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.
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Ken Auer Paradyne Corporation
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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.