Two paradoxes: $\pi = 2$ and $\sqrt 2 = 2$ 
Possible Duplicate:
Is value of $\pi = 4$? 

Can anyone explain how to properly resolve two paradoxes in this YouTube video by James Tanton?
 A: First: It is not a paradox: it is just wrong. The reasoning is wrong.
About $\pi = 2$ he says: "Well clearly we are approaching the diameter of the circle". That is a statement that he doesn't prove and which is false.
The same problem arises with the $\sqrt{2} = 2$ when he says: "Well clearly this geometric construction approaches the diagonal of the square". How does he know that?
All that this proves is that we have to be careful when we talk about finding limits from purely looking at pictures.
"Just because the sun sets in the west doesn't mean that it has to rise in the west as well.
Edit: There are plenty of example of proofs that seem right, but turn out to be wrong when we go over them in more detail. Take for example the proof that for complex numbers
$$
1 = \sqrt{1} = \sqrt{(-1)\cdot(-1)} = \sqrt{-1}\sqrt{-1} = i\cdot i = -1$$
Here again, the argument is invalid because the rule $\sqrt{ab} = \sqrt{a}\sqrt{b}$ doesn't hold for complex numbers.
A: If the semi circle is divided into $2k$ semicircles  , the total length is always $$\cfrac 12 \cdot 2\pi\left(\cfrac r{2k}\right)\cdot 2k=\pi r$$
Which means we have a constant sequence $a_{2k}=\pi r\rightarrow \pi r$ when $k\rightarrow +\infty$.
It is true that $\pi$ is an unknown constant $>0$. From the first semicircle, it is quite obvious that $\pi r > 2r$  which implies that $\pi r \ne 2r$ no matter the value of $\pi$ as long as $r$ stays positive. If you draw semi circles to infinity, we do approach the diameter. In this case, the geometric figure approaches the diameter while the total length remains constant. But no matter the number of semi circles drawn, there are always curves around the diameter and it is the total length of these curves that we are interested in, not the diameter so when this number gets large enough that we have no curves to measure, one simply measures the diameter and assume that the total length is that of the diameter without needing $\pi$ at all. If one now says $\pi r = 2r$, this contradicts the first assertion that $\pi r > 2r$
The geometric constructions might have approached the diameter, but the total lengths stays constant and doesn't approach the length of the diameter.
Applying the same logic to the one of $\sqrt 2$, you find out that his argument is also invalid. The semicircles don't converge to the diameter and the stairs don't converge to the diagonal.
A: At 1:00 in the video, it is stated that $2(\frac{1}{2}\times\frac{\pi r}{2})=\pi r$, which is incorrect; the former expression simplifies to $\frac{\pi r}{2}$, not $\pi r$. The $\pi r$ line of reasoning is used to prove that $\pi=2$.
As for the proof that $\sqrt{2}=2$, bear this in mind.

(Great, now $2=4$. We're reinventing even more things than were anticipated now.)
It doesn't actually approach anything; it just looks like it does. Maybe it would help if the staircase were less crude; like, actually a reasonable approximation of the diagonal of the square, in which the median averages relatively near the diagonal.
A: If the limit of a sequence of curves $\{\mathcal{C}_n\}$ is a curve $\mathcal{C}$, that does not mean that the limit of the lengths of the $\mathcal{C}_n$ will be the length of $\mathcal{C}$. The presentation makes the assumption that it will, but what validates that assumption, other than it feels intuitively to be true? This is a great example of how intuition, while usually helpful, can occasionally be hurtful. This example can serve as a poster child to champion mathematical rigor.
Take the circle example. After $N$ iterations, the extremely bumpy curve that you have is still much longer than the diameter.
And the staircase example. After $N$ iterations, the extremely jagged curve that you have is still much longer than the diagonal.
As a function from curves of finite length to $\mathbb{R}$, $\operatorname{length}(\phantom{x})$ is not continuous. It may feel like it should be, until you think about examples like the kind brought up here.
