查看“︁整数集”︁的源代码

'''整数集'''('''the set of integers''')是由全体[[整数]]构成的[[集合]]。

== 记号 ==

{{Identity
|name=整数集
|symbol=<math>\mathbb{Z}</math>
|latex=\mathbb{Z}
|type=集合
}}
整数集通常记作 <math>\mathbb{Z}</math> 。

{{CharMetaInfo
|char=&#x2124;
|unicodeCodePoint={{UnicodeCodePoint|U+2115|Double-Struck Capital Z}}<ref>有别名{{UnicodeName|The Set of Integers}}。</ref>
|latex=\mathbb{Z}
}}

== 性质 ==

是[[可数集]]，势为 [[ℵ₀]] 。

代数结构：
* 关于整数集上的[[加法]]构成[[交换群]] <math>\langle\mathbb{Z},+,0\rangle</math> ，即[[整数加群]]，结构上是最典型的无限循环群。
* 非零部分（非零整数集 <math>Z^*</math>）关于整数集上的[[乘法]]构成[[交换幺半群]]。<math>\langle\mathbb{Z}^*,\times,1\rangle</math>。
* 关于整数上的[[加法]]和[[乘法]]构成一个[[交换幺环]]，即[[整数加乘环]]，是典型的[[整环]]（整环得名于整数集）。

序结构：
* 整数上的小于等于关系 <math>\leq</math> 是一个[[全序]]。

== 琐事 ==

=== 记号来源 ===

字母 Z 来自德语 {{Deu|Zahlen}} 。

<blockquote>
The use of the letter Z to denote the set of integers comes from the German word {{Deu|Zahlen}} ("numbers")<ref>Miller, Jeff (29 August 2010). [https://web.archive.org/web/20100131022510/http://jeff560.tripod.com/nth.html "Earliest Uses of Symbols of Number Theory"]. Archived from the original on 31 January 2010. Retrieved 20 September 2010.</ref><ref>Peter Jephson Cameron (1998). [https://books.google.com/books?id=syYYl-NVM5IC&pg=PA4 Introduction to Algebra]. Oxford University Press. p. 4. ISBN 978-0-19-850195-4. Archived from the original on 8 December 2016. Retrieved 15 February 2016.</ref> and has been attributed to David Hilbert.<ref>[https://books.google.com/books?id=Z-7kAAAAMAAJ The University of Leeds Review]. Vol. 31–32. University of Leeds. 1989. p. 46. Incidentally, Z comes from "Zahl": the notation was created by Hilbert.</ref>

-- Integer - Wikipedia<ref>https://en.wikipedia.org/wiki/Integer</ref>
</blockquote>