Notes on the complex roots of unity

One of the most wonderful aspects of math is how deep we can get by studying seemingly trivial questions. Consider the roots of unity, where "unity" simply means the number 1. Basics: roots of unity for real numbers Studying the roots of unity usually begins in middle school: what is the square root of 1? Equivalently, what are the solutions to this equation? \[x^2=1\] Well, there are two answers: 1 and -1. Similarly, we can ask the question for higher roots; the third roots are solutions to…