This Video Will Make You Better At Math
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- Опубліковано 16 січ 2022
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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
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It seems at the end you justified that this is not true by stating that the limit is equal to 2, which is different from the true value that we already know. I don't understand how this resolves the problem. It just seems that you are saying this is wrong because we know it is wrong. If we have a shape that we don't know the true length of, how can we tell if we are doing a correct approximation? Is the rule is that if the limit shouldn't be a constant and Is there a justification that because the length path is a limit of a constant then the approximation is incorrect?
@@hbxit1888 Would you like us to discuss this step by step, perhaps you can figure out what's going wrong on your own? 🙂 (the long way).
Or, just give you a video that explains the problem? 😄 (the short way).
As an engineer, I stand by the fact that π=3 and e=3, and thus π=e
Lmao
g=e²
@@woobilicious. No, because g is 10.0 precisely 😅
I’m an idiot so I just conclude that since the second example shows the step approximation seems to approach 2 but actually approaches sqrt(2), the circumference approximation that seems to approach 4 actually approaches sqrt(4) = 2, so pi = 2.
π= 3.14
e= 2.71
π≠ e
I actually came here to get instantly better at maths, all this did was bring more questions
The less you understand maths the better you are. I have a maths degree and all I learned is that I don't know anything. One time, we logically discussed that to prove logic is real you have to use logic, which is circular reasoning. So we showed that maths might not even be real.
Edit: we = maths friends on lsd
@@putinsgaytwin4272 if john have 750.000.000 apple........
@@ravenghost-kh9gy lmaoooo
@@ravenghost-kh9gy u mean 7.5E7 apples
@@ravenghost-kh9gy And you take away 1 how many apples does you and John have? John has 749.999.999 and I have one. (I almost wrote 749.000.000)
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
Thank you, your comment prevented me from going crazy
You speak common sense
What do you mean by rigid? Do you mean that no matter how much divisions in the square you make if you zoom in far enough there will always be some imperfection where it doesn't perfectly line up with the circle? Because that's my understanding of why the pi=4 method doesn't work
yes
@@pxolqopt3597
but just imagine 1m line that is straight and has no ridges,you cant count any ridges because there are no ridges.but if the 1m line has infinite ridges and the length of each ridge is 2*1/n where n=number of ridges . so if you did this infinite times, n=infinity so the lenth of each ridge is 2*1/infinity.we know that 1/infinity is 0 so 2*0=0 so there are no ridges and the line is straight
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!
Well said!
@@BriTheMathGuy isn't it merely due to the fsct that the approximation toninfinty method is flawed?
In the limit, it is a 'circle' with no curves, only infinitely many line segments.
The gradient of the segments never changes, so any line segment is either horizontal or vertical.
A squared circle.
You guys have favourite examples?
@@edomeindertsma6669 I don’t see how it’s related to a circle when the line segments approach a straight line
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.
It couldntve been _this_ video though, as this one was posted in january
@@Anonymous4045 note that he said "this problem" not "this video". This is not the first nor will it be the last video to cover this.
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
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.
this is actually incorrect. the path traveled by an object does not follow the hypotenuse, it follows the path of the sides. pi = 4 for motion, where the time variable is introduced. it's only 3.14... when there are only two dimensions. so the video is misleading.
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.
Well this video is more like keep the area but extend the perimeter.
Well its actually the same with integral approximation, but there it does work. Ok, its areas there, but they never become "smooth".
@@Pyriold Except the most basic idea of integration is using trapeziums, which actually DO change direction.
My thoughts exactly. It has infinitely small teeth, giving the line more length. Think of a normal brain and a smooth brain. They can both be the same size, but a normal brain has wrinkles which increases it's surface area compared to the smooth brain.
Thanks, i pretty much learned more with this than the vid
This video is:
✔ Life changing ✔ Informative
✔ Inspiring ✔ Heartwarming
✔ Useful ✔calming ✔Enjoyable
✔ Other
😀
⩗ exploding my brain
@@sparkingdude9942 *√*
Oo ee aa aa ting tang walla walla bing bang
You copy paste this everywhere, stop this. :(
Created a doubt which i never had in the first place and then failed to solve the doubt. Good job✅
lol same here bro
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.
that's what i thought he was going to say at first, then he dropped the limit theory which also works
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!
Happy to hear that!
françe françe baguette napoleon
@@blueyellowtube5825 oui oui
Mdr ton clavier t'a trahi même sans dire que t'étais français, le petit é du correcteur toujours présent.
