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What are extended real numbers

  1. Jul 23, 2014 #1
    Definition/Summary

    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.

    Equations



    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

    [tex]a+(+\infty)=+\infty[/tex]

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

    [tex]a+(-\infty)=-\infty[/tex]

    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

    [tex]\frac{1}{+\infty}=\frac{1}{-\infty}=0[/tex]

    But division by 0 is still undefined.

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

    [tex]-\infty<a<+\infty[/tex]

    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}\\
    \end{array}\right.[/tex]

    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
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