$\sqrt {e ^{2 \pi i}} = 1$ or $-1$?

Question

$$\sqrt{e^{2\pi i}} = \sqrt{1} = 1 $$

However,

$$(e^{2\pi i})^\frac {1}{2} = e^{\pi i} = -1$$

By substituting $e$ for $\lim_{x\to 0} (1 + 1/x)^x$, it proves that the result in the first equation is accurate. Hence, I would like to know what is wrong with the second equation.

Answer 1

That depends upon the meaning of $\sqrt{\ }$:

• If, when $a\in[0,+\infty)$, $\sqrt a$ is the only non-negative square root of $a$, then, since $e^{2\pi i}=1$, the answer is $1$.
• If $\sqrt a$ is a square root of $a$ (a bad choice of notation), then both answers are correct, since both $1$ and $-1$ are square roots of $1$.

Answer 2

If you take $\sqrt x$ as a complex function, then you have two sqaure roots of $1$ namely $\pm1$.

If you just write $\sqrt 1$ then it is $1$ and if you want $-1$ you write $-\sqrt 1$

Answer 3

Technically, both are correct. You could ask he question: $$\sqrt{9} \stackrel{?}{=} 3 \ \text{or}\ \sqrt{9} \stackrel{?}{=} -3$$ Since the square root symbol asks the question: "What number raised to the second power results in this number?"

And since $3^2 = 9$ and $(-3)^2 =9$, both could be accepted as the correct answer.

However, in mathematics, we have defined $\sqrt{\ \ }$ to be a funtion with a $1$ to $1$ correspondance, so we define $\sqrt{x^2} := |x|$, and not $\sqrt{x^2} :\ne -|x|$. But when expanding on the idea of $\sqrt{\ \ }$, this can be quite misleading, so in the case of complex numbers, we have to go back to the notion of $\sqrt{x^2} = \pm |x|$.