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 $\mathbb{C}/R$.

Question

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

Answer 1

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.

Answer 2

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.