what is the infinite number?

In mathematics, "infinity" is often used in contexts where it is treated as if it were a number (i.e., it counts or measures things : "an infinite number of terms") but it is clearly a very different type of "number" than the integers or reals. Infinity is relevant to, or the subject matter of, limits, aleph numbers, classes in set theory, Dedekind-infinite sets, large cardinals, Russell's paradox, hyperreal numbers, projective geometry, extended real numbers and the absolute Infinite.



Mathematical infinity



Infinity is the state of being greater than any finite (real or natural) number, however large.







Infinity in real analysis

In real analysis,e symbol
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as the same, leading to the one-point compactification of the real numbers, which is the real projective line. Projective geometry also introduces a line at infinity in plane geometry, and so forth for higher dimensions.



Infinity is often used not only to define a limit but as if it were a value in the extended real numbers in real analysis; if f(t) ≥ 0 then






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means that f(t) does not bound a finite area from 0 to 1


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means that the area under f(t) is infinite.


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means that the area under f(t) equals 1

Infinity is also used to describe infinite series:






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means that the sum of the infinite series converges to some real value x.


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means that the sum of the infinite series diverges, i.e., its value is undefined.





Infinity in complex analysis

As in real analysis, in complex analysis the symbol
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at the poles. The domain of a complex-valued function may be extended to include the point at infinity as well. One important example of such functions is the group of Möbius transformations.







Infinities as part of the extended real number line

Infinity is not a real number but the extended real number line adds two elements called infinity (
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), less than all other extended real numbers, in which arithmetic operations involving these new elements may be performed. In this system, infinity, and minus infinity have the following arithmetic properties:







Infinity with itself




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Operations involving infinity and a real number x




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If 0 \," src="http://upload.wikimedia.org/math/7/f/4/7f487cbdbc8528bd14a1f971ef68def6.png"> then








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If
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then








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Indeterminate operations




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Notice that
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.



Also, by L'Hôpital's rule, limits of indeterminate solutions to an equation can be found if the equation can be put in the form of
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, often giving a finite answer.

There is no actual value of "infinite number", because "infinite" mean endless, "infinite number" represent an ultimate value that do no has an actual value, as any numbers (x) have a number larger than it (x+1).

Actually "infinite number" is a concept rather than a number.