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Why square root of a complex number is never purely imaginary?

Assume there is a complex number whose square root is purely imaginary.
$$\sqrt{a+ib} = ic$$
Squaring both sides,
$$a+ib = (ic)^2 = -c^2$$ The RHS is a real number, whereas the LHS is a complex number.
The derived equation is false. Therefore, our original assumption that there can be a complex number whose square root is purely imaginary is false.
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