@@whannabi aha ouai
You can explain pi in different ways, but this must've been my favourite. Well done!
Glad you enjoyed it!
@@BriTheMathGuy I interpreted this as "the square is in taxicab distance, the circle is in euclidean distance"
by explaining what it isn't?
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.
You dont know how mentaly balancing thous videos are and how calming it is to watsh it
Thank you
This is a very interesting demonstration. Well done!
Thanks a ton and thanks for watching!
How are neither of these channels verified?
@@davinfriggstad9731 To be "verified" a channel needs at least 100K subscribers. This is typical of UA-cam's "rich get richer" approach.
@@BriTheMathGuy that’s confusing because pi = 3.14159
@@mathflipped stay pooor
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.
That is 1/(2n). You need the grouping symbols because of the Order of Operations.
This was entertaining and ultimately an answer was offered, but after following the pace of expanded illustrations leading to the resolution, the answer offered was a disappointing stand-alone generalized conclusion. Blink and you miss it.
The essential concept resolving the integration of circles and rectangles is the difference between slope and right angles, and was completely missing.
This argument would have probably failed academically.
@@robertveith6383 I agree that would help clarify, but most mathematicians would know what I meant. They don't always use parentheses or the elementary school taught order of operations. Often they do simple multiplication of neighboring elements before division. Plus, that isn't really what was important for this video.
No it's not the case. It is not because segments are always non-ziro. It is because the error percentage between sum of length of segments we have been adding up as the approximation for the length we are trying to calculate will never decrease this way.
@@ZiroOne-hw7iw I'm fairly confident that I was correct. Whether that's the case or not though, I would need you to clarify what you meant in your comment. What do you mean by "It is because the error percentage between sum of length of segments we have been adding up as the approximation for the length we are trying to calculate will never decrease this way."
Love the way he just started the video without any intro
Thank you for this video! I have been wondering about this question for 4 years! Ever since i found it in high school!
e = 3
pi = 3
g = 3 squared
2 = 3
4 = 3
Everything is 3. Those are the rules.
6 = 5 but 5 = 4 and 4 = 3 so 6 = 3
Rule of three from English. English is maths
“Everything is 3”
But neither Half-Life, nor Portal, nor Dota, nor Left4Dead, nor Team Fortress
You know the rules and so do I
Mines too lmao it’s every three aha
Of course, the problem is that we have to precisely define what it means that a curve "approaches" another curve.
He did, it’s when the area between them tends to 0.
Exactly.
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.
This is very helpful 👌
Thank you for your video 😊
I had kinda hoped for a “so what can we do to make it work” section at the end, where you explain exactly how the length of a curvy path (like the circle) is even defined: By increasingly fine approximations with line segments _whose endpoints lie on the path_. That’s the crucial thing that is missing in the staircase approximations.
I think that if you zoom in the image to a large extent .. you will notice that the perimeter of the previous square will never match perfectly the circle perimeter... cause you will find sharp curves in every very small distance .. Or in other words... To the side of any infinitesimal horizontal piece you will find a vertical piece of the same length
...... That's an other way to explain this topic
@@soysamia that's not the problem, the goal was to get as close a possible to pi by infinitely doing this. I believe mathematicians in the past did this by hand but it becomes pointless at some point since having all these decimals closer and closer to pi...well you'll never use the whole number anyway. Just use π or an approximation, that's enough imo.
@@soysamia I believe mathematicians in the past did this by hand but it becomes pointless at some point since having all these decimals closer and closer to pi...well you'll never use the whole number anyway. Just use π or an approximation, that's enough imo
A good explanation and a good video (as always).
To introduce a squeeze theorem you could also plot the inner path. This looks like pixels turning into a smooth line. Every circle drawn on a screen (like this one) has this property.
Can make it something practical. Maybe easier to digest, or maybe not, hard to tell unless try.
I didn't understand it at the beginning...and I definitely didn't understand it at the end hahaha
Respect to the people that actually listened to him and drew it
As a person who has watched this video to it's end, I can confirm that I'm now better at math.
I conclude am still da same👁️👄👁️💀
This has to be the mathematical equivalent of "correlation doesn't equal causation"
Then what is it then?
More generally, "necessary but not sufficient condition".
I didn't understand, he said the path length is 2 regardless of the ineration number but then it's the square root of two
@@caiofabio4989 path length and diagonal are different, length of the diagonal is root 2
@@caiofabio4989one is a series of jagged lines and other is a perfectly straight line.
