Is $e^{i\pi}+1=0$ all it's cracked up to be? While it is beautiful and elegant and all that, isn't it true that Euler's identity is really just an artifact of how we define the radian? I'm speaking of those who say that it's great because it contains five important constants in one elegant identity.
If we had stuck with degrees, it would be $e^{180^\circ i}+1=0$, not quite as breathtaking.
I understand that the radian is a natural and dimensionless unit to use for angles, but I feel like there is still some element of arbitrariness to it which, to me at least, somewhat tarnishes the beauty of this identity.
Am I missing something that makes this relationship between $e, \pi, i, 1,$ and $0$ built-in to our universe? Or do we as mathematicians just like to think poetically?
 A: I think Euler must have been shocked when he discovered the identity.  I think he was just as shocked by the identity $e^{2i\pi} = 1$.
This is the point.  The history of $e$ and $\pi$ were rather separate until that time.  $\pi$ came about because of considering circles, whereas $e$ arose out of considering how to differentiate exponentials and logarithms.  So a priori, the numbers $e$ and $\pi$ would be thought to be not related.
Thus $e^{i\pi} = -1$ is a kind of unifying theory in mathematics in much the same way as Maxwell's equation unify electricity and magnetism in physics.  It means that many statements about $e$ carry over to $\pi$ and vice versa.  For example, if one can prove that $e^\alpha$ is transcendental for any algebraic $\alpha$, you can argue by contradiction that $\pi$ must be transcendental.
If we were to find non-trivial relationships between these and other mathematical constants, like the Euler–Mascheroni constant, or the roots of the Bessel functions, we would be similarly amazed.  After all, there is a reason why $\zeta(2)$ doesn't have a special name, since it is $\pi^2/6$, whereas $\zeta(3)$ is called Apéry's constant.
A: $e^{i \pi} = -1$ is just a poetic elegance.  However, the fundamental meaning behind it is more than just poetic.
The equation that really matters there is $e^{i\theta} = cos(\theta) + i sin(\theta)$, from which $e^{i \pi} = -1$ is nothing more than a quick substitution.
I can define any number system I please, I can add i, j, and k, or do anything I want.  However, what makes some systems of numbers "special" is that they follow the "rules of arithmetic."  These are special because mathematicians can use them without having to remember the identities for a whole suite of number systems... they can just remember the identities for addition and multiplication... and assume that they still apply (with just a handful of special cases to memorize)
There is one system, complex numbers, which gained great interest because it made a lot of equations simpler.  It also happens to follow a lot of conventions that mathematicians like.  For example, there is an operation called $+$ which allows you to assume $a + b = b + a$, without having to remember if that operation was valid on your particular number system.
For a while, it was easy to show complex numbers defined two operations which met all of the same rules as $a + b$ and $ab$ in real numbers.  However, there are a whole suite of transformations mathematicians love to use, but they needed more power -- particularly they needed a concept of a power tower.  Power towers are the natural extension of the pattern $ab = a + a + ... a(b times)$ $a^{b} = aaaaaaa(b times)$.  In common notation, $^{b}a = a^{a^{a...}}(b times)$ and $_{b}a = ^{^{^{^...}a}a}a(b times)$.  And clearly that pattern can be done endlessly.
Mathematicians wanted to find a definition for exponentiation with complex numbers.  Euler used a neat tool called analytic continuation to bring logarithms to complex numbers, which naturally provided exponentiation.  Without going into the details (I'm fuzzy on them myself), analytic continuation can either show that there are any number of possible "meanings" for an operation, or it can show that they all converge, such that there is only really one "meaning" for an operation, with several ways to get there (just like how $ab$ only has one value, even if you calculate it with repeated addition).
Euler not only showed that $e^{i\theta} = cos(\theta) + i sin(\theta)$ provided a valid definition for complex logarithms, he showed that all possible equations must converge on the THE value, such that $e^{i\theta} = cos(\theta) + i sin(\theta)$ can be thought of as the ONLY definition of exponentiation you need.
By doing this, he let us expand the list of things we can do with complex numbers from addition and multiplication, to exponentiation, logarithms, and then proved that, if one continued to define things like power towers, the method he used would generate a meaningful operation.
Phrased another way, with no math but waxing poetic instead, when it came to complex numbers, mathematicians were breathing through a soda straw, until Euler taught them how to breath fully.
A: $\newcommand{\Exp}{\operatorname{Exp}}$Suppose that for $x,y\in\mathbb R$ we define
$$
\Exp(x+iy) = e^x(\cos(y^\circ)+i\sin(y^\circ)).
$$
That has the same nice algebraic properties as the definition using radians; for example $\Exp(z+w)=\Exp(z)\Exp(w)$ for $z,w\in\mathbb C$.
