real number system 7

We address how to handle $\pm \infty$ in real number system. It is said to be a extended real number system.

In general, $x\in \mathbb{R}$ means $-\infty<x<+\infty$. However, if $x\in \mathbb{R}$ is not bounded above, we understand $x=+\infty$. It is convenient to make the rule of the fictitious number $\infty$.

["$\infty$"] for any $x\in \mathbb{R}, (x\ne 0)$ , that is, $-\infty<x<+\infty, (x\ne 0)$,
$x+\infty=\infty$, $x-\infty=-\infty$, $\infty+\infty=\infty$,
$(\infty)\cdot (\infty)=\infty$, $(-\infty)\cdot (-\infty)=\infty$, $(-\infty)\cdot (\infty)=-\infty$,
If $x>0$, then $x\cdot (+\infty)=\infty$, $x\cdot (-\infty)=-\infty$,
If $x<0$, then $x\cdot (+\infty)=-\infty$, $x\cdot (-\infty)=\infty$,
$\frac{x}{+\infty}=0$, $\frac{x}{-\infty}=0$,

Unfortunately we are not able to define the following forms.
$\infty-\infty$, $\frac{\pm\infty}{\pm\infty}$, $0\cdot\infty$

Therefore, the extended real number system is ordered, but it is not a field. Do not take any notice of these definitions.