Infinity/Negative Infinity and Zero

[Sikz]

Infinity and negative infinity are widely known to be equal. So, with that premise, I shall continue.

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.

What differentiates infinity from zero then? What reason exists that prohibits its use as a base like zero? Here is a diagram:

http://www.flatface.net/~comfox/mone.jpg

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.

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

 

[Hurkyl]

Infinity and negative infinity are widely known to be equal.

That is incorrect.

What is correct is that one extension to the real numbers adds a single point at infinity that turns the real line into a "circle".

By "circle" I mean something that has the topology of a circle; it is a mistake to think of this too literally as being a real circle embedded in a plane.

At one pole is zero, at the other infinity.

Which you did. The fact that zero and infinity are on opposite points of the circle in your diagram is merely an accident of the way you plotted the circle; any real number could be placed opposite infinity and it would be equally "valid".

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?

You need to be more clear about what you're trying to do here.



However, there is one interesting thing about this particular extension of the real numbers: if we exclude 0, we can then define a new addition:

$$a \oplus b = \frac{ab}{a+b}$$

and extend multiplication and addition to $\infty$ by $\infty \oplus x = x$ and $\infty x = \infty$ for all x.

It turns out that this new structure is isomorphic to the ordinary real numbers; for instance, the distributive law holds: $a (b \oplus c) = a b \oplus a c$.

 

[Sikz]

What is correct is that one extension to the real numbers adds a single point at infinity that turns the real line into a "circle".

Yes... there we go

Which you did. The fact that zero and infinity are on opposite points of the circle in your diagram is merely an accident of the way you plotted the circle; any real number could be placed opposite infinity and it would be equally "valid".

Indeed it would, and is- in my diagram addition is clockwise and subtraction counterclockwise (meaning that positive numbers are to the right of zero and the left of infinity, and negative numbers on the opposite sides. Negative infinite numbers are connected to positive zero numbers, while positive infinite numbers are connected to negative zero numbers.). The only numbers I have labelled on the diagram are Zero and Infinity because they constitute "different" numbers; 0*0=0, 0+0=0, 0-0=0. I*I=I, I+I=I, I-I=I (as far as this idea is concerned, at least). Also, if we take the "number line" as an actual circle, the distances between intergers max out at zero to negative/one and infinity to negative/one, decreasing as we approach the "sides" of the circle (since counting by negative/one from zero will never allow you to reach infinity, in the circle model the numbers get closer together as you approach the "sides").

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

I-0 = I
0-I = 0

Really simple stuff, I havn't done anything complicated or revolutionary (or useful, most likely ) simply added another "number system" counting "down" from infinity towards zero instead of the other way around and connected it with the zero-based system.[

 

[matt grime]

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

 

[Sikz]

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.

Very well, I called it a premise not a truth. Philosophicly "positive" and "negative" infinity are widely (very often) thought to be equal- I apologize for saying they were "widely known" to be equal.

That is not how one defines the one point compactification topologically.

If this was the standard one point compactification I would have no reason to post it here. While it very well may not be exactly a circle, I was posting the basic idea and hoping you people might have some refinements, comments, counterexamples, etc.

In what sense are you using base? Zero isn't a base in the usual mathematical sense for numbers.

Yes, I am aware and I should have clarified my meaning. I am using base in the sense of a "foundation", a point against which others are measured. Our numbers are all given in relation to zero; 1 is one past (beyond in the positive direction) zero, -1 is one before (beyond in the negative direction) it, and so on. Infinity as a base in this sense means that we can count up and down (positively and negatively) from infinity rather than from zero. For instance, we can say "One less than Infinity". That is meaningless in relation to zero, but in relation to infinity it has meaning.

? You want some new metric on the real numbers using the standard metric from the circle.

What exactly do you mean by that?

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

Perhaps you could assign those points values, but they could never actually be reached from zero or infinity. If they could then counting by ones from zero would eventually get to a number halfway between zero and infinity and then evenentually get to infinity itself. If this was the case infinity would not be infinity; it would have a finite value. The numbers much continually approach the middle points but never quite reach them. We can represent this as a circle, with the distances between numbers decreasing as we get closer to the middle points- we could also represent it as a sort of infinite oval, two semicircles connected by opposing rays; we can continually move along one ray but we will never reach a point where the rays meet. I prefer the former because in actuality two opposing rays would meet, pass each other's endpoints, and continue to form a line.

 

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