class letters of numbers

It is useful to define the following elements using symbols listed below.

[Def-2]
$\mathbb{N}$ : $1,2,3,\cdots $ are called natural numbers.

$\mathbb{Z}$ : Integers consist of natural numbers and negative natural numbers, including 0 and $\infty $ (i.e. $0, \pm 1, \pm 2,\cdots \pm \infty $).

$\mathbb{Q}$ : Rational numbers are numbers expressed by the fraction $p/q$  in which $p$ and $q$ are relative prime numbers and integers ($p,q\neq\pm\infty , q\neq 0$).

$\mathbb{R}$ : Real numbers consist of  rational and irrational numbers.

Although '0' does not fall in any class of numbers as it means empty, we decide to include 0 among integers, for convenience. $\pm \infty $ is not a number, too. However, I am going to use $n\rightarrow \infty $ in the explanation of epsilon-delta techniques. Therefore, we also include $\pm \infty $ among integers at this time. (Later, we will extend real number system including $\pm \infty $. )

Now, we need to make rules for calculations in which $p$ and $q$ are $\pm \infty $. I think you already know that. If necessary, then I will provide additional comments on that issue. The largest class is real numbers. The relation of inclusion is as follows:
\[ \mathbb{N} \subset \mathbb{Z} \subset  \mathbb{Q} \subset  \mathbb{R} \]
Perhaps you think the definition of irrational numbers is unclear. But please understand in this step that irrational numbers are not rational numbers, or in other words, they are numbers that cannot be expressed by dividing any $p$ by any $q$. In this case, $p$ and $q$ should be integers. You are familiar with $\sqrt{2}$, $\pi$, $e$, and so on. Nobody denies the existence of such irrational numbers.