Could you please explain to me why this simple equation is true
$$-i * i = 1$$
or
$$-\sqrt{1}*\sqrt{1}=1$$
I know basic properties of roots $$\sqrt{a} * \sqrt{b} = \sqrt{a*b}$$
but what I get is $$-\sqrt{-1} * \sqrt{-1} = -\sqrt{(-1)*(-1)}= -\sqrt{1} = -1$$
Answer
The equality $-i\cdot i=1$ is equivalent to $i\cdot i=-1$.
But $i^2=-1$ is a property of $i$, so you're done.
$\sqrt{-1}$ is not one number, but two numbers:
$\sqrt{-1}=a\iff a^2=-1$, there are two such $a$: $a=i$ and $a=-i$.
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