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Correct me if I'm wrong but (1+i)^2 = 2i is false? I thought it is = (i^2)+2i+1

i² = -1, so it all works out

(1+i) = scale by sqrt2 and rotate by 45deg

rotate45(rotate45(0deg)) = 90deg

sqrt2 * (sqrt2 * (1)) = 2

Length 2 in 90deg direction is 2i