# + infinity and - infinity join ?

hey

in advanced maths , does +infinity and -infinity join at some point ?
a bit like if the axis was a cilinder

considering -inf<0<+inf....if they joined then either +inf=-inf or +inf<-inf
then it would invalidate the first expr.

HallsofIvy
Homework Helper
"+ infinity" and "-infinity" are not numbers and trying to deal with them depends upon what method you use of extending the numbers.

For example, there is a perfectly valid method, called the "Stone-Czek compactification" that makes the real numbers topologically (geometrically) equivalent to a finite interval (but arithmetic doesn't work). In that case, +infinity and -infinity are distinct.

You can also use the "one point compactification" that makes the real numbers topologically equivalent to a circle. Although we would not use the terms "+"infinity and "-"infinity in that case (just the single "infinity"), you could think of that as +infinity and -infinity "joining".

"Advanced maths" has a lot of structures. In some of them you could say something like that, though it's not nearly precise enough to be useful. For example, you can imagine a natural almost-correspondence between the complex plane and the unit sphere, using the map

$$(\theta, \phi) \rightarrow \tan(\phi/2) e^{i \theta}$$

(where by $\phi$ I mean the polar angle and $\theta$ the azimuthal angle in polar coordinates)

But when $\phi \rightarrow \pi$ (ie. at one pole of the sphere, under this coordinate system), you find that $\tan (\phi/2)$ diverges to infinity.

This naturally suggests adding a single point at infinity to the complex plane (forming the extended complex plane). The sphere I've considered above is (one version of) the Riemann sphere.

On the other hand, there are two commonly considered versions of the extended real numbers; One involves doing something similar, and adding a single point at infinity (the projective extension), and one involves adding two new points - one at $\infty$ and one at $-\infty$ (the affine extension).

What construction you use depends on what you want to do!

Hurkyl
Staff Emeritus