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This is a (fairly) famous mathematical fallacy. Can you figure out what's going wrong? Does Pi really equal 4?

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Disclaimer: This video is for entertainment purposes only and should not be considered academic. Though all information is provided in good faith, no warranty of any kind, expressed or implied, is made with regards to the accuracy, validity, reliability, consistency, adequacy, or completeness of this information.

Pi=4

#math #brithemathguy #pi

[BriTheMathGuy]

🎓Become a Math Master With My Intro To Proofs Course!

[vit.budina]

As an engineer, I stand by the fact that π=3 and e=3, and thus π=e

[tracym.m]

I actually came here to get instantly better at maths, all this did was bring more questions

[Rackcoon]

For people wondering, yes you could “infinitely” do the staircase misconception for pi = 4, but if you think about it, you will realise that the finer the staircase is; the more C (sqrt a^2 + b^2) will be included which “C” overall reduces the perimeter and increases accuracy

[DrTrefor]

This has long been one of my favorite examples of why you have to be careful with approximations and limits. Sometimes we see the "nice" examples from calculus and it sort of gives the impression these types of things always work, but no you have to be careful!

[William-Nettles]

Brought this problem to my analysis professor last year and he ended up using it as the motivating example for our study of uniform convergence. Love it.

[eclipse]

As you make it smaller and smaller, you can zoom into the edge of the circle and it appears to be more of a straight line than a curve. The steps form the base and height of a triangle, and the edge of the circle is like the hypotenuse. No matter how many times you break up the steps and zoom in, this will always be the case. Since the hypotenuse is smaller than the sum of the other sides, the circumference of the circle will always be smaller than the perimeter of the staircase shape.

[struglemufin]

Here's a faster explanation for those that are lost.

4 cannot equal pi by this method, because if you zoom in far enough, the path that =4 will be rigid, while the path that = pi, will be straight

[dojelnotmyrealname]

A thing to note in the diagonal example: The staircase path never actually travels in the diagonal direction. Just because the up and right portions look like they do, they don't actually become diagonal. So instead of a diagonal, you have an infinite amount of really tiny squares. So you don't really have a diagonal, you're traveling by taxicab distance.

On a different note: this also demonstrates that you can alter the area of a shape and keep the perimeter the same by introducing concavities.

[cristiannicolas]

This video is:
✔ Life changing ✔ Informative
✔ Inspiring ✔ Heartwarming
✔ Useful ✔calming ✔Enjoyable
✔ Other

[lexshelton]

another way you can think about it is that even though it stretches out to infinity, there are infinitely small spaces between each step, so it may seem like a straight line, but it's still just steps with holes between each one, which adds distance.

[prakharjain]

Created a doubt which i never had in the first place and then failed to solve the doubt. Good job✅

[davidpasquier]

I'm a french math student and i actually enjoy your vidéos, your pronounciation and simple explaination makes your vidéos easy to understand without subtitle.
Thanks a lot for your work!

[Tletna]

The key things to note here are (a) The area 1/2n while approaching zero never reaches zero and (b) no matter how small the segments in the zigzag that we're adding up get down to, they're still always non-zero. Essentially, you have infinite elements adding up to 2 in one case and then infinite adding up to sqrt(2) in the other. Yes, you said this but I think you could have illustrated it better.

[sai_beo]

The main idea here is that we are after arclength convergence of these continuous functions. The limiting process shown here only gives uniform convergence. Although this is a strong mode of convergence, uniform convergence does not imply arclength convergence (it only implies area convergence). Arclength convergence is a bigger deal, requiring a much stronger condition than uniform convergence. For differentiable functions on a compact domain, one needs some strong condition on the derivatives.

[feelfree]

You dont know how mentaly balancing thous videos are and how calming it is to watsh it
Thank you

[chiefchili]

You can explain pi in different ways, but this must've been my favourite. Well done!

[mathflipped]

This is a very interesting demonstration. Well done!

[BraggestSole]

To make it easier on a lot of people, you can also just zoom in all the way and see that the steps are not just one line, there are multiple. But with the line, it is just a line.

[filippocontiberas]

And the reason is that the lenght of the path is always proportional with the diagonal length no matter how many times you reduce and zig-zag the sides or if you change the side lenght!
If I define L as lenght of the path and D as the diagonal, the proportion have this formula:

L / D = cos( x ) + sin( x )

this x is the angle that diagonal forms with the bottom side. This lets you to compute it also with rectangulars. The example with the square you used in the video:

The square with 1 of side has the lenghts traveled on the two sides L = 1 + 1 = 2 and has diagonal D = sqrt(1+1) = sqrt(2): The proportion in any square of any lenght is:

L / D = 2 / sqrt( 2 ) -->
L / D = cos(45°) + sin(45°) = 1 / sqrt(2) + 1 / sqrt(2) = 2 / sqrt(2)

The formula i find out can be used also to calculate the unknown value of the angle x using trigonometry proprieties of any rectangular knowing its base and hight. The existence of this proportion is the reason why lenght path are always different from diagonals.