📂Analysis

Extended Real Number System

Definition

The set defined as follows is called the extended real number system.

$$\overline{ \mathbb{R} } := \mathbb{R} \cup \left\{ -\infty, +\infty\right\}$$

Explanation

In fields such as analysis, for convenience, the set $\mathbb{R}$ is often replaced with $\overline{ \mathbb{R} }$. $\pm \infty$ is not a number, but for convenience, it is treated as one and added to $\mathbb{R}$. Within the extended real number system, the rules for comparison and operations are as follows.

---

For all $x \in \mathbb{R}$,

$$-\infty < x <+\infty$$

$$(\pm \infty) + (\pm \infty) = \pm \infty$$

$$x + (\pm \infty)=\pm \infty+x=\pm \infty$$

$$\dfrac{x}{+\infty}=0=\dfrac{x}{-\infty}$$

$$(\pm \infty)(\pm\infty)=+ \infty$$

$$(\pm \infty)(\mp \infty)=- \infty$$

$$x(\pm \infty)=(\pm \infty)x=\begin{cases} \pm \infty & x>0 \\ 0 & x=0 \\ \mp \infty & x<0 \end{cases}$$

Note that $(\pm \infty)+(\mp \infty)$ is undefined.

Theorem

• $\overline{ \mathbb{R} }$ is a complete ordered set.

• For a given $A \subset \overline{ \mathbb{R} }$, there exists $\sup A$ and $\inf A$.

• For $a \in \mathbb{R}$, $(a, +\infty]$ is a neighborhood of $+\infty$.