But it lacks differentiability.  What is its derivative at $0$?  As $x+iy$ increases in the real direction, just as it's passing through $0$, then $\Exp(x+iy)$ is changing $1$ times as fast as $x+iy$ is changing.   As $x+iy$ changes in the imaginary direction, just as it's passing through $0$, then $\Exp(x+iy)$ is changing $180/\pi$ times as fast as $x+iy$ is changing.  Should the value of the derivative at that point be $1$ or $180/\pi$?
A: You can define $e^{r+i\theta}$ as 
$$e^r(\cos(\theta)+i\sin(\theta))$$
where the real exponentiation is raising a constant to the power $r$ and the trig functions can be degrees, radians, or whatever other unit you want. This satisfies the exponential law either way, and this definition of the exponential function may be somewhat arbitrary. 
However, only when you consider the argument in radians does the identity:
$$e^x = \sum_{n=0}^\infty \frac {x^n} {n!}$$
hold, and similarly for the functions $\sin$ and $\cos$. The power series for $\sin$ (which as you know is closely related to the one for $e^x$) is one way $\pi$ can be defined; that is, the smallest non-zero root of the function.
The beauty therefore comes from the fact that this constant $\pi$ is the ratio of the diameter of a circle to its circumfrence. And that many historically separate topics - trigenometry, complex variables, power series - all come together to describe the same thing.
I think it's pretty cool.
A: Yes, it is, but for what it's worth, I like $e^{i \pi} = -1$ better. You see, $-1$ is a very underrated number; $0$ and $1$ are important, but $-1$ also deserves fame, too.
But no, that identity is NOT just an artifact of how we define the radian. The thing is, $e^{180 i}$ is wrong, what you should have written was $e^{180^\circ i}$. But then, what is $1^\circ$? Yeah, you guessed it, $\frac{180}{\pi}$. It comes back to $\pi$.
Furthermore, suppose that Michael Hartl convinces everyone that we should chuck $\pi$ in favor of $\tau = 2\pi$. Then Euler's identity would become $e^{i \frac{\tau}{2}} = -1$.
A: I don't view this as cracked at all. My view on this is the following:
The point is that the Euler identity is not about radians at all. The Euler identity basically says the following: $e^{ix}$ is a point on the unit circle
$|z|=1$ and $x$ represents the arclenght of the sector of the unit circle from $z=1$ to this point.
Therefore, in the identity 
$$e^{ix} =\cos(x) +i \sin(x) \,.$$
$x$ doesn't represent an angle, is it an arclenght function/parametrization. 
Also, this is the sense in which the derivatives of $\sin(x)$ and $\cos(x)$ are nice.
An then the identity is just an identity about numbers, there are no units involved.
$$e^{i \pi} = -1$$
Radians are indeed artificial, they are an artificial way of expressing the arclenght function on the unit circle as angles. And this is only needed because people artificially consider the trigonometric functions as functions of angles, in reality when we define them on the unit circle, we really define them as the coordinates of a point on the unit circle with a give arclenght, and we call this arclenght, which is a real number, "an angle in radians"... And then we go backwards, emphasize that we should think about this as a real number not angle...
If you think back to Analysis, in any of the fundamental proofs of limits/derivatives of trig functions the key is that the angle at the center is the arclenght on the unit circle. The proof actually never uses angles explicitly, it uses arclenghts.
A: I suppose you could say that the mystery/beauty is that the exponential function, which is monotone increasing as a function of a real variable, becomes periodic as a function of a complex variable, and has period $2 \pi i.$
A: The equation $e^{\pi i} + 1 = 0$ has nothing to do with our conventions for angle measure.  Whether I measure my angles in radians or degrees or minutes or seconds, the equation $e^{\pi i} + 1 = 0$ is still true.  Here, $\pi$ is the "dimensionless" number we all know and love, defined as the ratio of the circumference of a circle to its diameter.
We can define $e(z) = e^z$ to be the unique complex-differentiable function that satisfies $e'(z) = e(z)$ and $e(0) = 1$.  (But is the input in radians or degrees?  Neither!  The inputs are numbers, not angles nor masses nor temperatures.)
With some work, one can show that the number $\pi$, the ratio of circumference to diameter, satisfies the equation $e^{\pi i} + 1 = 0$.
Now, the second that we want to give a geometric interpretation of the above formulas involving angles, only then do we have to worry about whether our angles are in radians or degrees. In that case, yes, our geometric interpretations are cleanest when we use radians.
Finally, I should point out that $e^{180 i} \neq -1$.  You might protest that the question was about $e^{180^\circ i}$, not $e^{180 i}$.  My question is then: what exactly do you mean by plugging in $180^\circ$ into an algebraic formula?  There are two different things you could mean, and they will lead to different answers.
