Hi Kevin,
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 e^{x} is
e^{x} = 1 + x + ^{x2}/_{2!} + ^{x3}/_{3!} + ^{x4}/_{4!} + ...
and this can be applied to complex values:
e^{ib} = 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 e^{i} = cos() + i sin() = 1, hence ln(1) = i .
Claude
