(sorry I don't know how to add pictures)
Two friends argue if larger circles have more points than smaller circles
Friend number 1 (a well known argument)
Say the circles are concentric. you cannot draw a line from the centre that cuts the bigger circle while that doesn't cut the inner circle so they have the same amounts of points.
Friend number 2 ( a dissenting voice)
Ok lets take concentric circles, he draws a line to the centre ( say along the x axis)
adds another line parallel to this line it cuts both circles 2 times)
adds another (cuts both circles 2 times) and so on
two parallel lines are only tangent to the smaller circle while still cuting the bigger at two points. and some paralel lines only cut the larger cirle.
Therefore all points on the smaller circle are related to some point on the bigger one, but some points on the bigger one are not related to point on the smaller one.
So the larger circle has more points.
Which friend is right or how do you convince them that they are both right?