查看“︁可計算數”︁的源代码

{{expert|subject=數學|time=2018-08-22T14:06:05+00:00}}
{{NoteTA
|G1 = IT
|G2 = Math
|1 = zh-cn:域; zh-tw:體;
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{{Numbers}}
'''可計算數'''（{{lang-en|computable numbers}}），是[[数学]]名詞，是指可用有限次、會結束的[[算法]]計算到任意精確度的[[实数]]。可計算數也被稱為'''遞迴數'''、'''遞迴實數'''或'''可計算實數'''。

等效的定義可以用[[递归函数]]、[[图灵机]]及[[Λ演算|λ演算]]等演算法的形式表示法而得。可計算數形成[[實閉域]]，可以在許多數學應用上取代[[实数]]。

== 定義 ==
如果一個實數<math>a</math>能被某個[[可計算函數]] <math>f:\mathbb{N}\to\mathbb{Z}</math> 以下述方式來近似，那麼 <math>a</math> 就是一個可計算數：給定任何正整數<math>n</math>，函數值<math>f(n)</math>都滿足：
:<math>{f(n) - 1 \over n} \leq a \leq {f(n) + 1 \over n}</math>

== 不可計算數 ==
非可計算的實數即為'''不可計算數'''。1975年，計算機學家{{link-en|格里高里·柴廷|Gregory Chaitin}}做了一個有趣的實驗：選擇任意一種[[程式語言]]，隨意輸入一段[[程式碼]]，該程式碼能夠成功運行並且能夠在有限時間內終止的[[機率]]即為[[柴廷常數]]，這個數為一個經典的不可計算數。<ref>{{Cite news|url=https://www.guokr.com/article/11964/|title=比根号2更“无理”的数 {{!}} 科学人 {{!}} 果壳网 科技有意思|date=2011-03-09|accessdate=2018-06-30|archive-date=2019-06-05|archive-url=https://web.archive.org/web/20190605160428/https://www.guokr.com/article/11964/|dead-url=no}}</ref>

==相關條目==
* [[可定义数]]
* [[可计算函数]]、{{link-en|半可計算函數|Semicomputable function}}
* {{link-en|超计算问题|Transcomputational problem}}

== 相關書籍 ==
* {{cite book|title=科学技朮的哲学反思|publisher=清华大学出版社有限公司|year=2004|isbn=978-7-3020-8560-7|url=http://www.worldcat.org/title/ke-xue-ji-shu-de-zhe-xue-fan-si/oclc/56603547|accessdate=2018-06-30|pages=119|archive-date=2019-06-17|archive-url=https://web.archive.org/web/20190617104453/https://www.worldcat.org/title/ke-xue-ji-shu-de-zhe-xue-fan-si/oclc/56603547}}

== 參考資料 ==
=== 引-{}-用 ===
{{Reflist}}

=== 來源 ===
{{Reflist|list=
* Oliver Aberth 1968, ''Analysis in the Computable Number Field'', Journal of the Association for Computing Machinery (JACM), vol 15, iss 2, pp 276–299. This paper describes the development of the calculus over the computable number field.
* Errett Bishop and Douglas Bridges, ''Constructive Analysis'', Springer, 1985, {{ISBN|0-387-15066-8}}
* Douglas Bridges and Fred Richman. ''Varieties of Constructive Mathematics'', Oxford, 1987.
* Jeffry L. Hirst, Representations of reals in reverse mathematics, Bulletin of the Polish Academy of Sciences, Mathematics, 55, (2007) 303&ndash;316.
* <!-- [[Marvin Minsky]] -->[[马文·闵斯基]] 1967, ''Computation: Finite and Infinite Machines'', Prentice-Hall, Inc. Englewood Cliffs, NJ. No ISBN. Library of Congress Card Catalog No. 67-12342. His chapter §9 "The Computable Real Numbers" expands on the topics of this article.
* E. Specker, "Nicht konstruktiv beweisbare Sätze der Analysis" J. Symbol. Logic, 14 (1949) pp.&nbsp;145–158
* {{Citation | last = Turing | first = A.M. | publication-date = 1937 | year = 1936 | title = On Computable Numbers, with an Application to the Entscheidungsproblem | periodical = Proceedings of the London Mathematical Society | series = 2 | volume = 42 | issue = 1 | pages = 230–65 | url = http://www.abelard.org/turpap2/tp2-ie.asp | doi = 10.1112/plms/s2-42.1.230 | accessdate = 2018-08-22 | archive-date = 2004-04-03 | archive-url = https://web.archive.org/web/20040403210049/http://www.abelard.org/turpap2/tp2-ie.asp | dead-url = no }} (and {{Citation | last = Turing | first = A.M. | publication-date = 1937 | title = On Computable Numbers, with an Application to the Entscheidungsproblem: A correction | periodical = Proceedings of the London Mathematical Society | series = 2 | volume = 43 | issue = 6 | pages = 544–6 | doi = 10.1112/plms/s2-43.6.544 | year = 1938 }}). Computable numbers (and Turing's a-machines) were introduced in this paper; the definition of computable numbers uses infinite decimal sequences.
* Klaus Weihrauch 2000, ''Computable analysis'', Texts in theoretical computer science, Springer, {{ISBN|3-540-66817-9}}. §1.3.2 introduces the definition by nested sequences of intervals converging to the singleton real. Other representations are discussed in §4.1.
* Klaus Weihrauch, ''[http://eccc.uni-trier.de/static/books/A_Simple_Introduction_to_Computable_Analysis_Fragments_of_a_Book/ A simple introduction to computable analysis]''
* H. Gordon Rice. "Recursive real numbers." Proceedings of the American Mathematical Society 5.5 (1954): 784-791.
* V. Stoltenberg-Hansen, J. V. Tucker "Computable Rings and Fields" in ''Handbook of computability theory'' edited by E.R. Griffor. Elsevier 1999
}}

{{數的系統}}
[[Category:递归论]]
[[Category:計算理論]]
[[Category:数论]]