Projectively extended real line: Difference between revisions

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In [[real analysis]], the '''projectively extended real line''' (also called the [[one-point compactification]] of the [[real line]]), is the extension of the set of the [[real number]]s, <math>\mathbb{R},</math> by a point denoted {{math|∞}}. It is thus the set <math>\mathbb{R}\cup\{\infty\}</math> with the standard arithmetic operations extended where possible, and is 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 [[bounded set|unbounded]].

The projectively extended real line may be identified with the [[real projective line]] over the reals in which three points have been assigned specific values (e.g. {{math|0}}, {{math|1}} and {{math|∞}}). The projectively extended real line is distinct from the [[extended real number line]], in which {{math|+∞}} and {{math|−∞}} are distinct.

==Dividing by zero==