Extended real number line

Description

The extended real number line is the set $\overline{\mathbb{R}} = \mathbb{R} \cup \{-\infty, +\infty\}$. It has two new elements, $-\infty$ and $+\infty$, that are not real numbers. Its order extends the order of $\mathbb{R}$, with $-\infty < x < +\infty$ for each real number $x$.

Remarks

In $\overline{\mathbb{R}}$, each subset has a supremum and an infimum. For example, $\sup \mathbb{N} = +\infty$ and $\sup \emptyset = -\infty$. Some operations extend: $x + \infty = +\infty$ for each real $x$. Other expressions, such as $\infty - \infty$, stay undefined. The set $\overline{\mathbb{R}}$ is not a field.

Explanations

L1 human

The extended real line adds two end points to the number line, one at each end. This lets you write $\lim_{x \to \infty}$ and $\sup$ of unbounded sets with symbols instead of words.