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In mathematics, the extended real number system [a] is obtained from the real number system by adding two elements denoted + and [b] that are respectively greater and lower than every real number. This allows for treating the potential infinities of infinitely increasing sequences and infinitely decreasing series as actual infinities .
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 ∞. In contrast, the affinely extended real number line (also called the two-point compactification of the real line) distinguishes between +∞ and ...
The standard part function can also be defined for infinite hyperreal numbers as follows: If x is a positive infinite hyperreal number, set st(x) to be the extended real number +, and likewise, if x is a negative infinite hyperreal number, set st(x) to be (the idea is that an infinite hyperreal number should be smaller than the "true" absolute ...
By definition, real analysis focuses on the real numbers, often including positive and negative infinity to form the extended real line. Real analysis is closely related to complex analysis , which studies broadly the same properties of complex numbers .
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].
the affinely extended real number system and order-isomorphic sets, for example the unit interval; the set of real numbers with only +∞ or only −∞ added, and order-isomorphic sets, for example a half-open interval; the long line; The set I × I (where × denotes the Cartesian product and I = [0, 1]) in the lexicographic order is a linear ...
Some authors define metrics so as to allow the distance function d to attain the value ∞, i.e. distances are non-negative numbers on the extended real number line. [4] Such a function is also called an extended metric or "∞-metric". Every extended metric can be replaced by a real-valued metric that is topologically equivalent.
The real numbers can be generalized and extended in several different directions: The complex numbers contain solutions to all polynomial equations and hence are an algebraically closed field unlike the real numbers. However, the complex numbers are not an ordered field. The affinely extended real number system adds two elements +∞ and −∞.