- #1

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in advanced maths , does +infinity and -infinity join at some point ?

a bit like if the axis was a cilinder

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- Thread starter JPC
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- #1

- 201

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in advanced maths , does +infinity and -infinity join at some point ?

a bit like if the axis was a cilinder

- #2

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then it would invalidate the first expr.

- #3

HallsofIvy

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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".

- #4

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[tex](\theta, \phi) \rightarrow \tan(\phi/2) e^{i \theta} [/tex]

(where by [itex]\phi[/itex] I mean the polar angle and [itex]\theta[/itex] the azimuthal angle in polar coordinates)

But when [itex]\phi \rightarrow \pi[/itex] (ie. at one pole of the sphere, under this coordinate system), you find that [itex]\tan (\phi/2)[/itex] 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

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

- #5

Hurkyl

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You're thinking of the extended real line; the Stone-Cech compactification is immensely more complicated.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.

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