-infinite sequence of digits . This is because the Euler summation
meaning that
and for every positive number  with bijective numeration digit representation  is represented by . For base , negative numbers  are represented by  with , while for base , negative numbers  are represented by . This is similar to how in signed-digit representations, all integers  with digit representations  are represented as  where . This representation is no longer bijective, as the entire set of left-infinite sequences of digits is used to represent the -adic integers, of which the integers are only a subset.
== 性质 ==
对于进制：
* 表示非负整数n的双射k进位位数是，与相比，k进制如果是「1」，位数就是n。
* 最小可表示为长度的双射k进制数字的非负整数是。
* 最大可表示为长度的双射k进制数字的非负整数是，相当于或。
* 非负整数n的双射k进制可和普通进制k相同，当且仅当普通进制不含数字「0」，或者等效地，双射进制既不是空字符串也不包含数字k。
对于进制，
* 会有个双射进制，长度为的k。1995
* 双射进制k的列表.。用λ表示空串，1、2、3、8、10、12、16为底的数如下（这里列出普通的表示方式以供比较）：
== 例 ==
* 双射五进制的34152 = 3×54 + 4×53 + 1×52 + 5×51 + 2×1 = 2427（十进制）
* 双射十进制119A（A代表数值10） = 1×103 + 1×102 + 9×101 + 10×1 = 1200（十进制）
** 双射11进制B = 11（十进制）
** 双射35进制Z = 35（十进制）
== 双射十进制 ==
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.
== 双射26进制 ==
双射26进制可全用字母写作（A, B, C...X, Y, Z, AA, AB, AC...ZX, ZY, ZZ, AAA, AAB, AAC...）。
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 ABC represents the value 1 × 262 + 2 × 261 + 3 × 260 = 731 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. It can be applied to any problem where a systematic naming using letters is desired, while using the shortest possible strings.
== 历史 ==
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 1947 for the case k = 10, and 1961 and 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′′. 1969 mentions the special case of k = 10, and 1973 discusses the cases k ≥ 2.  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.