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matt grime
#4
Jan28-04, 06:19 AM
Sci Advisor
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P: 9,398
Infinity/Negative Infinity and Zero

Originally posted by Sikz
Infinity and negative infinity are widely known to be equal. So, with that premise, I shall continue.
No that is not true. You would need to define your infinity. Here we can see you mean the one point compactification of the real line. There is a two point compactification where they are different.




To extend the number line infinitely along the positive direction results in infinity. To extend it in the negative direction results in negative infinity. These two infinities are equal- therefore the number line can be visualized as a circle rather than a line. At one pole is zero, at the other infinity.
That is not how one defines the one point compactification topologically.

What differentiates infinity from zero then? What reason exists that prohibits its use as a base like zero?
In what sense are you using base? Zero isn't a base in the usual mathematical sense for numbers.



My idea is this: Infinity can be used as a base for a number stystem just as validly as zero can. In the diagram, addition is shown as clockwise, subtraction counter-clockwise. No matter how much we add to zero, we never reach infinity. No matter how much we add to infinity, we never reach zero. The same goes for subtraction. Why is this?

Technicly the point that is never reached is that point halfway between zero and infinity. Perhaps the distance between numbers actually grows smaller the larger the numbers one deals with- the distance between 20,000 and 20,001 for instance would be smaller than the distance between 5 and 6.
?????? You want some new metric on the real numbers using the standard metric from the circle.

As we are using the one point compactification, surely there are two numbers halfway between 0 and infinity?

As a last idea (it's getting rather late), here are some "undefined" equations defined. I=infinity. Note that they are defined on the infinity-based number system, but still have no meaning to the zero-based system (those that's results have to do with infinity, that is).

0/0 = I
I/I = 0

x/0 = I-x
x/I = 0-x