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.
I think what happens when you keep doing that cutting the corner thing , the length remains the same until certain point but once it get's small enough and there isn't much room to actually shrink the "stairs" they overlap thus the length of the diagonal is less than those sum.
My understanding is that no matter how much times you cut the corner if you zoom in far enough there will always be some imperfection where the square's lines don't match up with the circles. Another way of understanding it is that in the case of the triangle while the area 1/2n does approach zero, it never actually reaches zero because if you solve for 1/2n=0 you either get that 1=0 or 1/0=2n both if which of course are not mathematically correct answers
At no point do the steps stop existing. The total length will always add back up the the original number.
Bro are you even School passed? Cuz the thing you just mentioned literally makes no sense lmao.. tbh i doubt blud is school dropout.. tho atleast should have paid a little attention
The explanation for this is no matter how small you shrink the right angles, they will still be there, even though they are microscopic :)
thank you soo much for this i was thinking how he proved that...
But how do you distinguish between a curve and a straight line on atomic level?
@@SyNcLife You just need an atomic sized protractor!
@@SyNcLife good question!
I'm not sure that's how infinity works
Very informative but I had hoped for an actual proof of why this is wrong
Goldplatedgoof gives a full explanation on what exactly went wrong. Check it out if you want a better explanation
It's not really a proof, but you can think of the process as glueing points from one curve to another. At the end you only connect a countably infinite number of points, dense in the curve, but the curve is a continuum, with an uncountably infinite number of points.
Thanks for watching anyhow. Have an awesome day!
You can't really prove *why* something doesn't work. You can prove *that* it doesn't work, and you can *explain* why it doesn't work. This video kind of does both; whether or not the presentation was satisfying to you is another story.
This actually is a complete proof. I agree that a full explanation as to why it doesn't work that somehow would fit our intuition would be desirable. However, this video basically gives a proof that if one path converges on to another by the area between them going to 0, the lengths are not necessarily the same. It is a proof by counterexample.
wow its really wonderful video with an excellent explanation. what software did you use to make videos?
Thank you for making this video! This video says slowly, so other people can know more easily. Please make the video more and more! I will wait for you
Just an addition. What converges when you modify the boundary is not the length of the figure but the area
I am a simple engineer
I see π=4, I click
ua-cam.com/video/b1fXcnnCAbA/v-deo.html
π = 2 in Riemann Paradox and Sphere Geometry System
So Tau = 4 in Riemann Paradox and Sphere Geometry System Incorporated
Fact: If you rotate the screen at 0:03 the circle will rotate.
Cool!
Saving this for future use
Simple proof that polygon and circle (and other "staircase" cases) are not the same is: polygon does not have derivative, while circle has. Moreover you can define polygon with any perimeter greater than circle. In order to get circle perimeter you should take minimum of perimeter limits of all possible polygons that converging to circle. And that minimal polygon will be converging from inside.
For area - yes, it's the same, because same way we define integral.
Title: This video will make you better at Math
Thumbnail: π=4
*Visible Confusion*
One of the biggest things that people don't seem to properly understand is this:
Just because f(x) approaches some value as x approaches infinity, does not mean that f(x) *is* the approached value when x actually equals infinity.
For example, take a box. Put the first n natural numbers into a box. If n^2 is in the box, remove n.
As n approaches infinity, the "output" of our function, the number of integers left in the box, approaches infinity too. But *at* infinity, once the entire set of natural numbers has been put in (to whatever extent on can operate on "the entire set of natural numbers"), there will be no pieces of paper left in the box, as every number's square has already been put in.
This is why dealing with infinity requires a person to put away intuition and trust the logic.
This is the same thing as the “infinite coastline” problem
What a theory! Love this channel so much. Keep it up, Mr.Bri!
Thanks! Will do!
I'm so confused
If you don’t understand basically its not diagonal it’s parallel lines witch doubles the length of the diegonal
You guys probably won't believe me but I have been thinking about the triangle part of the video for a couple days now... I did the same thing of dividing the two sides of the square continuously to make it the hypotenuse but just couldn't make things add up all together.. I am so happy this video came up on my feed 😭
Edit: I just realised this video is 1 year ago.. So this video existed all this while but decided to show up when I was struggling with this problem
if you approach this with an integral, you get dx and dy instead of ds. You have to convert it with ds²=dx²+dy² and so you get the factor sqrt(2). But of corse this is trivial because you could calculator the length that way in the first place. :D
That's like the fallacy of, "How can one cat have three tails? No cat has two tails, and one cat has one more tail than no cat, so one cat has three tails".
