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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} <br /> -\frac{\pi}{2} & \text{if} & x=-\infty\\<br /> \frac{\pi}{2} & \text{if} & x=+\infty\\<br /> atan(x) & \text{if} & \text{otherwise}\\<br /> \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!
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} <br /> -\frac{\pi}{2} & \text{if} & x=-\infty\\<br /> \frac{\pi}{2} & \text{if} & x=+\infty\\<br /> atan(x) & \text{if} & \text{otherwise}\\<br /> \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!