Is Infinity Just an Inverse Zero?

[Coqui]

Hey, I'm new here. I wanted to ask a question about infinity - I've been playing with something in my mind.

If you define infinity as a number that is greater than all numbers except itself, and negative infinity as a number that is less than all numbers except itself...

Couldn't both infinity and negative infinity be two "aspects" of an unsigned number - some kind of inverse zero, where:

For zero:

All negative numbers < 0 < All positive numbers

For "inverse zero"

All positive numbers < "Inverse Zero" < All negative numbers

It's probably just a crackpot idea...can someone show me if this is even remotely possible or not?

 

[mathman]

Not completely. When dealling with complex numbers, people sometimes talk about "the point at infinity", reflecting the fact that the complex plane can be mapped (bijection) onto a sphere, where one pole is the image of the origin and the other pole is the image of the point at infinity.

 

[Hurkyl]

It sounds like you're arriving at the concept of the real projective line. One way to form it is by taking the extended real line (that is, the real numbers adjoined with -∞ and +∞) and then identifying -∞ with +∞.

Topologically, this is the same thing as a circle. (Think about it)

Anyways, it no longer makes sense to talk about an ordering. A suitable replacement concept is separation. For example, one would say: "0 and 2 separate 1 and 3", or that "0 and &infin; separate -3 and 5".

(In the projective line, &infin; is occasionally used to denote the point at infinity. However, $\omega$ is more common)

 

[Vanes63]

Definition of Infinity

But really infinity isn't an actual number, it's a concept.

So it doesn't really belong as a point on the number line but as a direction on the number line.

The Wolfram Website defines it like this:
http://mathworld.wolfram.com/Infinity.html

The point at infinity really is a projective geometry bit isn't it though? No lines are parallel because lines that would be parallel meet at the point at infinity.
http://mathworld.wolfram.com/PointatInfinity.html
http://mathworld.wolfram.com/ProjectiveGeometry.html

But it's an interesting thought.

- Vanes.

 

[arildno]

But really infinity isn't an actual number, it's a concept.

Sure, infinity is a number AND a concept.
It just isn't either a real number, for exampe.

And, BTW, please give an example of a utilized non-concept in maths.

 

[Coqui]

Why can't you create a class of numbers that includes infinity/negative infinity/"inverse zero", like when they created complex numbers to deal with square roots of negative numbers?

 

[Hurkyl]

The major problem is that:

0 = (0*x) + (-(0*x)) = ((0 + 0)*x) + (-(0*x)) = ((0*x) + (0*x)) + (-(0*x))
= (0*x) + ((0*x) + (-(0*x))) = (0*x) + 0 = 0*x


1 = 0 * 0^-1 = (0 * 0) * 0^-1 = 0 * (0 * 0^-1) = 0

So, you cannot have all of the following properties:
(1) 0 is the additive identity.
(2) Every number has an additive inverse.
(3) Multiplication distributes over addition.
(4) Addition is associative.
(5) 0 is invertible.
(6) Multiplication is associative
(7) 1 is different from 0.
(8) You can multiply any two numbers.