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 numbers, ${\displaystyle \mathbb {R} }$, by a point denoted ∞. It is thus the set ${\displaystyle \mathbb {R} \cup \{\infty \}}$ with the standard arithmetic operations extended where possible, and is sometimes denoted by ${\displaystyle \mathbb {R} ^{*}}$ or ${\displaystyle {\widehat {\mathbb {R} }}.}$ The added point is called the point at infinity, because it is considered as a neighbour of both ends of the real line. More precisely, the point at infinity is the limit of every sequence of real numbers whose absolute values are increasing and unbounded.

The projectively extended real line may be identified with a real projective line in which three points have been assigned the specific values 0, 1 and ∞. The projectively extended real number line is distinct from the affinely extended real number line, in which +∞ and −∞ are distinct.

Unlike most mathematical models of numbers, this structure allows division by zero:

for nonzero a. In particular, 1 / 0 = ∞ and 1 / ∞ = 0, making the reciprocal function 1 / x a total function in this structure. The structure, however, is not a field, and none of the binary arithmetic operations are total – for example, 0 ⋅ ∞ is undefined, even though the reciprocal is total. It has usable interpretations, however – for example, in geometry, the slope of a vertical line is ∞.

The projectively extended real line extends the field of real numbers in the same way that the Riemann sphere extends the field of complex numbers, by adding a single point called conventionally ∞.

In contrast, the affinely extended real number line (also called the two-point compactification of the real line) distinguishes between +∞ and −∞.

The order relation cannot be extended to ${\displaystyle {\widehat {\mathbb {R} }}}$ in a meaningful way. Given a number a ≠ ∞, there is no convincing argument to define either a > ∞ or that a < ∞. Since ∞ can't be compared with any of the other elements, there's no point in retaining this relation on ${\displaystyle {\widehat {\mathbb {R} }}}$. However, order on ${\displaystyle \mathbb {R} }$ is used in definitions in ${\displaystyle {\widehat {\mathbb {R} }}}$.

Fundamental to the idea that ∞ is a point no different from any other is the way the real projective line is a homogeneous space, in fact homeomorphic to a circle. For example the general linear group of 2 × 2 real invertible matrices has a transitive action on it. The group action may be expressed by Möbius transformations (also called linear fractional transformations), with the understanding that when the denominator of the linear fractional transformation is 0, the image is ∞.