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'Complex Squares' printed from http://nrich.maths.org/
What do you get if you square $a+ib$?
if $(a+ib)^2$ is real, what relationships must $a$ and $b$ satisfy?
if $(a+ib)^2$ is imaginary, what relationships must $a$ and $b$ satisfy?
What does this look like on an Argand diagram?