Hi. I am a brazilian student of statiscs.
My caculus professor (he is VERY eccentric/crazy) gave us a challange.
I will prove that Pi is equal 2!!!
Letīs imagine we have a semicircle with diameter equal 2.
as the ray is one, the perimeter is pi.
now imagine inside this semicircle two smaller semicircles, with ray equal 1/2.
The sum of the perimeter wold be 2*pi/2. So we would have pi again. (And the sum of the diameters would be 2)
imagine now 4 semicircles with ray equal 1/4. we would have the sum of the perimeter equal 4*pi/4=pi
imagine now 16 semicircles with ray equal 1/16. we would have the sum of the perimeter equal 16*pi/16=pi
and so on. no matter how many semicircles, the total perimeter would be pi and the sum of the diameters would be 2.
The smaller the rays are, the more the circles approach to the segment made by the sum of the diameters (equal 2).
He sai that if we had rays = 1/ infinite, the limit of the sum of the perimeters is the segment = 2....
So, falsely, proving that pi is equal 2. Itīs easy to see that this is the mistake. No mather how smal the rays are the circles NEVER "touch" the botton.
What he wants know is:
What is the limit of this curve/function...
I told him the limit is not the segment, he agrees but wants me to tell what is this limit.
I tried a lot but I can't prove anything.
check this video
Two Paradoxes: Pi equals 2 and SQRT(2) equals 2 (TANTON: Mathematics) - YouTube