Please check my answer and tell me if im thinking in proper way.
I have to find all equivalence classes in equivalence relation $\mathcal R$ in set $A = (k \in \Bbb Z: -230\le k \le 2003)$ defined as follows:
$m\mathcal R l \iff m^2 = l^2$ for $m,l \in \Bbb Z$
Equivalence class will look as follows: $[m]$ = {$l \in A: m\mathcal R l$}
Let's take a look numbers in this set. First of all I'm checking negative number:
$[-230]$ = {$\emptyset$} because here doesn't exist number such that any number can be power to negative number. It's contradiction.
$[0]$ = {$0$} I think that is easy and obvious.
Let's take positive number for example:
$[1]$ = {$-1,1$} because if we want have result 1, we can increase the positive and negative number (I mean $(-1)^2$ = $1$ and $1^1$ = $1$ ).
Summing up all possible equivalence classes we get 2004, because we have 2004 representative of class (from 0 to 2003).