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