Extended Real Numbers

Last updated Nov 1, 2022

We define the Extended Real Numbers as

$$\bar{R} := \mathbb{R} \cup {-\infty, \infty}$$ where $-\infty < x < \infty$ $\forall x \in \mathbb{R}$. If we want to work with an interval in $\bar{\mathbb{R}}$ that includes $\infty$, we put a square bracket on $\infty$. For example, to write the Closed Interval from $x \in \mathbb{R}$ to (including) $\infty$, we write $[x, \infty]$. Likewise for $-\infty$.

Some arithmetic rules

• $\infty + x = \infty$ $\forall x \in (-\infty, \infty]$
• $- \infty + x = - \infty$ $\forall x \in [-\infty, \infty)$
• $\pm \infty * x = \pm \text{sgn}(x) \infty$ $\forall x \in \bar{\mathbb{R}} \setminus {0}$
• $\frac{x}{\pm \infty} = 0$ $\forall x \in \mathbb{R}$

The remaining arithmetic possiblities are undefined:

• $\infty - \infty$
• $\pm \infty * 0$
• $\frac{\infty}{\infty}$ for any choice of signs for $\infty$

We can view the Extended Real Numbers as a Compactification of $\mathbb{R}$. We want to attach points ${- \infty, \infty}$ to $\mathbb{R}$ so that TODO