What is angle subtended by two consecutive points on the circumference at the centre?

Question

A circle is continuous and yet when you take two consecutive points it seems as if the angle subtended at the centre is zero. If there was some angle between them how could they be consecutive? Tell me if am thinking wrong.

Answer 1

Yes, your "thinking is wrong", but in an old and respectable way.

The usual version of this (wrong) way of thinking is that on the number line there's a "next number" after any particular number. But that's not the case. There is no "next number" after $0$. To see why, suppose that next number is $t$. Then where would you put $t/2$, which is in between $0$ and $t$?

Mathematicians and philosophers have struggled with this problem since Zeno and the Greeks: you can read about Zeno's paradox many places. Mathematicians have figured out over the years how to deal with questions like yours.

Your concern about "adjacent points" on a circle is essentially the same. It's a good question.

Answer 2

For calculation of circumference C we calculate the distance between two subsequent point which the angle between their connecting lines to the center( two radii) is$d t$.We have:

$ C =\int^{2\pi}_0 r dt=r [ t ]^{2\pi}_0 =2\pi r$

How can we calculate C if there is no distance $rdt$ between two points or the angle between two connecting lines to center is zero?If there is no distance between them then they are coincident not consecutive!