Interpretations of Exponents

Question

I have been reading one of the proofs of Euler's identity, $e^{i\theta}=\cos(\theta)+i\sin(\theta)$.

I have always thought that exponents can be interpreted as its base being multiplied its exponent times (i.e. $3^5$ multiplying 3 5 times together).

But, this interpretation breaks down when the exponent is not a rational number. ($2^{1/2}$ can be interpreted using this logic. $2^1$ is 2 multiplied once. $\left(2^{1/2}\right)^{2}=2^1$ So $2^{1/2}$ is a number that can be multiplied twice to get 2).

Why does this intuition break down when we multiply complex numbers ($a+b\textbf{i}$) and irrational numbers? And also is there some other geometric intuition for complex and irrational numbers?

Answer 1

For exponents to a complex number’s power, it is as if you are rotating a vector. For example, $2^{2+i}$ can’t be broken down to $2^2\cdot 2^i$. $2^2$ Scales your vector and the $2^i$ part rotates your vector 1 radian. For example in the case of $e^{i\pi}$, it rotates the unit vector $\pi \ rad$ to make $e^{i\pi} = -1$

Answer 2

The rate at which $e^z$ changes from $1$ to something else as $z$ changes from $0$ to something else, is the same as the rate at which $z$ changes. Thus if $z$ is changing at a rate of $i,$ then so is $e^z.$