# Given the equivalence relation on $\mathbb{C}$ defined by $a+bi\ R\ c+di \iff \sqrt{a^2+b^2}=\sqrt{c^2+d^2}$, describe $\mathbb{C}/R$. [closed]

How can I describe properly the set of all equivalence classes?

• Are you familiar with polar coordinates? – Joe Aug 30 '19 at 15:09

You can write the relationship as $$z_1Rz_2\iff|z_1|=|z_2|$$. Is now very easy to understand that $$\mathbb{C}/R$$ is the set of all origin centered circumferences of the complex plane plus the origin. Each one is the equivalence class of any of it's points.

As other have noted, each circle centred on $$0$$ is an equivalence class, since $$R$$ is equivalent to modulus equality. However, there is a subtlety: while circles are uncountable equivalence classes, there's another equivalence class, $$\{0\}$$, with only one element.