What does $\mathbb{Z}_2$ mean?

Question

Wich number space is ment by: $\mathbb{Z}_2$

(I know that $\mathbb{Z}$ stands for Integer)

Answer 1

$\mathbb{Z}_n$, read "the integers mod $n$", essentially partitions the integers into $n$ "residue classes". The division algorithm provides a natural framework for assigning an arbitrary integer $x$ to a particular residue class. That is, by dividing that integer $x$ by $n$, we can rewrite $x$ as $an + r$ for some unique $a, r \in \mathbb{Z}$. While $a$ is known as the quotient of this division, we are really interested in the remainder, $r$.

When the division algorithm is carried out, the remainder $r$ lies strictly in the range $0 \leq r<n$, and hence there are $n$ possible values for $r$. These are the residue classes. So for example, $\mathbb{Z}_2 = \{0, 1\}$, since any number, when divided by $2$, will yield a remainder of either $0$ or $1$. This is intuitive because any given integer is either even or odd.

For a concrete example, let's take a look at $\mathbb{Z}_{10}$. By the above, we are partitioning the integers into $10$ residue classes. So to which does the integer $47$ belong? Well, the division algorithm tells us that $47 = 4 \cdot 10 + 7$, where $4$ is the quotient and $7$ the remainder. Hence, $47$ corresponds to the residue class "$7$" in $\mathbb{Z}_{10}$. Typically, this is written as $47 \equiv 7 \pmod{10}$.