There are a pair of linked paradoxes that tend to do the rounds on the internet: one that proves that pi is equal to two, and one that proves that the square root of two is equal to two. Obviously this isn’t true which is why they are called paradoxes. What follows is my rather small addition to the paradox.
The Staircase Paradox
Imagine a 2-dimensional square of side-length 1. A diagonal line drawn between opposite corners has an easily calculable length using Pythagoras’ Theorem:
Of course there are other ways to get from one corner to the other… We could go halfway along one side, then take a ninety-degree turn, go halfway again (to the middle), then repeat to get to the far corner. Or we could take even more steps! Figure 1 shows a couple of these “staircase” options, in addition to the direct diagonal path.
The paradox then goes on that as the number of steps in the staircase (n) is increased it converges on the diagonal line. Therefore as n approaches infinity the staircase still has length 2, but it also converges infinitely close on the diagonal line.
Does it Converge?
From this, we can calculate the total area between the staircase and the diagonal line as:
And now applying limits we see that as the number of steps increases, the area of the difference between the two paths does actually converge on zero, while the distance traversed by each path remains different!
Same Thing With Circles!
The length of the n=2 red semicircle is:
If these calculations are repeated for n=4 and n=8:
At this point the paradox argument goes (like in the staircase above) that the semicircles eventually converge on the straight line. Since the straight line has a fixed length of 2 and the circles always have a length of π, that therefore π = 2!
In fact, the same area-based logic can be applied to this problem as well. First we find a general formula for the total area under the semicircles in terms of the number of semicircles along the line, n.
This is actually the same form of equation as in the staircase-case, except that the triangle coefficient in the numerator (1/2) is replaced by the circle coefficient (π)! Once again the limit shows that this area converges to zero as we increase the number of semicircles:
So How Does This Resolve Anything?!
It doesn’t, I’ve just made it more confusing… But Gödel did point out that paradoxes have to exist… For that matter, we can always prove that 0.9999[…] is exactly equal to 1: