# 可計算數

${\displaystyle \mathbb {N} \subseteq \mathbb {Z} \subseteq \mathbb {Q} \subseteq \mathbb {R} \subseteq \mathbb {C} }$

## 定義

${\displaystyle {f(n)-1 \over n}\leq a\leq {f(n)+1 \over n}}$

## 參考資料

### 引用

1. ^ 比根号2更“无理”的数 | 科学人 | 果壳网 科技有意思. 2011-03-09 [2018-06-30]. （原始内容存档于2019-06-05）.

### 來源

• 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) 303–316.
• 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. 145–158
• Turing, A.M., On Computable Numbers, with an Application to the Entscheidungsproblem, Proceedings of the London Mathematical Society, 2 42 (1), 1936, 42 (1): 230–651937 [2018-08-22], doi:10.1112/plms/s2-42.1.230, （原始内容存档于2004-04-03） (and Turing, A.M., On Computable Numbers, with an Application to the Entscheidungsproblem: A correction, Proceedings of the London Mathematical Society, 2 43 (6), 1938, 43 (6): 544–61937, doi:10.1112/plms/s2-43.6.544). 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, 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