eli5: Riemann Sphere

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So if I’m looking at this correctly, if you keep counting in the positive direction, it will eventually go to the negatives and then back to positive?

Is this used just as a way to do x/0? Or is this how numbers actually work?

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> So if I’m looking at this correctly, if you keep counting in the positive direction, it will eventually go to the negatives and then back to positive?

There is a sense in which that’s sort of true. More properly, as you divide by positive numbers closer and closer to zero, you “hit infinity” at zero, and then come back around from the other side. If you’re dividing a positive real by a number close to zero, you go through large positive reals, the point at infinity, and then come back from large negative reals. If you’re dividing a positive pure imaginary number, you go through large positive imaginaries, the point at infinity, and then come back from large negative imaginaries. And if you divide, say, (2 – 3i) (in the fourth quadrant of the complex plane), you go through large-magnitude values in the fourth quadrant, hit the point at infinity, and then loop back around from the first quadrant.

> Is this used just as a way to do x/0? Or is this how numbers actually work?

I mean…there’s not really a distinction between those things in a formal sense, but it’s closer to the latter. It turns out that when you’re dealing with functions on the complex plane, infinity “acts like” a point on the plane in a way that results in a lot of theorems producing results like “either this property holds for some value z in C or it holds in some sense at infinity”. So it’s easier to just include the point at infinity and say “this holds for some value z on the Riemann sphere”. As a concrete example, it simplifies the way you think about [Liouville’s theorem](https://en.wikipedia.org/wiki/Liouville%27s_theorem_(complex_analysis)#Remarks), one of the most important theorems in complex analysis (along with its big brother Picard’s theorem).

If that feels like terrifying voodoo that really shouldn’t work: well, welcome to complex analysis, the most occult of mathematical disciplines.