I'm old 35 but starting just now with maths, so sorry if I ask non complex questions. 0 is the only number that just has one square root. Is the explanations for this simply that 0 in arithmetic does not have a sign. If this is the explanation this seems to me like a definition matter. Thank you.
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$\begingroup$For real or complex numbers, if $ab=0$, then either $a=0$ or $b=0$. This means that the complex numbers form an integral domain. It follows from this that if $a^2=0$, then $a=0$.
You could say this is the reason. If you want to use the fact that if $a$ is a root of $x^2=b$ then $-a$ is the only other root, then that is basically what you are saying : $0=-0$. But this presupposes the integral domain condition. When that fails, it could even happen that $x^2=b$ has infinitely many roots, even if $b=0$.
Every algebraic structure that can reasonably be called a set of numbers forms an integral domain. Except that this is not quite true! Suppose we are dealing with integers, but we declare that $m$ and $n$ are the same if they differ by a multiple of $4$. Then $2^2=0$, using this new notion of equality as differing by a multiple of $4$. So $2^2=0$ and $0^2=0$, but $0\neq - 2$.
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