28th January 2008 - 02:28 PM
QUOTE (StevenA+Jan 16 2008, 08:07 PM)
If 1/infinity=0, then by multiplying both sides by infinity we would get 0*infinity=1, which is false (or at least missing the context in which such an identity could be assumed true).
You can't do that, because infinity/infinity isn't defined and you assume it equals 1.
28th January 2008 - 02:32 PM
Number systems are constructed by mathematicians from sets of postulates. In the standard system, there is no number "infinity" and so since division is only defined for numbers within the system, division cannot be applied to infinity. Therefore, in the standard number system it is impossible to ask this question and still be consistent with the postulates.
There are several alternative number systems. One is the Non-Standard system, constructed by Abraham Robinson in the 1960s using Goedel's incompleteness theorem. In this system, there are infinite numbers of various sizes, and so within this system, 1/infinity is defined (but you have to specify which infinite number you meant) and it is not zero. Every infinite number has a reciprocal in this system, so there are lots of different infinitesimal numbers. The nice thing about this number system is that if you work in it, you can do all the tricks of calculus without ever having to take a limit.
There is also the system of Surreal numbers constructed by J. H. Conway, which is an even bigger system than the Non-Standard numbers. What is amazing about this system of numbers is that Conway derived it from a simple game that can be played by two people alternately removing lines from a picture. It's a very simple game like nim, where the last person able to remove a line from the picture wins. It turns out that if you study all the moves of this game, and try to predict who has a winning strategy, the pattern you find among the different ways the game can be played satisfies all the rules of a number system. Every game is a number, and there are lots more of these numbers than you get in the standard number system. The standard number system can be embedded in the Surreals but forms only a tiny part of them. The surreal numbers system also has many infinite numbers in it, and again, each one has a reciprocal which is not zero, and all the reciprocals are different from each other.
It seems that every time someone constructs a number system that includes infinity, they get not just one, but a whole lot of different infinite numbers. So the answer to the original question "what is 1/infinity" is that...
---in the standard system:
"you can't ask that."
---in the nonstandard system and the surreals:
"it will be one of the nonzero infinitesimal numbers, which infinity did you have in mind?"
Hope that helps!
6th October 2008 - 10:17 PM
I formally apologize to all brilliant people who made sensible, useful, knowledgeable answers here, and won't try to add anything better. I will therefore just slip away from the question. Sorry again.
Cutting in half again and again just reminds me of a novel in an excellent SF book by Philippe Cousin (the French title may be "Mange ma mort", or is it rather "H˘pital nord"?) where an infected patient was cut in half again and again to try to save the sound part from a really bad virus.
The limit, as this was no maths, was attained as the patient was composed of two remaining atoms ("A-Tome" meaning "impossible to split") and the surgeon separated the virus atom from the sound atom.
6th October 2008 - 10:59 PM
Thank you and welcome back, mr_homm.
6th October 2008 - 11:30 PM
QUOTE (MaxTrans+Jan 28 2008, 09:46 PM)
Sorry, but when you divide one apple to ANY number of pieces the answer is not how many pieces are there but how big is one of the equally divided pieces (you already know how many pieces will be there - n, if 1/n)
Several parameters here are fairyland.
1. If you could divide an apple into..........YOU CAN"T
2. by saying "it's an apple" you already disqualified it as an infinity BY DEFINING IT
3. quote: not how many pieces are there but how big is one of the equally divided pieces....HUH...it's infinity !!!.... You NEVER stop dividing. So there is never a size or quantity TO MEASURE.
(society against the propagation of infinity)....lol
7th October 2008 - 11:23 AM
Very cool Mr_H.
So what number system relates to the idea of larger and smaller infinities?
Or rather, how many relates to them:)
I remember a good friend and a gifted mathematican telling me once.
'I would like to develop a simple symbolic system to explain how symbolic math is used' :)
Like triangles * Squares ^3 etc etc.
The emphasis here being on simple.
I for one would really have liked it.
Btw: found this.
7th October 2008 - 12:10 PM
22nd October 2008 - 03:53 AM
QUOTE (Darren+Jan 15 2008, 10:09 PM)
1/infinity is simply zero
I must say that I disagree with that.
Although it may seem to be 0, it isn't...
Two alien civilizations (that have never had contact outside of their own civilization) using the same exact alphabet. The chances are basically 0, but there is still a chance.. 1/infinity)
Meh. Not the greatest example, but it'll do.
24th October 2008 - 07:28 PM
well i think 1/infinity could be 0 or infinitsimal thingy coz:
for 0: consider a line segment 1 unit long...infinite amount of points, and their all 0 size, so infinity times 0 is 1, and that means 1 over infinity is 0
for infinitsimal: well my geometry teacher says so so it must be true
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