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? Well, there are two answers: 1 and -1. Similarly, we can ask the question for higher roots; the third roots are solutions to the equation: And so on. When we’re only dealing with real numbers, that’s pretty much the whole story. For odd-degree roots of 1, the only answer is 1; for even-degree roots of 1, the answer is either 1 or -1. When we extend our domain to the complex numbers, things become much more interesting. Basics: roots of unity for complex numbers By the formal definition, a root of unity is any complex number that yields 1 when raised to a positive integer power . Rather than looking at all the possible roots of 1,…