complex numbers - Imaginary fraction square root?

I have a fraction -

$-\frac{1}{3}$

Which could either mean the value of fraction is $\frac{-1}{3}$ or $\frac{1}{-3}$ Note the minus sign

Now, what is the sqaure root of the fraction? I tried and I got this -

$\sqrt{-\frac{1}{3}}$

$=\frac{i}{\sqrt3}$ or $\frac{1}{\sqrt3i}$

Now what is the actual square root? Or is it both? Am I going wrong somewhere? Which one should I use in my calculations?

Answer

Good question! You have discovered that it's not possible to define a square root function in the complex numbers that obeys the rule $\sqrt{ab}=\sqrt{a}\,\sqrt{b}$ (or the equivalent $\sqrt{a/b}=\sqrt{a}/\sqrt{b}$, with $b\ne0$).

You get the same dilemma, in an easier way, by considering
$$

i=\sqrt{-1}=\sqrt{\frac{1}{-1}}=\frac{\sqrt{1}}{\sqrt{-1}}=\frac{1}{i}=-i
$$
Note that this is clearly wrong, which doesn't tell us that mathematics is contradictory, but that we have used an unproved (and unprovable) property, namely that we can define a square root function satisfying the rule above.

Note that the false argument produces both complex numbers whose square is $-1$, the same happens in your argument.

A suggestion: never use the symbol $\sqrt{-1}$, because it suggests the possibility to apply the wrong property. Neither use $\sqrt{z}$, for the same reason, unless $z$ is a real number with $z\ge0$.