"This video will make you better at math."
a) True
b) False ✔
I love this video, but the thing is… The moment you asked what the length of the diagonal was at 1:40, I instantly thought of that beautiful bastard Pythagoras
Now THAT was a fun ad to watch !
Had an argument about this problem with my mate, basically I was arguing this video's point except it was in a loud pub and I'd had a few drinks so it was very hard to convince him what my point was. He was basically trying to say that you CAN take the fact the squares perimeter approaches the circle's because it is visually getting closer, but I argued that while the points of each edge are indeed closer, there are more points of each edge that are away from the circle so as it approaches infinity, you have an infinite number of points that are 'away from' the circle. He argued you can't have that, because you can't have an infinitely small corner against the circle - it would just be 0 units away from the circle, but I argue that you can, because we basically made up the rules for this problem, and how infinity behaves, so if you can say that the circle zoomed in infinitely is flat, then I can say that a corner zoomed infinitely is still a corner to that flat line and thus the two perimeters can't be equal. Thinking about this more now, I would even say a truly infinitely small set of teeth-like corners around a circle would be 2 circles inside each other, the smaller being the same as the circle you're trying to match, but the other being just a tiny bit larger because of the points of each spike which are always further from the circle edge than the part touching it, then when you infinitely zoom into the two circles they may become flat, but they'll also be parallel and thus cannot be the same. Thinking about this more while typing, that actually is irrelevant because you're talking about perimeters, aka the length of these lines, which are both theoretically infinite because your zoomed in circle that becomes a straight line is just an infinite line which IS the same length as the other infinite line, and this could be used to claim everything is equal to everything else, much like the 0=1 'proofs' or 1=2. Goes back to my point about how you basically make up rules again because the lines not infinite, it's just infinitely zoomed so you can never follow it to the ends, but it still has the property of having an end of it somewhere.
man wrote a newyork best seller
@@PurpleAmalgam 😂😉
This is what a battle of nerds looks like, awesome.
your wronge
@@sarthakthakur7253 Care to explain why?
The first day i entered grade 7,i have been struggling with math,this video has heled me alot,thank you so much!
Shorter answer: if you replace the "L2" (Euclidean) distance metric with "L1", then Pi is indeed equals to 4. There is no paradox (or rather, this formally resolves the paradox).
In simple words it would be using an idea from calculus 2 Arc length formula. *No matter how small dx and dy are, the approximated small arc length is always greater than both. Due to Pythagorian theorem*
🤓🤓🤓🤓 calc 2 🤓🤓🤓🤓🤓🤓🤓🤓🤓🤓 hypotenuse 🤓🤓 GOD WHY WON'T YOU JUST F "Ock OFF
The orthogonal lines of the approximation formed by inverting the square corners will never be tangential to the circle.
@@mikezappulla4092 🤓🤓🤓🤓🤓🤓🤓🤓 "orthogonal" 🤓🤓🤓🤓 "never tangential" 🤓🤓🤓🤓🤓🤓🤓🤓🤓🤓🤓🤓🤓🤓🤓🤢🤮🤮
🤓
I have a history test tomorrow i don’t know why i’m watching this
who knew there were that much people who needed to be better in maths
since if the line turns into a skew angle then it is longer no matter what
I find it fascinating that this works
Also, In theory, with any shape you can do this theoretically?
Like the 1/n graph from say, 0.5 to 2 you can either somehow make a distance function for the path of this graph orpoo you can "approximate" the length with a square and doing this process
Incredible video!
One tiny thing I noticed is that the diameter at the beginning of the video is not centred. Nobody else seemed to mention it which is surprising, seeing as it is a very noticeable difference (about 4 percent off of the true centre). Maybe it's just one of those things that you only see once somebody tells you.
Keep making great videos!
I mean to measure the diagonal path of the square, instead of some other complex way to do it, isn’t using Pythagorean theorem faster?
If you take two parallel lines of the same length, total length is twice that of one of them. As you bring those lines closer and closer together, they start to resemble one line with the length of one. The reason it "feels right" is because of the imperfection of our perception and way of showing graphs with lines of *some* thickness
Shouldn´t it be that the number of triangles after nth iterration is 2^n instead of n and the length of their sides being 1/2^n?
