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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 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 −∞.
The IEEE standard employs (and extends) the affinely extended real number system, with separate positive and negative infinities. During drafting, there was a proposal for the standard to incorporate the projectively extended real number system, with a single unsigned infinity, by providing programmers with a mode selection option. In the ...
The extended real number line (real numbers together with +∞ and −∞) is a completion in this sense of the rational numbers: the set of rational numbers {3, 3.1, 3.14, 3.141, 3.1415, 3.14159, ...} does not have a rational least upper bound, but in the real numbers it has the least upper bound π. [4]
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 ...
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 ...
An example is obtained by projecting points in R 2 onto the unit circle and then identifying diametrically opposite points. In terms of group theory we can take the quotient by the subgroup {1, −1}. Compare the extended real number line, which distinguishes ∞ and −∞.