Square √2/2 + i √2/2 and see what you get.
Actually any complex number can be written in polar coordinates: z = r(cos(a) + i sin(a) ),
and it is then the square of √2 (cos( a/2) + i sin( a/2) ). You get two "square roots" of z that way,
one from selecting the angle a in [0,2) and the other by selecting a in [2,4).
Natural logarithms of negative numbers can be understood once imaginary exponents are explained:
The infinite series expansion of ex is
ex = 1 + x + x2/2! + x3/3! + x4/4! + ...
and this can be applied to complex values:
eib = 1 + ib + (ib)2/2! + (ib)3/3! + (ib)4/4! + ...
= (1 - b2/2! + b4/4! - ... ) + i(b - b3/3! + b5/5! - ...)
= cos(b) + i sin(b).
In particular ei = cos() + i sin() = -1, hence ln(-1) = i .