What is wrong with this fake proof $e^i = 1$?

Question

$$e^{i} = e^{i2\pi/2\pi} = (e^{2\pi i})^{1/(2\pi)} = 1^{1/(2\pi )} = 1$$

Obviously, one of my algebraic manipulations is not valid.

Answer 1

As said in the comments, the expressions $(a^b)^c$ and $a^{bc}$ are in fact multivalued functions; they are not a uniquely determined complex number. A classic example of a multivalued function is the complex logarithm denoted by $\log(z)$, $z\in\mathbb{C}$. The complex logarithm $\log(z)$ is any complex number $w$ satisfying $e^w=z$ (which has several solutions, see e.g. this), and hence $\log(z)$ gives rise to a whole set of complex numbers instead of just a single complex number.

Complex exponentiation such as $z^w$ for $z,w\in\mathbb{C}$ is usually defined as $$ z^w=\exp(w\log(z)), $$ where $\log(z)$ is the complex logarithm, and hence this is also a multivalued function. I hope this sheds some light on the problems with doing manipulations on complex numbers as if they were real numbers. See also this for other examples of identities which fail when using complex numbers as they were real numbers.

Answer 2

I'm going to add this here as the most general answer to this common problem in solving equations, a principle which the other answers do not cover:

When asked to solve for $y$, an equation of the form:

$$y=f(x)$$

(in your example $f(x)=e^i$)

The approach commonly taken is to solve by writing a sequence of equivalent equations working towards $y=\text{the desired answer}$, in your case as follows:

$y=e^{i}\\ y= e^{i2\pi/2\pi} \\y= (e^{2\pi i})^{1/(2\pi)}\\y = 1^{1/(2\pi )}\\y = 1$

Then this method is only reliable if it is truthful to write if and only if between every consecutive pair of equivalent equations in sequence as follows:

$y=e^{i}\\\iff\\ y= e^{i2\pi/2\pi} \\\iff\\y= (e^{2\pi i})^{1/(2\pi)}\\\iff\\y = 1^{1/(2\pi )}\\\iff\\y = 1$

Due to the periodicity of complex powers, the $\iff$ statement is contradictory to the laws of arithmetic in various places in the above chain.