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Concept
Extended real number line
Summary
In mathematics, the affinely extended real number system is obtained from the real number system by adding two infinity elements: and where the infinities are treated as actual numbers. It is useful in describing the algebra on infinities and the various limiting behaviors in calculus and mathematical analysis, especially in the theory of measure and integration. The affinely extended real number system is denoted or or It is the Dedekind–MacNeille completion of the real numbers.
When the meaning is clear from context, the symbol is often written simply as
There is also the projectively extended real line where and are not distinguished so the infinity is denoted by only .
It is often useful to describe the behavior of a function as either the argument or the function value gets "infinitely large" in some sense. For example, consider the function defined by
The graph of this function has a horizontal asymptote at Geometrically, when moving increasingly farther to the right along the -axis, the value of approaches 0. This limiting behavior is similar to the limit of a function in which the real number approaches except that there is no real number to which approaches.
By adjoining the elements and to it enables a formulation of a "limit at infinity", with topological properties similar to those for
To make things completely formal, the Cauchy sequences definition of allows defining as the set of all sequences of rational numbers such that every is associated with a corresponding for which for all The definition of can be constructed similarly.
In measure theory, it is often useful to allow sets that have infinite measure and integrals whose value may be infinite.
Such measures arise naturally out of calculus. For example, in assigning a measure to that agrees with the usual length of intervals, this measure must be larger than any finite real number. Also, when considering improper integrals, such as
the value "infinity" arises.
Official source
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