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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 .
An axiomatic definition of the real numbers consists of defining them as the elements of a complete ordered field. [2] [3] [4] This means the following: The real numbers form a set, commonly denoted , containing two distinguished elements denoted 0 and 1, and on which are defined two binary operations and one binary relation; the operations are called addition and multiplication of real ...
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 −∞.
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 ...
For example, an open real interval X = (0, 1) ... On the other hand, the extended real number line carrying the analogous topology is compact; ...