雙射記數

雙射記數系統。大多數數字系統，例如十進位制，都不是雙射的；因為不止一串數字可以表示同一個正整數：新增不會改變表示的值，例如「1」、「01」、「001」都表示數字「1」。而一進位制因為只有一個數字「1」所以必然「是」雙射的。雙射進位制-"k"記數系統是一個雙射進位制。使用集合{1, 2, ..., "k"}編碼正整數；值的位置定義為「k」的冪倍數。#重定向 稱此為k"-adic：用有限非零數字串表示普通整數的系統，而p"-adic數是包含整數作為子集的一個數學值系統，並且可能需要無限數字序列表示。定義.雙射進位制使用集合{1, 2, ..., "k"}來唯一編碼每個非負整數：當formula_1且formula_2formula_3是不小於的最小整數"x"相反，標準進位制可用類似遞迴演算法定義當formula_4擴充套件到整數.已隱藏部分未翻譯內容，歡迎參與[ 翻譯]。在formula_5進位制, the bijective base-formula_6 numeration system could be extended to negative integers in the same way as the standard base-formula_7 numeral system by use of an infinite number of the digit formula_8, where formula_9, represented as a left-infinite sequence of digits formula_10. This is because the Euler summationformula_11meaning thatformula_12and for every positive number formula_13 with bijective numeration digit representation formula_14 is represented by formula_15. For base formula_16, negative numbers formula_17 are represented by formula_18 with formula_19, while for base formula_20, negative numbers formula_17 are represented by formula_22. This is similar to how in signed-digit representations, all integers formula_13 with digit representations formula_14 are represented as formula_25 where formula_26. This representation is no longer bijective, as the entire set of left-infinite sequences of digits is used to represent the formula_6-adic integers, of which the integers are only a subset.性質.對於formula_28進位制：對於進位制formula_37，雙射十進位制.已隱藏部分未翻譯內容，歡迎參與[ 翻譯]。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.雙射26進位制.雙射26進位制可全用字母寫作。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.歷史.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 #重定向 for the case "k" = 10, and #重定向 and #重定向 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′′. #重定向 mentions the special case of "k" = 10, and #重定向 discusses the cases "k" ≥ 2. #重定向 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.