For some reason, the view has become standard that the most beautiful equation in all of mathematics is . “What a mystical coincidence! What on earth do , , and have to do with each other?”* and all that rot. The equation is then often established in a bizarrely roundabout manner with the use of Taylor series, all the more to heighten its appearance as coming out of nowhere and to dull attempts to sully its transcendent beauty with mere understanding. Of course, this involves sweeping under the carpet the forgotten matter of where the Taylor series came from in the first place and whether the proof could be carried out more directly in that original context to begin with.

Bah! Don’t swallow the hype, attractive though it may be. Understanding things is better than finding them amazing. And to understand this equation, to know what it means, is to see that ultimately, all it is claiming is a very simple, intuitively obvious fact: as one rotates a stick, the direction in which the end of the stick is moving is always 90 degrees rotated from the direction in which the stick is pointing. That is, the claim is little more than that the tangent to a circle is perpendicular to its radius.

Sure, it doesn’t look like that’s all it’s saying, but it is. Let’s break it down, using the relevant definitions (if you haven’t yet done so, you should first review how the arithmetic of negative and complex numbers is a way of talking about rotation). First, let’s recall our definitions of and : I’ll use to mean the action of one full revolution (360 degree rotation). , as we saw before, can be thought of as the action of turning halfway through a full revolution (that is, as a 180 degree rotation); in other words, . And is of course defined as the square root of ; thus, , a quarter revolution (90 degree rotation).

As for , it is the base of the natural logarithm; that is, is by definition equivalent to . The natural logarithm, in turn, is defined so that per unit of time is the constant ratio between an exponentially growing quantity’s velocity and the quantity itself, where the quantity multiplies by over every unit of time. (In calculus jargon, is the derivative of with respect to when )

And, at last, the matter of ; first, let’s introduce for rotation by one radian. Radians, of course, are the normalized unit of rotation in the sense that swinging through one radian is the same as swinging through an arclength equal to the radius; in other words, is , the purely directional component of with magnitude normalized to ; i.e., is the direction of the tangent to rotation. Finally, is the number of radians in a full revolution; that is, , which is to say, .

Ok, now for the unravelling: the claim is, by the definition of , just the claim that . In other words, that (the form the claim would be presented in if one cared more about (full revolution) than about (half revolution) [and, indeed, people should…]). By the definition of , this is just the claim that . By the definition of , this is just the claim that . Finally, by the definitions of and , this is just the claim that is the direction of the tangent to rotation. In other words, this is just the claim that as one rotates a stick, the direction in which the end of the stick is moving is always a quarter revolution (90 degrees) rotated from the direction in which the stick is pointing.

And if you understand that basic geometric fact about rotation, you understand that (and that in general**). That’s all there is to it.

*: What do , , and have to do with each other? Quite a lot! Both and are all about rotation (remember, is just a fancy name for 90 degree rotation), and is all about exponential growth, and rotation is one very simple example of exponential growth (multiplying by the same amount over any equal intervals). So the idea that these are disparate concepts far from having any obvious connections to each other is hogwash; its perpetuation is willful ignorance.

**: Of course, we can always split rotation into its parallel and perpendicular components (which go by the names cosine and sine respectively), and rewrite this as . This is just a change in the presentation of the same simple geometric fact; sometimes it’s helpful to split things into co-ordinates, and sometimes it’s distracting.

[TODO: General cleanup throughout the post, do more hand-holding on explanation of natural logarithm, perhaps more explicitly raise the issue of contexts in which to consider cognate rotations (e.g., 14 degree rotation vs. 374 degree rotation) the same and in which to consider them distinct and how this relates to complex exponentiation and logarithms, including perhaps using color-coding to more explicitly track how the two different perspectives are being used throughout]

(Need a re-summary of the ideas in the above post? Try here)