Thread: number circle theory... what branch of math is this?

Just looking for feedback on an idea. I'm sure I can't be the first to think of this... Who else has presented this idea? Probably a philosopher.

<-- - <--
0<1<[n=(∞-x)]<(∞-1)<∞<(∞+1)<[(∞+x)=-n]<-1<0
-->+-->

oversimplification:
n=any number other than 0, 1, ∞ as a function of 0 while x would act as it's compliment as a function of ∞. Forming a circle, the traditional infinite number line is now a number circle. ∞+ and ∞- would attract and come together to become it's own singularity, also explaining the size of points on the cirlce being one infinityith, infinith??? (1/∞). Something about the distances through the circle (z) [as opposed to arround (x) (and y when the circle becomes a sphere)] acting like a wormhole. The circle would (obviously) be infinitely large with ∞ at the top to accomidate the religious.
Remove the void by having ...-3, -2, -1, -1/∞, +1/∞, 1, 2, 3... could zero take up 1/∞ of space on the number 'line' 1/∞>0 or would zero just be an imaginary divider of the positive and negative taking up absolutely no space whatsoever.... 1-1≠(±1/∞) so zero must take up just as much room as any other number.

Originally Posted by TiGeRmOo

Just looking for feedback on an idea. I'm sure I can't be the first to think of this... Who else has presented this idea? Probably a philosopher.

<-- - <--
0<1<[n=(∞-x)]<(∞-1)<∞<(∞+1)<[(∞+x)=-n]<-1<0
-->+-->

oversimplification:
n=any number other than 0, 1, ∞ as a function of 0 while x would act as it's compliment as a function of ∞. Forming a circle, the traditional infinite number line is now a number circle. ∞+ and ∞- would attract and come together to become it's own singularity, also explaining the size of points on the cirlce being one infinityith, infinith??? (1/∞). Something about the distances through the circle (z) [as opposed to arround (x) (and y when the circle becomes a sphere)] acting like a wormhole. The circle would (obviously) be infinitely large with ∞ at the top to accomidate the religious.
Remove the void by having ...-3, -2, -1, -1/∞, +1/∞, 1, 2, 3... could zero take up 1/∞ of space on the number 'line' 1/∞>0 or would zero just be an imaginary divider of the positive and negative taking up absolutely no space whatsoever.... 1-1≠(±1/∞) so zero must take up just as much room as any other number.

What you have here is rather garbled, but it seems to reflect the one-pooint compatification of the real numbers or what is called the "real projective line".

http://en.wikipedia.org/wiki/Real_projective_line

There's some pretty wierd lookin' symbols at that website...maybe it's a wiki thing... maybe I have a few languages to learn if I have any hope of understanding how we currently talk mathematically with relation to.... (calculous?) recommended reading?
(BTW: "Zero: The biography of a dangerous idea" interresting and enjoyable)

Originally Posted by TiGeRmOo

There's some pretty wierd lookin' symbols at that website...maybe it's a wiki thing... maybe I have a few languages to learn if I have any hope of understanding how we currently talk mathematically with relation to.... (calculous?) recommended reading?
(BTW: "Zero: The biography of a dangerous idea" interresting and enjoyable)

Reading? What is your background ? Also, and particular areas of interest. Mathematics is a really big subject.

I find popularizations of mathematics, in constrast to those of physics, generally pretty worthless. So, any recommendations that I make will involve real mathematics books.

DrRocket thinks this thread relates to the projective real line. I am not convinced, but some of you might find the following interesting.

Suppose I take a segment of the real line and join the two ends together. This we recognize as a circle; grand folk call it the 1-sphere . Now since this is essentially the "same" as our line segment , we might expect to be able to find a one-to-one correspondence between them. This correspondence - technically a "homeomorphism", since these are topological spaces - will be given by an invertible function from one to the other (this is called a bijection, btw).

Of course no such bijection exists; the map sends the two ends of to same point of - our "join". And since well-behaved functions are not allowed to send a single point in the domain to distinct points in the co-domain, there can be no inverse .

Now there is a theorem (due I think to Dedekind and Pierce) that states that an infinite set is always isomorphic to a proper subset of itself. Since and since homeomorphism is an equivalence relation, by transitivity we might expect to be able to do something like the following.

Place our circle anywhere on the real line, and project each point of "down" to a point in in such a way that none of these projections intersect each other or .

All goes swimmingly until we get to the North pole, where now our projections run parallel to and seem never to meet it.

But we have the perfect construction specifically made for just this purpose, called the "projective real line". This is formed by adjoining what's called the "point at infinity" to to form .

So now our point at the North pole projects to this point at infinity, and a homeomorphism can be recovered, but at some cost; I cannot make this construction of 1-dimensional spaces without embedding them in, say a 2-dimensional space. And, if I am willing to do this, I have to say that parallel lines meet!

Worse still, since the two projections - "left" and "right" - from the North pole map to the same point in our projective line, this implies that, in , the "end-points" of the real line may be considered as the same point.

Which is no more, or less, than to assert that the projective real line has the same topology (up to homeomorphism) as the real circle.

This line of thought can be extended to the plane and its projective "partner" , but here the geometry is a little more interesting; it's called "projective geometry"