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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 .
Another extension field of the rationals, which is also important in number theory, although not a finite extension, is the field of p-adic numbers for a prime number p. It is common to construct an extension field of a given field K as a quotient ring of the polynomial ring K [ X ] in order to "create" a root for a given polynomial f ( X ).
In topology, the topos of right actions on the extended natural numbers is a category PRO of projection algebras. [ 4 ] In constructive mathematics , the extended natural numbers N ∞ {\displaystyle \mathbb {N} _{\infty }} are a one-point compactification of the natural numbers, yielding the set of non-increasing binary sequences i.e. ( x 0 ...
The addition x + a on the number line. All numbers greater than x and less than x + a fall within that open interval. In mathematics, a real interval is the set of all real numbers lying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negative infinity, indicating the interval extends without a ...
This extended plane represents the extended complex numbers, that is, the complex numbers plus a value for infinity. With the Riemann model, the point ∞ {\displaystyle \infty } is near to very large numbers, just as the point 0 {\displaystyle 0} is near to very small numbers.
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.
An interval can be defined as a set of points within a specified distance of the center, and this definition can be extended from real numbers to complex numbers. [2] Another extension defines intervals as rectangles in the complex plane. As is the case with computing with real numbers, computing with complex numbers involves uncertain data.
If the source of the operation is an unsigned number, then zero extension is usually the correct way to move it to a larger field while preserving its numeric value, while sign extension is correct for signed numbers. In the x86 and x64 instruction sets, the movzx instruction ("move with zero extension") performs this function.