Yes you are right. But maybe he wanted to keep video simple and thus avoided exponential
@@yashkrishnatery9082 I would think the video would be kept simpler if you avoided mistakes that a beginner might just happen to get hung up on.
@@yashkrishnatery9082 Seems like a mistake/oversight. If it was intentional, it would be natural to say that the formula holds for n triangles, instead of for n iterations.
I didn't understand a single thing from this video,hella confused
To be more precise, the path pointwise converges to the circumference of the circle, but it does not continuously converge.
Why are we measuring the areas of triangles when we need to find the value of half of the perimeter in order to get the path length?
me who got more confussed after watching this
Contradictions like this is why treating all infinities as equal is a bad idea. Just because both shapes have infinite points does not mean that they have the same 'ammount' of points, at least relative to their respective diameter. The 'square' has infinite corners, so it takes on the shape of a circle. However, a circle doesn't have infinite corners, instead having infinite points equidistance from a center. Taking this into account, alongside the fact that a corner requires at the very least 3 points, you realize that the 'square' technically has more points than the circle, even if both have infinite points. It's part of why infinity minus infinity is undefined, we don't know if one of the infinities is bigger or smaller, so saying that they cancel each other out because they're both infinity is incorrect.
This is honestly a much better explanation than "Pythagorean Theorem says no."
Everything after, *first draw a circle* has gone above my head.
For the diagonal, the path of a staircase will always be 2, because you are limited to going in 2 directions only, you cannot go into 1.
The direct (true) path will always be different because the angle is different.
I think this video perfectly explains the confusion most people have.
This is why we must not use approximation in some scenarios because lookalikes might give us a totally unreal answer.
Sometimes bringing all to the base size of 1 will best help us solve the problem.
I think the biggest problem here (without watching the video yet) is the fact that you can't do the same "if you can map one set to another exactly, they must be the same set (or of the same length). Which while fine for finite sets, doesn't work with infinite ones. For example the set of all numbers between 0 and some infinitely small fraction, say 1/10^10^10^10000, and the set of all numbers have the same amount of elements.
The limit of rectangle approximations and the circle are the same object. The real issue is that you can not derive, in general, the properties of a limit from the properties of a sequence converging towards it.
That's something really interesting and informative.. thanks a lot 😊😊🙏👍✌️
Glad you liked it!
As a 6th grader i consider myself good in math currently but this video is so good and learned so many stuff from it .
Excellent video 👏🏼
Also if you zoom in when looking at the line (that was folded in) there’s a bunch of tiny staircase lines. Put them back together, and you get what you started with
yes kind of. You can think of it like that only if you are confortable with the fact that those tiny triangles (staircase lines) would have area 0
Yeah kinda like those old paint circles or diagonal lines on low res screen it looks like a line but it has lots of right angles if you look carefully
@@axelperezmachado3500which is why taking the limit to infinity if pointless. If you do that at any point in the process, the areas will be a real number and the length will still be two.
Wrong. π is equal to 2 according to the fundamental theorem of engineering, since e=2 and e=π.
I know you preserved pi =4 but this just narrows it down to line segment thus in this situation you have distance ^°A,B which =distance A,4,B but including formula it stands out as 1 of 8 then in staircase misconsumption is made growing infinitley through the method , as you still have to calculate parenthisis sides; (^2)+(^2)+(^2). Im not 100% about this since we dont learn about it because about complex math as im in year 5 (10year old)
Title: This video will make you better at math!
Me: *i need it now*
I take issue with the phrase "Path A approaches Path B". The conceptualization and reasoning that purports to show that is glaringly incomplete. Paths don't have areas, so what's the connection between the areas of the two figures and whatever "one path approaching another" is supposed to mean?
1:45
bruh.
1^2 + 1^2 = 2
2 = c^2
square root of 2 = c
(c is the diagonal/hypotenuse)
theres already an equation for this called the pythagorean theorem
Yes. Thanks for saying this. Thank you. Almost made me scream if you didn’t mention the pythagorean theorem
He knows. It was a demonstration of why pi doesn’t equal 4.
The diagnol line is √2 by pythagorus Theorem
If you take a isosceles right triangle and need the hypotenuse, just multiply a by the square root of two.
