Why are radians treated like a quantity while degrees are treated like a unit?

Question

According to this answer about using Euler's Identity in Degrees, radians and degrees are interchangeable.

Why is it that:$$(e^i)^{\pi/2}=i=(e^i)^{90^\circ}$$ But: $$i^{2/\pi}\neq i^{1/90}$$

Edit: Im well aware $2/\pi \neq 1/90$

Answer 1

To summarise the comments given by myself and littleO, let me provide an answer that I hope you will find thorough enough.

When you make the statement $$ (e^i)^{90^\circ}=i $$ you require to clarify what you mean by $90^\circ$. Really, there are three flavours of answer.

1. $^\circ$ is a unit, in which case you cannot take its exponent and the statement does not make sense.

2. $^\circ$ is a dimensionless quantity in its own right, in which case $90^\circ := 90$ and we have $ (e^i)^{90^\circ} \ne i $.

3. $^\circ$ is a dimensionless quantity defined as $a^\circ := \frac{\pi}{180}a$, in which case $ (e^i)^{90^\circ} = i $.

To add further clarification, I will briefly comment on why $e^{\frac{\pi}{2}i}=i$. Raising a real number by an imaginary number has the effect of rotating it around the complex plane and scaling it to magnitude $1$. 3Blue1Brown has an excellent video of why this is https://www.youtube.com/watch?v=mvmuCPvRoWQ. The number $e$ is special as it is the unique number such that the arclength of rotation is exactly equal to the exponent (i.e. $e^{ai}$ rotates the number $1$ through an arclength of $a$). The reason for this is to do with the property that $\frac{d}{dx} e^x=e^x$. The arclength from $1$ to $i$ is $\frac{\pi}{2}$ and thus $e^{\frac{\pi}{2}i}=i$. This is why radians is useful as it is a measure of arclength. Degrees were used historically as 360 is highly divisible, however calculus doesn't care about this property.

I understand that argument is a little rough, but you should watch the video for a better idea of what is going on.