Projectively extended real line: Difference between revisions

Revision as of 04:18, 15 June 2020

Revision as of 08:21, 15 June 2020

Line 1:

{{about|the extension of the reals by a single point at infinity|the extension by <{{math>|+\infty</math>∞}} and <math>-\infty</{{math>|–∞}}|Extended real number line}}

{{unreferenced|date=December 2014}}

[[Image:Real projective line.svg|right|thumb|

The projectively extended real line can be visualized as the real number line wrapped around a circle (by some form of [[stereographic projection]]) with an additional point at infinity.]]

In [[real analysis]], the '''projectively extended real line''' (also called the [[one-point compactification]] of the [[real line]]), is the extension of the [[number line]] by a point denoted <{{math>\infty</math>|∞}}. It is thus the set <math>\mathbb{R}\cup\{\infty\}</math> (where <math>\mathbb{R}</math> is the set of the [[real number]]s) with the standard arithmetic operations extended where possible, sometimes denoted by <math>\widehat{\mathbb{R}}.</math> The added point is called the [[point at infinity]], because it is considered as a neighbour of both [[End (topology)|ends]] of the real line. More precisely, the point at infinity is the [[limit (mathematics)|limit]] of every sequence of real numbers whose absolute values are increasing and unbounded.

The projectively extended real line may be identified with the [[projective line]] over the reals in which three points have been assigned specific values (e.g. {{math|0}}, {{math|1}}, and <math>\infty</{{math>|∞}}). The projectively extended real line must not be confused with the [[extended real number line]], in which <{{math>|+\infty</math>∞}} and <math>-\infty</{{math>|−∞}} are distinct.

==Dividing by zero==

Line 14:

:<math>\frac{a}{0} = \infty</math>

for nonzero ''a''. In particular <{{math>|1/0</math> {{=}} <math>\infty</math>∞}}, and moreover {{math|1/<math>\infty</math>∞ {{=}} 0}}, making [[Multiplicative inverse|reciprocal]], {{math|1/''x''}}, a [[total function]] in this structure. The structure, however, is not a [[field (mathematics)|field]], and none of the binary arithmetic operations are total, as witnessed for example by 0 • <{{math>\infty</math>|0⋅∞}} being undefined despite the reciprocal being total. It has usable interpretations, however – for example, in geometry, a vertical line has ''infinite'' [[slope]].

==Extensions of the real line==

Line 181:

As [[projectivity|projectivities]] preserve the harmonic relation, they form the [[automorphism]]s of the real projective line. The projectivities are described algebraically as [[homography|homographies]], since the [[real number]]s form a [[ring (mathematics)|ring]], according to the general construction of a [[projective line over a ring]]. Collectively they form the group [[PGL(2,R)]].

The projectivities which are their own inverses are called [[involution (mathematics)#Projective geometry|involutions]]. A '''hyperbolic involution''' has two [[fixed point (mathematics)|fixed point]]s. Two of these correspond to elementary, arithmetic operations on the real projective line: [[additive inverse|negation]] and [[multiplicative inverse|reciprocation]]. Indeed, 0 and <math>\infty</math>∞ are fixed under negation, while 1 and &minus;1 are fixed under reciprocation.

== Notes ==