Infinity

:For the automobile brand, see Infiniti. For the radio company, see Infinity Broadcasting.

Mathematical infinity

Infinity in real analysis

In real analysis, the symbol infty, called "infinity", denotes an unbounded limit. x ightarrow infty means that

Related Topics:
Real analysis - Limit

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x grows beyond any assigned value, and x ightarrow -infty means x is eventually less than any assigned value. Points labeled infty and -infty can be added to the real numbers as a topological space, producing the two-point compactification of the real numbers. Adding algebraic properties to this gives us the extended real numbers. We can also treat infty and -infty 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.

Related Topics:
Topological space - Compactification - Extended real number - Real projective line - Projective geometry - Line at infinity - Plane geometry

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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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• $int\_\{0\}^\{1\}\; ,\; f(t)\; dt\; =\; infty$ means that f(t) does not bound a finite area from 0 to 1
• $int\_\{0\}^\{infty\}\; ,\; f(t)\; dt\; =\; infty$ means that the area under f(t) increases without bound as its upper bound increases limitlessly
• $int\_\{0\}^\{infty\}\; ,\; f(t)\; dt\; =\; 1$ means that the area under f(t) approaches 1, though its upper bound increases limitlessly.

Arithmetic properties of infinity

Infinity is not a real number but may be considered part of the extended real number line, in which arithmetic operations involving infinity may be performed.

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Infinity with itself

1) infty + infty = infty cdot infty = (-infty) cdot (-infty) = infty

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2) (-infty) + (-infty) = infty cdot (-infty) = (-infty) cdot infty = (-infty)

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Operations involving infinity and real numbers

1) -infty < x < infty

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2) x + infty = infty

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3) x - infty = -infty

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4) x - (-infty) = infty

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5) {x over infty} = 0 and {x over -infty} = 0

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6) If 0

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7) If -infty

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Undefined Operations

1) 0 cdot infty and 0 cdot (-infty)

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2) infty + (-infty) and (-infty) + infty

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3) infty over infty

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4) infty^0

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5) 1^infty

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Notice that otequiv . This is because zero times infinity is undefined.

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Infinity in set theory

A different type of "infinity" are the ordinal and cardinal infinities of set theory. Georg Cantor developed a system of transfinite numbers, in which the first transfinite cardinal is aleph-null (leph_0), the cardinality of the set of natural numbers. This modern mathematical conception of the quantitative infinite developed in the late nineteenth century from work by Cantor, Gottlob Frege, Richard Dedekind and others, using the idea of collections, or sets. Dedekind's approach was essentially to adopt the idea of one-to-one correspondence as a standard for comparing the size of sets, and to reject the view of Galileo (which derived from Euclid) that the whole cannot be the same size as the part. An infinite set can simply be defined as one having the same size as at least one of its "proper" parts; this notion of infinity is called Dedekind infinite.

Related Topics:
Ordinal - Cardinal - Georg Cantor - Transfinite number - Aleph-null - Cardinality - Natural number - Gottlob Frege - Richard Dedekind - Sets - One-to-one correspondence - Euclid - Proper - Dedekind infinite

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Cantor defined two kinds of infinite numbers, the ordinal numbers and the cardinal numbers. Ordinal numbers may be identified with well-ordered sets, or counting carried on to any stopping point, including points after an infinite number have already been counted. Generalizing finite and the ordinary infinite sequences which are maps from the positive integers leads to mappings from ordinal numbers, and transfinite sequences. Cardinal numbers define the size of sets, meaning how many members they contain, and can be standardized by choosing the first ordinal number of a certain size to represent the cardinal number of that size. The smallest ordinal infinity is that of the positive integers, and any set which has the cardinality of the integers is countably infinite. If a set is too large to be put in one to one correspondence with the positive integers, it is called uncountable. Cantor's views prevailed and modern mathematics accepts actual infinity. Certain extended number systems, such as the hyperreal numbers, incorporate the ordinary (finite) numbers and infinite numbers of different sizes.

Related Topics:
Ordinal number - Cardinal numbers - Well-ordered - Sequence - Integers - Mappings - Number - Hyperreal number

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Our intuition gained from finite sets breaks down when dealing with infinite sets. One example of this is Hilbert's paradox of the Grand Hotel.

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Mathematics without infinity

Leopold Kronecker rejected the notion of infinity and began a school of thought in the philosophy of mathematics called finitism, which led to the philosophical and mathematical school of mathematical constructivism.

Related Topics:
Leopold Kronecker - Philosophy of mathematics - Finitism - Mathematical constructivism

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