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extended real numbers∗ matte† 2013-03-21 16:14:28 The extended real numbers are the real numbers together with +∞ (or simply ∞) and −∞. This set is usually denoted by R or [−∞, ∞], and the elements +∞ and −∞ are called plus and minus infinity, respectively. (N.B., “R” may sometimes mean the algebraic closure of R; see the special notations in algebra.) The real numbers are in certain contexts called finite as contrast to ∞. 0.0.1 Order on R The order relation on R extends to R by defining that for any x ∈ R, we have −∞ < x, x ∞, < and that −∞ < ∞. For a ∈ R, let us also define intervals (a, ∞] [−∞, a) 0.0.2 = {x ∈ R : x > a}, = {x ∈ R : x < a}. Addition For any real number x, we define x + (±∞) = (±∞) + x = ±∞, and for +∞ and −∞, we define (±∞) + (±∞) = ±∞. It should be pointed out that sums like (+∞) + (−∞) are left undefined. Thus R is not an ordered ring although R is. ∗ hExtendedRealNumbersi created: h2013-03-21i by: hmattei version: h34441i Privacy setting: h1i hDefinitioni h28-00i h12D99i † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. 1 0.0.3 Multiplication If x is a positive real number, then x · (±∞) = (±∞) · x = ±∞. Similarly, if x is a negative real number, then x · (±∞) = (±∞) · x = ∓∞. Furthermore, for ∞ and −∞, we define (+∞) · (+∞) = (−∞) · (−∞) = +∞, (+∞) · (−∞) = (−∞) · (+∞) = −∞. In many areas of mathematics, products like 0 · ∞ are left undefined. However, a special case is measure theory, where it is convenient to define 0 · (±∞) 0.0.4 (±∞) · 0 = 0. = Absolute value For ∞ and −∞, the absolute value is defined as | ± ∞| = +∞. 0.0.5 Topology The topology of R is given by the usual base of R together with with intervals of type [−∞, a), (a, ∞]. This makes R into a compact topological space. R can also be seen to be homeomorphic to the interval [−1, 1], via the map x 7→ (2/π) arctan x. Consequently, every continuous function f : R → R has a minimum and maximum. 0.0.6 Examples 1. By taking x = −1 in the product rule, we obtain the relations (−1) · (±∞) 2 = ∓∞.