Is it appropriate to apply Euclidean Distance to Complex Numbers?

Question

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?

Answer 1

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 $\mathbb R^2$.

Answer 2

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.