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     Complex numbers are a powerful tool for studying the two-dimensional plane: each point corresponds to a unique complex number . The beauty of this correspondence is that it allows you to add, subtract, and multiply points in the plane
I wish someone would have told me this five years ago, it is such a neat way of thinking!

My introductory physics class in college included a problem that investigated the effects of "2d" gravity (a generic attractive force proportional to 1/r instead of 1/r^2) on a point mass in the plane.

When the professor mentioned that I could write all the vectors as complex numbers, and do regular old algebra with them, my brain almost melted.

if you are interested in new insights about complex numbers and analysis from a geometric POV, this book is wonderful: http://www.amazon.com/Visual-Complex-Analysis-Tristan-Needha... and quite accessible.

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