Projectively extended real line

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} ,}{\displaystyle \mathbb {R} ,} by a point denoted ∞. It is thus the set {\displaystyle \mathbb {R} \cup {\infty }}\mathbb {R} \cup {\infty } with the standard arithmetic operations extended where possible, and is sometimes denoted by {\displaystyle {\widehat {\mathbb {R} }}.}{\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.

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.

1 Dividing by zero

Unlike most mathematical models of the intuitive concept of ‘number’, this structure allows division by zero:

{\displaystyle {\frac {a}{0}}=\infty }{\frac {a}{0}}=\infty
for nonzero a. In particular 1/0 = ∞, and moreover 1/∞ = 0, making reciprocal, 1/x, a total function in this structure. The structure, however, is not a field, and none of the binary arithmetic operations are total, as witnessed for example by 0⋅∞ being undefined despite the reciprocal being total. It has usable interpretations, however – for example, in geometry, a vertical line has infinite slope.

2 Extensions of the real line

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 {\displaystyle \infty }\infty .

In contrast, the affinely extended real number line (also called the two-point compactification of the real line) distinguishes between {\displaystyle +\infty }+\infty and {\displaystyle -\infty }-\infty .

3 Order

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

4 Geometry

5 Arithmetic operations

5.1 Motivation for arithmetic operations

5.2 Arithmetic operations that are defined

5.3 Arithmetic operations that are left undefined

6 Algebraic properties

7 Intervals and topology

8 Interval arithmetic

9 Calculus

9.1 Neighbourhoods

9.2 Limits

9.2.1 Basic definitions of limits

9.2.2 Comparison with limits in ’“UNIQ--postMath-0000004C-QINU”'

9.2.3 Extended definition of limits

9.3 Continuity

10 As a projective range

11 Notes

12 See also