In mathematics, the extended real number system is obtained from the real number system R {\displaystyle \mathbb {R} } by adding two elements denoted +...

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...

the number line corresponds to a unique real number, and every real number to a unique point. Using a number line, numerical concepts can be interpreted...

real number line (applied to a function or sequence that "diverges to infinity" or "increases without bound"), or as an extreme point of the extended...

hyperreal number, set st(x) to be the extended real number + ∞ {\displaystyle +\infty } , and likewise, if x is a negative infinite hyperreal number, set st(x)...

example of a real projective line is the projectively extended real line, which is often called the projective line. Formally, a real projective line P(R) is...

numbers) requires both +0 and −0. Real arithmetic with signed zeros can be considered a variant of the extended real number line such that ⁠1/−0⁠ = −∞ and ⁠1/+0⁠ = +∞;...

topology Extended real number line Finite topological space Hawaiian earring Hilbert cube Irrational cable on a torus Lakes of Wada Long line Order topology...

{\displaystyle +\infty } and − ∞ {\displaystyle -\infty } . However, the extended real number line would be compact, since it contains both infinities. There are...

typically sought, where f {\displaystyle f} is valued in the extended real number line [ − ∞ , ∞ ] = R ∪ { ± ∞ } . {\displaystyle [-\infty ,\infty ]=\mathbb...

{\displaystyle \mathbb {N} ^{\infty }} . It is a subset of the extended real number line, which extends the real numbers by adding − ∞ {\displaystyle -\infty } and...

quotient by the subgroup {1, −1} under multiplication. Compare the extended real number line, which distinguishes ∞ and −∞. Adding a point at infinity to the...

{\displaystyle \lim _{n\to \infty }u_{n}(x)} automatically exists in the extended real number line for every x. Harnack's theorem says that the limit either is infinite...

the extended real number line, dividing any real number by infinity yields zero, while in the surreal number system, dividing 1 by the infinite number ω...

In mathematics, a real number is a number that can be used to measure a continuous one-dimensional quantity such as a duration or temperature. Here, continuous...

real number line), it is always considered open and adjoined to a parenthesis. The endpoint can be closed when considering intervals on the extended real...

real numbers Natural number Integer Rational number Irrational number Completeness of the real numbers Least-upper-bound property Real line Extended real...

see extended real number line. Apostol, Tom M. (1974). Mathematical Analysis (2nd ed.). Addison–Wesley. ISBN 978-0-201-00288-1. Gerald Folland, Real Analysis:...

(that is, the set of all pairs (x, y) with x, y belonging to the extended real number line R = [−∞, +∞], endowed with the product topology), which will contain...

model of the extended complex plane (also called the closed complex plane): the complex plane plus one point at infinity. This extended plane represents...

topological subspace of the extended real line, into the space (the closure of N in [−∞,∞], the extended real number line, is N ∪ {∞}.) The power set...

{\displaystyle \textstyle \limsup _{x\to a}} (at least on the extended real number line) always exists. In computer science, a slightly more restrictive...

the extended real number line. Such a function is also called an extended metric or "∞-metric". Every extended metric can be replaced by a real-valued...

unit interval is a complete metric space, homeomorphic to the extended real number line. As a topological space, it is compact, contractible, path connected...

_{k=0}^{\infty }2^{k}} is divergent in the real numbers. However, it is convergent in the extended real number line, with + ∞ {\displaystyle +\infty } as its...

{\displaystyle \mu } from Σ {\displaystyle \Sigma } to the extended real number line, that is, the real number line together with new (so-called infinite) values +...

the real line R {\displaystyle \mathbb {R} } is also the statement used to define convex functions that are valued in the extended real number line [ −...

usual real-number line consists of a countable number of line segments [ 0 , 1 ) {\displaystyle [0,1)} laid end-to-end, whereas the long line is constructed...

value +∞, in other words, f takes non-negative values in the extended real number line. We define ∫ E f d μ = sup { ∫ E s d μ : 0 ≤ s ≤ f ,   s   simple...

exact number (u = 0), or an interval between consecutive exact unums (u = 1). In this way, the unums cover the entire extended real number line [−∞,+∞]...