## Euler’s Formula and the Fundamental Theorem of Algebra

### August 26, 2012

During my vacation time, I found a couple days for an old project I’ve been wanting to complete for a while now:

**Euler’s Formula and the Fundamental Theorem of Algebra
**

For a few years now, I have wished that the Fundamental Theorem of Algebra had been written up at a much lower level, and that someone had given me such a write-up at, say, age 16. I am not sure whether this first draft is a success or not, but I have attempted to write what sort of thing I had in mind, complete with remarks that are included in undergraduate lectures, but not always present in undergraduate textbooks. The audience for this is supposed to be relatively general, though familiarity with Taylor Series would be an advantage.

The interplay between Euler’s Formula and the FToA is strong. Although students are often introduced to both in high school, the FToA is never proved (but Gauss sure was smart! He proved it five different ways! Wish you were him, don’t you?), and Euler’s Formula is reduced to the status of a bit of numerological trivia I like to call Bieber’s Theorem.

It’s not longish because it’s that hard. It’s long because I’m attempting to make it that easy. Though I suppose there’s always a little bit more to it than I thought that there was. Depending on the audience, a paper 2-3x as lengthy can often take a quarter as long to read, depending on how useful the expanded explanations are.

The theorem has a few other proofs, one of which in particular I want to sketch in a later draft. The proof presented in the first draft is much closer to a fully rigorous complete proof than the others. Some content has been left out, but it is content that is easier to fill in details for in a self-contained sort of way, when I get around to it.

[...] Euler’s Formula and the Fundamental Theorem of Algebra – To understand how complex polynomials work, you must first understand what it looks like when you multiply two complex numbers. The deep but easy-to-use identity called Euler’s Formula allows us to do this. One common proof of the FToA follows from Euler’s Formula without the need for extra machinery. [...]