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To the algebraist, $\mathbb{Z}$ is just the free group with one generator. To the algebraic topologist, $\mathbb{Z}$ is just the fundamental group of the circle. To be glib, what do $\mathbb{Z}$ mean to you?

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Please check Community Wiki. –  Jonas Meyer Jan 22 '10 at 7:18
    
He he: everyone's answering the wrong question! When translated into English, the question reads "To be glib, what do the integers mean to you?". –  Andrew Stacey Jan 22 '10 at 10:44
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Why I don't like this question: (1) $\mathbb Z$ is not "just" anything. An "algebraist" very well might study "rings and algebras", for example. (2) It's discussion-y. How are you going to pick a "right" answer? –  Theo Johnson-Freyd Jan 22 '10 at 16:29
    
I'm not sure how appropriate this is to MathOverflow, but I've certainly wondered the same thing and I'm interested in seeing people's answers. –  Zev Chonoles Jan 22 '10 at 16:40
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I agree with Theo. I don't really see what you expect to get out of asking this question. I don't like any of the answers, and it's hard to imagine somebody giving a good one. –  Anton Geraschenko Jan 22 '10 at 18:36
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closed as not a real question by Pete L. Clark, Mariano Suárez-Alvarez, Theo Johnson-Freyd, S. Carnahan, Anton Geraschenko Jan 22 '10 at 18:36

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

5 Answers

An infinite discrete subset of $\mathbb R$.

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From the coarse geometry perspective, it is R! –  Andrew Stacey Jan 22 '10 at 21:17
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Final object in the category of schemes.

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To a number theorist, shouldn't $\mathbb Z$ be the world?

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Initial object in the category of commutative rings.

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(Or the category of rings.) –  Kevin H. Lin Jan 22 '10 at 7:22
    
(Or the category of Z-algebras) –  Andrew Stacey Jan 22 '10 at 10:42
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