1. THE CONTEXT OF THE PROBLEM
This question came to me when I was exploring complex exponents. The key identity to computing expressions with complex exponents is the Euler's identity:
$$e^{i\theta}=\cos\theta+i\sin\theta$$
This enables us to compute, for example what $2^i$ is by some algebraic manipulation. The computation is as follows:
$$2^i=e^{\ln2^i}=e^{i\ln2}=\cos\ln2+i\sin\ln2\;\approx\;0.769+0.639i$$
2. THE PROBLEM
A question arises in my head. We compute sines and cosines with radian values. And since $\ln2\approx0.693$ then $\sin\ln 2$ will be the $y$ coordinate of the unit circle when the angle is $0.693$ radians. But if we use turns instead of radians, then $\sin\ln 2$ will be the $y$ coordinate of the unit circle when the angle is $0.693$ turns. So the value of sine when using turns or radians is different. Similarly, the value of cosine is different.
But that creates a problem. When using radians, $2^i$ computes to about $0.769+0.639i$. But when using turns, $2^i$ computes to about $0.994+0.110i$.
3. POSSIBLE IMPLICATIONS
The problem above illustrates that $2^i$ and any expression with complex exponent is a generalisation that directly depends on what units we use for angles.
There are only two possibilities of "the state of truth" that this fact implies. Either complex exponents are a concept completely made up by humans which would mean that expressions like these are really undefined to the code of the universe (we only make it defined because we think that radians are the true angle units), or there must be a unit that is the truest unit for measuring angles, whether that would be radians or something else.
If the first statement is true, then this would mean that we can accept $2^i$ to be $0.769+0.639i$. After all, this concept would be defined by radians only because we defined it like this.
However, if the second statement is true, then there are even more questions to be asked. If there is a "truest" unit for measuring angles then what is it? Perhaps radians are really the truest unit, meaning that $2^i$ is unambiguously $0.769+0.639i$, but if so, what justifies this fact? What makes radians truer than turns?
Best Answer
The actual definition of the complex exponential map is equivalently via an ODE ($f'=f$, $f(0)=1$) or via power series ($f(z)= \sum_{k\geq 0}\frac{z^k}{k!}$).
Similarly, cosine and sine functions are defined via power series above all. If you are to change the way you measure angles and therefore define a new, distorted version of sine and cosine, then the most natural thing to say would be that the Euler formula simply doesn't hold in your new settings. Or you could use it as a new definition for a distorted exponential map, but then what is more important for you: the fact that Euler holds, or the fact that $\exp$ satisfies the above ODE?
All of mathematics rely on the fact that we agree on axioms, language and definitions. How natural these are depend on how beautiful the resulting theorems seem to the trained eye.