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For example, the set of real numbers consisting of 0, 1, and all numbers in between is an interval, denoted [0, 1] and called the unit interval; the set of all positive real numbers is an interval, denoted (0, ∞); the set of all real numbers is an interval, denoted (−∞, ∞); and any single real number a is an interval, denoted [a, a].
For example, both [] ... So, given the fact that an interval number is a real closed interval and a complex number is an ordered pair of real numbers, ...
The central question to be posed is the nature of the intersection over all the natural numbers, or, put differently, the set of numbers, that are found in every Interval (thus, for all ). In modern mathematics, nested intervals are used as a construction method for the real numbers (in order to complete the field of rational numbers).
The unit interval is a subset of the real numbers. However, it has the same size as the whole set: the cardinality of the continuum . Since the real numbers can be used to represent points along an infinitely long line , this implies that a line segment of length 1, which is a part of that line, has the same number of points as the whole line.
A weaker three-sigma rule can be derived from Chebyshev's inequality, stating that even for non-normally distributed variables, at least 88.8% of cases should fall within properly calculated three-sigma intervals. For unimodal distributions, the probability of being within the interval is at least 95% by the Vysochanskij–Petunin inequality ...
The Poisson distribution, which describes a very large number of individually unlikely events that happen in a certain time interval. Related to this distribution are a number of other distributions: the displaced Poisson, the hyper-Poisson, the general Poisson binomial and the Poisson type distributions.
A nontrivial example of a compact space is the (closed) unit interval [0,1] of real numbers. If one chooses an infinite number of distinct points in the unit interval, then there must be some accumulation point among these points in that interval.
As a less trivial example, consider the space of all rational numbers with their ordinary topology, and the set of all positive rational numbers whose square is bigger than 2. Using the fact that 2 {\displaystyle {\sqrt {2}}} is not in Q , {\displaystyle \mathbb {Q} ,} one can show quite easily that A {\displaystyle A} is a clopen subset of Q ...