Editing Projectively extended real line (section)

== Arithmetic operations ==

=== Motivation for arithmetic operations ===
The arithmetic operations on this space are an extension of the same operations on reals. A motivation for the new definitions is the [[limit of a function|limits]] of functions of real numbers.

=== Arithmetic operations that are defined ===
In addition to the standard operations on the [[subset]] <math>\mathbb{R}</math> of <math>\widehat{\mathbb{R}}</math>, the following operations are defined for <math>a \in \widehat{\mathbb{R}}</math>, with exceptions as indicated:<ref>{{Cite book |last=Lee |first=Nam-Hoon |url=https://books.google.com/books?id=l3HgDwAAQBAJ&dq=%22Projectively+extended+real+line%22+-wikipedia&pg=PA255 |title=Geometry: from Isometries to Special Relativity |date=2020-04-28 |publisher=Springer Nature |isbn=978-3-030-42101-4 |language=en}}</ref><ref name=":1" />
:<math>\begin{align}
a + \infty = \infty + a & = \infty, & a \neq \infty \\
a - \infty = \infty - a & = \infty, & a \neq \infty \\
a / \infty = a \cdot 0 = 0 \cdot a & = 0, & a \neq \infty \\
\infty / a & = \infty, & a \neq \infty \\
a / 0 = a \cdot \infty = \infty \cdot a & = \infty, & a \neq 0 \\
0 / a & = 0, & a \neq 0
\end{align}</math>

=== Arithmetic operations that are left undefined ===
The following expressions cannot be motivated by considering limits of real functions, and no definition of them allows the statement of the standard algebraic properties to be retained unchanged in form for all defined cases.{{efn|An extension does however exist in which all the algebraic properties, when restricted to defined operations in <math>\widehat{\mathbb{R}}</math>, resolve to the standard rules: see [[Wheel theory]].}} Consequently, they are left undefined:
:<math>\begin{align}
& \infty + \infty \\
& \infty - \infty \\
& \infty \cdot 0 \\
& 0 \cdot \infty \\
& \infty / \infty \\
& 0 / 0
\end{align}</math>
The [[exponential function]] <math>e^x</math> cannot be extended to <math>\widehat{\mathbb{R}}</math>.<ref name=":1" />