How is it possible to divide by zero in the Reimann’s sphere? In other words how come there is no such thing as a negative infinity?


I recently did some (really short) research on dividing by zero and heard it is possible to divide by zero in the Riemann’s sphere, because it only defines a positive infinity and not a negative one. Is that true? And how does that work? Can’t we always find a negative of a certain number?

In: Mathematics

Dividing by zero is not allowed in standard algebra because it leads to undefined answers. 1/0 = 2/0, but under standard algebra that implies that 1 = 2 which breaks the rules of algebra.

The Riemann Sphere behaves like typical complex algebra, except with the addition of a *valid answer to 1/0*, namely positive infinity. It explicitly says there is no negative infinity. Operations involving infinity are explicitly defined; x + inf = inf, for example. Other operations are not allowed, same as in normal algebra; inf – inf is undefined and an illegal operation.

By handling infinity in these specific ways, any paradoxes that would allow us to “prove” 1=2 are removed, and logical consistency is maintained.

Riemann’s sphere pretty much introduces a wrap-around at infinity, so infinity is like zero in terms of not having a sign, being its own negative (well, except for the fact that adding two of them gets you infinity instead of zero) and so on. Poles are no longer undefined and are in fact not even discontinuities anymore.