points on a circle ARE countable???

Question

I always thought that the points on a circle would be uncountable, like the real numbers between 0 and 1. But I read or saw something that made me wonder. It said you could form a one to one pairing between the integers and the points on a circle of radius one. Start at any point and pair that to the integer 1. Move along the circumference a distance of one unit and pair that point with the integer 2. Keep doing this. Since the length of the circumference is irrational, it was claimed, you will not pick the same point ever again and you have a one to one pairing. Can this be right? It used to be thought that rational numbers were uncountable until someone came up with the clever idea to zip back and forth through a grid of all the fractions. Is this the clever idea that shows that the points on a circle are countable or is there something wrong with the claim? And if it turns out to be correct, aren't then the real numbers between 0 and 1 countable by wrapping that line segment into a circle and moving along that circle by a distance of some irrational number?