>>10935780For any complex number k (which includes real k), there are two solutions to z^2=k. ?k denotes the "principal" square root of k, which is the one whose argument is in (-?/2,?/2], i.e. the one closest to the positive real line. The other square root is -?k with an argument in either (-?,-?/2] or (?/2,?].
Multiplying complex numbers adds their arguments. If you multiply two complex numbers with arguments in the range [0,?/2] the argument of the product doesn't necessarily lie in that range; it may be in (?/2,?]. So while the product of the principal square roots of two complex numbers is *one of* the square roots of their product, it isn't necessarily the principal square root; it may be the negation.
The square roots of a negative real are imaginary, i.e. they have arguments of -?/2 and ?/2, so the one with argument ?/2 is the principal root. So if a and b are both negative reals, ?a?b = -?(ab), i.e. the principal square root of their product is the negation of the product of their principal square roots.
For positive reals, the principal square root is always a positive real, and the product of two positive reals is a positive real, so ?a?b is always equal to ?(ab).
