1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

What are extended real numbers

  1. Jul 23, 2014 #1

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

    [tex]\overline{\mathbb{R}}:=\mathbb{R}\cup \{+\infty,-\infty\}[/tex]

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


    Extended explanation

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

    [tex]\overline{\mathbb{R}}:=\mathbb{R}\cup \{+\infty,-\infty\}[/tex]

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


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


    Multiplication is defined very similarly, for example

    [tex]2\cdot (+\infty)=+\infty[/tex]

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


    But division by 0 is still undefined.

    We can extend the order of [itex]\mathbb{R}[/itex] by setting


    for [itex]a\in \mathbb{R}[/itex].

    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

    [tex]\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}\\

    Now, we define the following metric;

    [tex]d:\overline{\mathbb{R}}\times\overline{\mathbb{R}}\rightarrow \mathbb{R}:(x,y)\rightarrow|\overline{atan}(x)-\overline{atan}(y)|[/tex]

    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 [itex]\lim_{x\rightarrow +\infty}{f(x)}=a[/itex] in the extended reals if and only if

    [tex]\forall \varepsilon >0:~ \exists P:~\forall x\geq P:~|f(x)-a|<\varepsilon[/tex]

    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!
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted