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?

Solution 1:

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.

Solution 2:

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.