Projectively extended real line

with the standard arithmetic operations extended where possible,[1] and is sometimes denoted by

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.

[1] It has usable interpretations, however – for example, in geometry, the slope of a vertical line is ∞.

[1] 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 −∞.

Since ∞ can't be compared with any of the other elements, there's no point in retaining this relation on

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 ∞.

The detailed analysis of the action shows that for any three distinct points P, Q and R, there is a linear fractional transformation taking P to 0, Q to 1, and R to ∞ that is, the group of linear fractional transformations is triply transitive on the real projective line.

The terminology projective line is appropriate, because the points are in 1-to-1 correspondence with one-dimensional linear subspaces of

A motivation for the new definitions is the limits of functions of real numbers.

, with exceptions as indicated:[3][2] 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.

[a] Consequently, they are left undefined: The exponential function

However, since it is not an ordered set, the interval has a slightly different meaning.

):[2][additional citation(s) needed] With the exception of when the end-points are equal, the corresponding open and half-open intervals are defined by removing the respective endpoints.

[b] The open intervals as a base define a topology on

Sufficient for a base are the bounded open intervals in

The tools of calculus can be used to analyze functions of

The one-sided limit of f (x) as x approaches p from the right (left) is L, denoted if and only if for every neighbourhood A of L, there is a right-sided (left-sided) punctured neighbourhood B of p, such that

In particular, this is the case of polynomial functions, which take the value

Many discontinuous functions that become continuous when the codomain is extended to

When the real projective line is considered in the context of the real projective plane, then the consequences of Desargues' theorem are implicit.

In particular, the construction of the projective harmonic conjugate relation between points is part of the structure of the real projective line.

The projectivities are described algebraically as homographies, since the real numbers form a ring, according to the general construction of a projective line over a ring.

Two of these correspond to elementary, arithmetic operations on the real projective line: negation and reciprocation.