Are the points on a unit circle a countable set?

Question

I was watching this video: https://youtu.be/s86-Z-CbaHA

Near 4:10, he says that all the real numbers between 0 and 1 are uncountable. But near 8:38, he says that all the points on that circle are countable because they can be in one-to-one correspondence with natural numbers. But, isn't that just another way of saying that all the real numbers between 0 to 2pi are countable? Because , on a unit circle, each point can also correspond to one real number between 0 and 2pi. So, how can that be countable?

Answer 1

I don't hear any claims about circles around 6:38.

At 8:50 the video does say "this set [of points on the unit circle] is never ending but countable". But it doesn't claim that the set it speaks about consists of all points on the unit circle, and it doesn't.

Answer 2

The numbers on the unit circle with finite order are countable, i.e. all complex numbers $z$ such that there is a natural number $n$ with $z^n=1$. Geometrically these points correspond to all points on the unit circle such that there is a regular $n$-gon with that point and the point $(1,0)$ as a vertex and the point $(0,0)$ as its center (not every point has this property).

The set of all points of the unit circle is uncountable.

Answer 3

The points on the unit circle are in one to one correspondence with real numbers so they are not countable . On the other hand the set of all $n^{th} $ roots of unity is a countable subset of the unit circle as claimed in videos.