What are extended real numbers

[Greg Bernhardt]

Definition/Summary

Let $\mathbb{R}$ be the set of all real numbers. We can extend $\mathbb{R}$ by adjoining two elements $+\infty$ and $-\infty$. This forms the extended real number system. In notation:

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

The extended real numbers are being introduced to give an interpretation to limits such as $\lim_{x\rightarrow +\infty}{f(x)}$. Without introducing extended reals, the notation $x\rightarrow +\infty$ would just be a notation, nothing more. But after introducing the extended reals, we can work with $+\infty$ like other numbers.

Equations



Extended explanation

As stated in the introduction, we define the extended real numbers as

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

We can define arithmetic on the extended reals. If $a\neq -\infty$ is an extended real, then we set

$$a+(+\infty)=+\infty$$

The operation $(-\infty)+(+\infty)$ is left undefined. Likewise, if $a\neq +\infty$ is an extended real, we set

$$a+(-\infty)=-\infty$$

Multiplication is defined very similarly, for example

$$2\cdot (+\infty)=+\infty$$

The only limitation is that $0\cdot (+\infty)$ and $0\cdot (-\infty)$ are left undefined. Note that this implies that

$$\frac{1}{+\infty}=\frac{1}{-\infty}=0$$

But division by 0 is still undefined.

We can extend the order of $\mathbb{R}$ by setting

$$-\infty<a<+\infty$$

for $a\in \mathbb{R}$.

All of this was fairly obvious, but the fun starts when we want to define a metric on the extended reals. For this we define the extended arctangent function as

$$\overline{atan}(x):\overline{\mathbb{R}} \rightarrow [-\frac{\pi}{2},\frac{\pi}{2}]:x\rightarrow \left\{\begin{array}{ccc} -\frac{\pi}{2} & \text{if} & x=-\infty\\ \frac{\pi}{2} & \text{if} & x=+\infty\\ atan(x) & \text{if} & \text{otherwise}\\ \end{array}\right.$$

Now, we define the following metric;

$$d:\overline{\mathbb{R}}\times\overline{\mathbb{R}}\rightarrow \mathbb{R}:(x,y)\rightarrow|\overline{atan}(x)-\overline{atan}(y)|$$

With this definition, the extended reals become a compact, connected metric space. This definition also allows us to define limits and continuity in the ordinary sense, and these limits correspond to the limits without extended reals. For example $\lim_{x\rightarrow +\infty}{f(x)}=a$ in the extended reals if and only if

$$\forall \varepsilon >0:~ \exists P:~\forall x\geq P:~|f(x)-a|<\varepsilon$$

As a topological space, the extended reals are homeomorph to [0,1].

* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!

 

[fresh_42]

See also:
https://www.physicsforums.com/threads/why-non-standard-analysis-is-not-used.864899/https://www.physicsforums.com/threads/infinity-a-concept-or-number.605367/#post-3909986https://www.physicsforums.com/threa...gh-field-axioms-rules-of.638756/#post-4088365