If a number is chosen from $[-n,n]$ is that the same as being chosen from $\mathbb{Z}_n$? Since the operations are modular, what meaning does the negative part of the range have?

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    $\begingroup$ Can you provide us with context to this? (maybe a link to the paper or title + author?) $\endgroup$
    – SEJPM
    May 26 '16 at 18:22
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    $\begingroup$ Are you sure the operations are modular? For many programming languages (Java, C etc.) the % operator is the remainder operator, not the modulo operator. $\endgroup$
    – Maarten Bodewes
    May 26 '16 at 18:27

We can present $\mathbb{Z}_m$ in different manner. For example, bellow sets are some equivalence classes of $\mathbb{Z}_3$ $$\{0,1,2\}, \{3,4,5\},\{-3,-2,-1\}. $$ Now, about your question

  • If you select the numbers of $[-n,n]$ module a positive integer $m$ which $m\leq 2n+1$, you can construct $\mathbb{Z}_m$:

$$\mathbb{Z}_m=\{i \pmod m \mid i\in[-n,n]\}$$ but, in above set, we may have some repetitive numbers. So we can construct $\mathbb{Z}_m$ as follows: $$\mathbb{Z}_m=\{i \pmod m \mid i\in[0,m-1]\}=\{i\mid i \in[0,m-1]\}=\{0,...,m-1\}.$$

  • We have $2n+1$ numbers in the range $[-n,n]$, so we can say that $[-n,n]$ is same as $\mathbb{Z}_{2n+1}$ (not $\mathbb{Z}_n$).

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