And got this intresting theory to show-off myself before friends 😂btw good explanation 👏
Glad you liked it!
i didnt understand a single thing u said💀💀
Same bro 😭😁
"just because path A approaches to B, doesnt mean path length should too." i got it, they didnt take the slope, slopes change while square side slopes dont. they need to consider something that can change while they change something different
つまり、isosceles right triangleのhypotenuseのlengthは2で、Pythagoras said " 1^2 + 1^2 = 2^2 "ということか。なるほど。
As someone like me who is facing Insomnia, videos like these really help to fall asleep.Thank you :)
lol
Finally a comprehensive explanation for the diagonal staircase "paradox". Thanks for this.
Glad you enjoyed it!
It is even simpler. By drawing a shape around the circle to "approximate" the perimeter of the circle, you are just providing an upper bound for the perimeter. This staircase paradox actually proves that pi is less than 4. If you draw 5-gon, 6-gon ... you can get better upper bounds for pi. But upper bound alone is not enough. You also need a lower bound. You can draw 5-gon, 6-gon ... inside the circle and get a lower bound for the perimeter.
To calculate a quantity, you need upper and lower bounds, and also these bounds should get closer and closer to each other.
this is related to fractals and measuring the length of a country borders for example
Hey sir the perimeter does change as you increase the sides
I still can't fully understand this. If path is a set of points, then the exact same set of points should have the same legth. I think the real problem here is with the definition of approach and length of a path. Without talking about a more precise definition of those trying to explain this problem is just turning this problem into a same problem that at first glance doesn't look like a problem, but when you actually think about it you're getting a paradox, not an explanation.
Well there is another explanation quite simple...
As you said, a path is an infinite set of points. Thing is, with this process, to our eyes, it does seem the two path merge together. But if you zoom, you'll find out that they still aren't the same
Basically, you can do the process an infinite amount of times, you just have to zoom an infinite amount of times to see that the two paths cannot be the same set of points, and therefore you can't conclude they are the same length
True. Your point is valid. This is yet another infinity issue. Someone can write 3 x infinity = 4 x infinity. But that doesn't mean 3 equals 4. Similarly, there is infinitesimal number points in each line, but relating them doesn't mean they are equal in finite length.
Think of it like this: The points of the path get closer and closer to the diagonal line, but they never actually travel diagonally. The tangent line will always be vertical or horizontal, so even though it looks like a diagonal, you're actually still using taxicab distance.
Try thinking of it like this... Take the square/triangle example at step 1, when the top left corner has been cut away. No matter how many times you do this, if you zoom in far enough you'll eventually reach that same shape/situation. Which means every 'x' distance along the truly straight line will always be 1.414 times shorter than the 'staggered' line due to replication.
The problem is that the convergence is uniform. For every ε>0, you can find an n-th iteration such that for all other iterations and for all points the distance from the limit circle will be less than ε. That's what still keeps my mind occupied.
Maybe uniform convergence of same-length curves does not mean that the limit curve has the same length as well? If that holds, it's a very non-trivial theorem.
Bri, can you explain why π^π^π^π might be an integer. This is a mathematical debate so it'll be pretty interesting.
It's not a debate, we know it's possible and can't prove it isn't yet.
pi ^ pi ^ pi can be shown to be a decimal number simply by calculating its later digits with enough precision, to know that it falls in-between two integers.
To show the same for pi ^ pi ^ pi ^ pi, i.e. that it isn't an integer, requires the use of sooo many digits of pi that it is simply not possible with the computational power we have today.
It COULD be an integer for all we know, however it most likely isn't.
The original Archimedes‘s method to calculate Pi, has to give a "lower estimate" that is increasing, and a "upper estimate" that is decreasing, then we can conclude the value of Pi is sitting somewhere between the two. With this "method" however, there is only a single "upper estimate", so it could not work. At most we can only say Pi < 4 in all iterations.
*π COULD NEVER =4.*
So, if the square's perimeter = 4, and the perimeter (which we could call θ) approaches π, π=θ. Just because the angles go from 360º to ∞º (because there is no end on a circle, so the angles on a circle will never end), θ cannot be π because if it could be, then one of the rules of maths (just because 1 path approaches another path (which in our case is π»θ), then it still does not mean that the path that is approaching the other one can ever be equal to it) are blatantly broken.
What differentiates (😉) differentiation from this then? Is it literally that differentiation is applied monoaxially along the 'line', whereas in this example, it doesn't matter how close you approach zero, there's no changing the fact that you're just simulating a straight line with lots of right angles?
"This video will make you better at math"
Me who not a native english speaker and bad at English: Yeah great video!
look its one thing to say cut out the corner of the figure, but how have you quantitatively defined the said corners ?
for the circle there would still be spaces in between. Just making sure everyone knows this