93 lines
3.6 KiB
Plaintext
93 lines
3.6 KiB
Plaintext
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Article 410 of misc.misc:
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Xref: puukko sci.math:453 misc.misc:410
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Path: puukko!santra!tut!enea!mcvax!uunet!seismo!sundc!pitstop!sun!decwrl!decvax!ucbvax!jade!ig!uwmcsd1!csd4.milw.wisc.edu!markh
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From: markh@csd4.milw.wisc.edu (Mark William Hopkins)
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Newsgroups: sci.math,misc.misc
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Subject: Re: Properties of Infinity
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Summary: Laissez-Faire
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Keywords: Infinity properties
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Message-ID: <4235@uwmcsd1.UUCP>
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Date: 17 Jan 88 02:49:51 GMT
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References: <1990@pdn.UUCP>
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Sender: daemon@uwmcsd1.UUCP
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Reply-To: markh@csd4.milw.wisc.edu (Mark William Hopkins)
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Organization: University of Wisconsin-Milwaukee
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Lines: 74
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In article <1990@pdn.UUCP> ken@pdn.UUCP (Ken Auer) writes:
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>For reasons which I'd rather not explain, I need to find out several
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>properties of infinity and negative infinity which I'm sure are in some
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>8th grade math book (which I don't have immediate access to).
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>
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>I've got lots of educated guesses, but I really need concrete answers
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>for things like the following (concrete meaning I can call a routine
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>which can supply me with a concrete answer).
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>
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> infinity is not even,
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> infinity is not odd,
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> infinity + infinity = infinity
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> infinity - infinity = ?
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> .
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> .
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> .
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>
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>I really don't want to start any highly theoretical discussions here, I
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>just want to know what to do when some one tries to use infinity as s/he
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>would use a finite number in an equation, etc.
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>
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>--------------------------------------------------------------------------
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>Ken Auer Paradyne Corporation
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>{gatech,rutgers,attmail}!codas!pdn!ken Mail stop LF-207
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>Phone: (813) 530-8307 P.O. Box 2826
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> Largo, FL 34649-9981
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>
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>"The views expressed above do not necessarily reflect the views of my
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>employer, which by no means makes them incorrect."
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Addition: Multiplication:
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Infinity + Finite = Infinity Infinity x Infinity = Infinity
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Infinity + Infinity = Infinity Infinity x Finite = Infinity,
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but Infinity x 0 is undefined
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Infinity + -Infinity can be
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absolutely anything finite or not Infinity x -Infinity = -Infinity
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-Infinity + Finite = -Infinity -Infinity x Finite = -Infinity,
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with the same exception for 0 as before
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-Infinity + -Infinity = -Infinity
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-Infinity x -Infinity = Infinity
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Subtraction:
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Same as addition, with u-v treated as u+(-v):
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where
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-(Infinity) = -Infinity
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-(-Infinity) = Infinity
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Division:
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Same as multiplication, with u/v treated as u x (1/v):
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where
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1/(-Infinity) = -0
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1/(Infinity) = +0
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1/(-0) = -Infinity
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1/(+0) = Infinity
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You'll need to make the distinction between +0 and -0, if you're going to say
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anything useful about division with infinity.
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These rules are made in such a way that all the properties (+,x,-,/) will
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remain true when infinite limits are included. It is possible for a limit
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to be infinite without its positive or negative sign being determined. This
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limit will represent the unsigned infinity. Its negative is itself and its
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reciporical is 0 (without the + or - sign). You'll need to use all three
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kinds of infinity. Much of Calculus is devoted to resolving those limits
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involving the undefined operations above, like
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Infinity - Infinity, Infinity x 0, Infinity/Infinity
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There is a theory of infinitesimals based on what is known as Non-Standard
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Analysis. Its content is completely equivalent to Calculus. In fact, it is
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a reformulation of Calculus that matches very closely the original formulation
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of Calculus as a calculation system for infinite and infinitesimal numbers.
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