Would complex numbers be considered as part of Euclidean Space? Would this measurement give an accurate result? If not, what would be a more appropriate distance measurement/similarity measure with respect to complex numbers?
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You can certainly use the Euclidean idea of distance between two complex numbers $$ |z_2 - z_1| = \sqrt{(x_2-x_1)^2 + (y_2 - y_1)^2} $$
However, the complex plane contains more algebraic structure than plain euclidean space $R^2$ so calling complex numbers "part of the euclidean space" is not quite accurate. Here is a link to another question with a discussion of the differences between the two.
user_of_math
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Euclidean distance is exactly what we get when we consider the modulus of a complex number: $$|a+bi| = \sqrt{a^2 + b^2}$$
so it makes complete sense to use Euclidian distance, and this is in some sense the most natural distance to choose, since $\mathbb C$ is isomorphic to $\mathbb R^2$.
Mathmo123
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