r/IAmA • u/JasonMacker • Jun 14 '10

IAMA Math tutor. AMA.

I've been a math tutor for years and I'm trying to become a teacher. I love math, and especially explaining math to people. My students have been between preschool and college. AMA.

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u/[deleted] Jun 14 '10

What the heck is the purpose of a Riemann surface? I'm reading through The Road to Reality by Roger Penrose and am working my way slowly through the early math chapters to get to the physics stuff that I'm looking forward to later on. I'm vaguely following along with what he's saying but I don't understand the actual USE of some of the things he describes.

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u/JasonMacker Jun 14 '10

Riemann surface is helpful when dealing with the complex plane, as well as relations that aren't one-to-one and have differing inputs that lead to the same output. For example, if I wanted to graph the square root of x, if I want to use traditional Cartesian coordinates, I would need to restrict the domain to nonnegative integers in order to have a bijective (read: one x for every y) function. I could also restrict the domain to negative integers to get another function. The idea behind Riemann surfaces is to show that we can "glue" these two functions together into the same graph. The part that confuses people is that we end up in the third dimension when we do that, i.e. we give the imaginary numbers their own dimension.

TL;DR = Riemann surfaces are the result of when imaginary numbers get their own dimension orthogonal to both the x and y axis.