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| + | [[File:AnEternalGrayscaleRain.gif|thumb|258x258px|This picture represents infinity in terms of an infinite loop and is also a metaphor for the paradoxical qualities the infinite can possess.]] |
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| − | '''Infinity''', usually represented by the symbol \(∞\), is a mathematical concept that indicates a "number" larger than any other number. It has different meanings across different branches in mathematics. |
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| + | '''Infinity''', usually represented by the symbol \(\infty\) (or a ''{{W|lemniscate}}''), is a mathematical concept that indicates a "quantity" larger than any other number. The word quantity has been placed in quotation marks to indicate that infinity goes beyond a definite quantity and in fact is not just vast, but ''vastless''.<ref>[https://sites.google.com/site/largenumbers/home/1-1/new_intro 1.1.1 Welcome to the Numberscape - Large Numbers]</ref> It has different meanings across different branches in mathematics. |
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| − | To [[googologist]]s, infinity is a |
+ | To [[googologist]]s, infinity is not a valid "[[large number|largest number]]". However, precisely defined mathematical infinities ''are'' useful to googologists. [[Ordinal|Ordinal infinities]] (transfinites) are vitally important in measuring the growth rates of functions, in particular the [[fast-growing hierarchy]], and serve to stratify the cornucopia of computable functions that exist. [[Cardinal|Cardinal infinities]] on the other hand can be used in set theory, which in turn can be used to generate massive numbers. |
| − | In traditional {{w|algebra}}, infinity is meaningful only as a symbol, not a number that can be legitimately manipulated. One use is the definition of open intervals such as \([5,\infty)\), or inequalities like \(n < \infty\). In differential and integral {{w|calculus}}, however, infinity is |
+ | In traditional {{w|algebra}}, infinity is meaningful only as a symbol, not a number that can be legitimately manipulated. One use is the definition of open intervals such as \([5,\infty)\), or inequalities like \(n < \infty\). In differential and integral {{w|calculus}}, however, infinity is one of the central concepts. An integral, for example, is the sum of an "infinite number of infinitely small parts" — but still infinity is merely symbolic of a process with an unnumbered amount of steps. |
| + | Infinity is vital when working with a field of real numbers: for example, [[exponentiation]] \(a^b\) for real numbers is defined as \(e^{\ln a \times b}\), and the {{w|exponential function}} is defined using either infinite limit or infinite series. |
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| + | In complex analysis, the {{w|Riemann sphere}} is defined as \(\mathbb{C} \cup \{\infty\}\), where \(\infty\) is an unsigned infinity. |
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| − | [[User blog:Commando Conceptor L5 5.12.159.141/How much is infinite?|Thanks to Logo for the formula!]] |
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| − | [[Category:Numbers]] |
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| + | Related to the "regular" mathematical concept of infinity is complex infinity, defined as an infinite quantity which has an undefined {{w|complex argument}}.<ref>[http://mathworld.wolfram.com/ComplexInfinity.html Complex Infinity - from Wolfram MathWorld]</ref> |
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| + | A supertask is a task that has an infinite amount of steps but can be completed in a finite length of time. The most famous examples of supertasks are the paradoxes proposed by Greek philosopher {{w|Zeno of Elea}}, one of which proposes that motion is impossible as in order to traverse a distance ''d'', you must travel half that distance, then half of ''that'' distance, and so forth infinitely. Clearly the reasoning is at fault with our common experience, but it goes to show just some of the contradictions that arise when super tasks are invoked. |
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| + | == Sources == |
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| + | <references /> |
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[[Category:Concepts]] |
[[Category:Concepts]] |
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Revision as of 22:27, 19 December 2020
This picture represents infinity in terms of an infinite loop and is also a metaphor for the paradoxical qualities the infinite can possess.
Infinity, usually represented by the symbol \(\infty\) (or a lemniscate), is a mathematical concept that indicates a "quantity" larger than any other number. The word quantity has been placed in quotation marks to indicate that infinity goes beyond a definite quantity and in fact is not just vast, but vastless.[1] It has different meanings across different branches in mathematics.
To googologists, infinity is not a valid "largest number". However, precisely defined mathematical infinities are useful to googologists. Ordinal infinities (transfinites) are vitally important in measuring the growth rates of functions, in particular the fast-growing hierarchy, and serve to stratify the cornucopia of computable functions that exist. Cardinal infinities on the other hand can be used in set theory, which in turn can be used to generate massive numbers.
In traditional algebra, infinity is meaningful only as a symbol, not a number that can be legitimately manipulated. One use is the definition of open intervals such as \([5,\infty)\), or inequalities like \(n < \infty\). In differential and integral calculus, however, infinity is one of the central concepts. An integral, for example, is the sum of an "infinite number of infinitely small parts" — but still infinity is merely symbolic of a process with an unnumbered amount of steps.
Infinity is vital when working with a field of real numbers: for example, exponentiation \(a^b\) for real numbers is defined as \(e^{\ln a \times b}\), and the exponential function is defined using either infinite limit or infinite series.
In complex analysis, the Riemann sphere is defined as \(\mathbb{C} \cup \{\infty\}\), where \(\infty\) is an unsigned infinity.
The term "Infinity Scraper", defined by Jonathan Bowers, refers to any number larger than tridecal. The term is of course hyperbolic.
Related to the "regular" mathematical concept of infinity is complex infinity, defined as an infinite quantity which has an undefined complex argument.[2]
A supertask is a task that has an infinite amount of steps but can be completed in a finite length of time. The most famous examples of supertasks are the paradoxes proposed by Greek philosopher Zeno of Elea, one of which proposes that motion is impossible as in order to traverse a distance d, you must travel half that distance, then half of that distance, and so forth infinitely. Clearly the reasoning is at fault with our common experience, but it goes to show just some of the contradictions that arise when super tasks are invoked.