The set of all points in a plane is ℝ², not ℂ. ℂ has many special properties that aren't shared by ℝ²,[1] so while you can think of ℝ as just the number line, ℂ

The set of all points in a plane is ℝ², not ℂ. ℂ has many special properties that aren't shared by ℝ²,[1] so while you can think of ℝ as just the number line, ℂ is more than just a plane. [1]: https://math.stackexchange.com/questions/444475/whats-the-difference-between-mathbbr2-and-the-complex-plane https://math.stackexchange.com/questions/444475/whats-the-di...