extended real numbers

The set R union {∞}, obtained by adjoining one improper element to the set R of real numbers, is the set of projectively extended real numbers. Although notation is not completely standardized, R^* is used here to denote this set of extended real numbers. With an appropriate topology, R^* is the one-point compactification (or projective closure) of R. As shown above, the cross section of the Riemann sphere consisting of its "real axis" and "north pole" can be used to visualize R^*. The improper element, projective infinity (∞), then corresponds with the ideal point, the "north pole."

affinely extended real numbers | compactification | interval arithmetic | real number | Riemann sphere | set closure