The difference between $\mathbb{Z}$ and $\mathbb{Z}^2$ I know that $\mathbb{Z}$ is the set of integers. But, what does $\mathbb{Z}^2$ mean? How is it different from $\mathbb{Z}$?
Thanks.
 A: $ \mathbb{Z} $ is defined as the set of integers, and in general for any set A, we can define $ A^2=A\times A=\left\{(a,b): a,b\in A\right\}$. According to this definition, $ \mathbb{Z}^2=\mathbb{Z}\times\mathbb{Z}=\left\{(a,b): a,b\in \mathbb{Z}\right\}$, so it's basically the set of vectors with 2 coordinates, when every coordinate is an integer.
A: If $A$ is a set then $A^2$ is a shorthand for $A\times A$, which is the set of ordered pairs with elements from $A$. That is: $$A^2=\{\langle a,b\rangle\mid a,b\in A\}.$$
So $\Bbb Z^2$ is the set of all ordered pairs whose elements are integers.
A: Instead of $\mathbb{Z}$ and $\mathbb{Z}^2$ I will draw for you $\mathbb{N},\mathbb{N}^2$.
The set $\mathbb{N}$ may be drawn as
$$\begin{array}{ccc}
*&*&*&*&*&*&*  &* &*&\cdots\end{array}$$, where the left most $*$ is the number zero. Now if you look at $\mathbb{N}^2$, you are looking at ordered pairs of numbers. We may visualize this as the following
$$\begin{array}{ccc}
.&.&.&.&.&.&.  &. &.\\
.&.&.&.&.&.&.  &. &.\\
.&.&.&.&.&.&.  &. &.\\
*&*&*&*&*&*&*  &* &*&\cdots\\  
*&*&*&*&*&*&*  &* &*&\cdots\\  
*&*&*&*&*&*&*  &* &*&\cdots\\ 
*&*&*&*&*&*&*  &* &*&\cdots\\ 
*&*&*&*&*&*&*  &* &*&\cdots\\ 
*&*&*&*&*&*&*  &* &*&\cdots\\ 
*&*&*&*&*&*&*  &* &*&\cdots\\ 
*&*&*&*&*&*&*  &* &*&\cdots\\ 
*&*&*&*&*&*&*  &* &*&\cdots\\   \end{array}$$
Where the lower left hand $*$ is the point $(0,0)$ and the point to the right of $(0,0)$ is $(1,0)$, and the point above the origin is $(0,1)$. Note that this is similar to the way we picture $\mathbb{R}$ as a line and $\mathbb{R}^2$ as  the $xy$-plane. What we did here was to look at only those points whose co-ordinates are positive integers. The picture for $\mathbb{Z}$ and $\mathbb{Z}^2$ is similar except that it is infinite in all four directions. As an exercise, how would one visulaize $[0,1]^2$ where $[0,1]$ is the set of reals between zero and one inclusive.   
