One of my favorite of all mathematical statements (and many people agree with me on this) is Euler’s identity:
eiπ + 1 = 0
It’s frustrating that I cannot explain to my “friends who are not into math” just why this simple statement is so extraordinary and beautiful. Basically, it opens the door to understanding why many things in the universe which at first seem complicated are actually very simple and elegant.
In my mind this statement is a kind of litmus test: Once someone gets to the point of understanding the profound idea that Euler’s identity embodies, and why that idea is so profound, simple and lovely all at once, then so much other understanding of the beauty of the universe opens up.
It would be interesting to design a fun and engaging course for “non-mathematicians” that specifically leads to this place of understanding.
I think I fall somewhere in between “friends who are not into math” and those who can claim an understanding of the profound idea. Anyone who has spent any time doing math (even calculating things without fully understanding all the deeper connections) can marvel at this equation that reveals a simple relationship between such seemingly unrelated (and important) quantities as the base of natural logarithms, the square root of negative 1, and the ratio of a circle’s circumference to its diameter. How about trying an explanation for us in-betweeners as a first step?
I’m another betweener, I have used the tools involved, but whenever I see this all I can say is that I see that it works, without appreciation for why it should work. The interpretation of complex powers of e has never yet made sense to me as more than a really bizarre notation, but I take it there’s as good a reason for it as for negative exponents being reciprocals and fractional exponents involving roots. I hate missing out on unifying ideas like this, and I think it’s one of the reasons my attempts at learning signal processing have all been frustrated at early conceptual levels.
I appreciate the idea of a course designed around one culminating thing of beauty. When I worked through the videos of Abelson and Sussman’s SICP lectures, the moment of seeing that the Lisp-in-Lisp actually works was both a pun and a revelation (something Hofstadter plays with as well in GEB, and with no coincidence). Would it be interesting for a game to conclude, having led the player to understand the methods by which the game was made, asking for the game’s code itself to be remade or reinterpreted?
I’ve always wondered if the existence of this formula suggests that we’ve reached the end of a certain kind of mathematical discoveries… in other words, if it’s true that all of our most important tools fit together in an especially elegant way, does that suggest that we’ve already uncovered all of the most important tools?