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).
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.
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
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!
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.
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
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
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
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.
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
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.
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.
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!
Eh.. how can you be so wrong. π=3 is first fundamental theorem of engineering Second is e=3 P.S. third theorem of engineering is every ducking number=3
@@yashkrishnatery9082 how can you be so wrong, this isn’t a theorem of engineering, it’s a mathematical fact made by MY 2 year old son. in fact everything is 3, doesn’t matter what could be infinity times infinity, it’s still 3
@@yashkrishnatery9082 technically pi=3 implies e=3. pi=3 implies pi-3=0 Multiplying both sides by (e-3)/(pi-3) gives e-3=0 so e=3. Thus there is only 1 fundermental theorem of engineering, everything else is a corollary
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.
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
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
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.
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."
You put my brain in much pain for the first 1 minute and 30 seconds of the video. Edit: nvm my brain was in pain for the length of the video, it was also screaming: "SHUT UP,SHUT UP"... I am in huge amounts of anguish
@@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 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)
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.
You calculate the limits for the area and the length ok but you do not really explain or give an intuition on why the lengths do not match the smooth lines lenghts when they visually appear to do so. What is it in that processes that make the areas match and not the lengths?
I think... The conclusion of this video is that even if the Path A approaches the Path B, the distance between them is still existing. 'Cause even by dividing these two sides infinitely, u create just Infinit stairs, and each of them ad a little space between them and the Path B, and so little space * infinity is equal to the difference between 2 and √2. Not sure if that is clear but u know, after 2 years maybe u will understand this 🤷
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 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".
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.
*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.
And afterwards 94 percent of the general public in the United States didn't understand the majority if not the entire video because of the higher form of math displayed. With only 6 percent of the American population understanding the video based on their being in a STEM field.
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
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.
it probably is the most important thing in the vid but i just don't understand what the difference is between path and path length. what do they signify?
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.
@@TheUnbearded well the resulting curve will be exactly like mathematically exactly equal to a circle. An easy way to verify this is by trying to think of any point on one curve that is not on the circle (there are none). If you want an actual answer as to where the error in the "proof" is I can explain that too but it's a bit harder to understand. If you want an explanation tho I'll provide
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.
Without knowing it, you actually proved the real nature of spacetime! Congratulations! In terms of your logical argument, your last part doesn't work. You take the pythagorean theorem as an axiom, so of course you get the conclusion that the argument for "pi" = 4 is false. In "real space", ie, "spacetime" (ie , when you take time into account), the metric to favour is the taxicab metric L1, in which the length of the diagonal of a unit square is actually "2" (d=1+1, no "irrationals"). In other words, real space-time is a fully symmetric manifold with a taxicab geometry,within which a circle is actually a "square"! Thanks for the proof!
I Know All Math But, Not Algebras, Magics, And Changes I Know: Addition, Powers, Subtraction, Division, Multiplication, Digits Of Pi, Square Roots, Sides, Units, Symmetry, Lines, And Fractions
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*
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.
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.
@@edgelernt4021 the area of each will be 1/2^(n+m) where n begins from 1, m begins from 0, and number of triangles=2^n. Unfortunately I can't figure out a way to start from the point where there is only 1 triangle, but we don't even need these values to prove the point in the video
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 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
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.
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.
Beautiful explanation! I saw the first demonstration on an odd1sout video a long time ago but I never really understood the mathematics behind it. You just earned another sub :)
π= 22/7 π²= 484/49 = 9.8 ............① Also, g = 9.8 m/s² ............② So... From ① & ② We get, π² = g Hence Proved that π² is nothing, but the gravitational constant !
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.
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
@@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.
This was great , I have newer seen this approach before and I will definitely not show it to my students. I teach them how to proof the area of a circle by dividing the circumference into infinitive many parts and adding them together to show the area being radius squared times π. This "proof" could easily be used against me.
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.
Endless enigmatic book in all languages. You can write a book with mirrors in all languages of the world. You can speak two languages at once, you just need to find the perfect reflection, same content, different translation. Infinite Mirrors. Pi 3.14 XBooks. Hybrid language!!!!!!!##!!!!!!
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.
The problem is that dots (connectors) are taken as with no length, so whenever a line is broken to two lines we're adding a length of one dot to calculus. For infinite number of line breaks the sum of these dots gets some measurable length, which added to Pi gives a sum to 4, and added to sqrt(2) gives a sum to 2 in examples.
imagine like this, if there is no area between them then they must be on top of each other. for the square's diagonal as you go to infinity you decrease the area infinitely but increase the number of triangles/areas infinitely so you will always have a significant total area.
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
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
I am about to end this man whole career with my 200 IQ. Okay so for the first point (the pai equal 4 thingy), if you do that you will end up with a zig zag line which if you straighten it up, will be longer than the curvy line of a circle. Nice way of thinking bro but everyone please remember this is for fun video, dont believe in any of this. I dip out of this video as soon as I saw the first point. Bye and have a great day.
It won't appear as circle ...like those who play Minecraft and once tried to make a circle like this .....by doing this it just become a square again but tilted
The value you are taking when you separate it infinity times ... Does not give you the exact value of the circle or the diagonal... It's like the value that tends to the value of them
