I never would have known about you if I was never incarcerated. I discovered Khan Academy on the tablets in jail and they had you under math lessons or whatever. Your mind is the most amazing, loveable, beautiful blessing. I discovered you a year ago, but I couldn't tell you until now. It really made everything worth it. Thank you for your music and the captivating topics and your work! I love you! You are so wonderful.... Thank you! Thank you! Thank you with a function that approaches infinity as an exponent to already numerous thanks!
Zigfinigons remind me a bit of how audio is encoded digitally. The higher the sample rate, the closer you can get to something that is LIKE the original waveform, but is actually a shape with a number of sides determined by the sample rate.
Kinda. When it's represented digitally sure, but when it's being played the sound passes through an antialiasing filter that smooths everything out. In theory, with a sample rate of 40,000kHz you can *perfectly* recreate any signal audible to the human ear (See: Nyquist sampling theorem). If you *did* try to play the weird polygon shape on a speaker, even with a high sampling rate, you'd get fantastically weird retro-sounding aliasing artifacts. (Note: this is ignoring the issue of sample depth)
@@antonliakhovitch8306 Aliasing only occurs during sampling. Because of this, playing back a sampled signal won't actually introduce anything in the audible range. This weird polygon shape would even have it's high-frequency components attenuated by the speaker itself, so it would be difficult to even tell that the signal is being played by a DAC without removing the speaker from the equation. The artifacts that do make it through would only be at multiples of the fundamental frequency of the sampling rate, so at worst, a 40 kHz sampled signal would have a 40 kHz ring on top of it (assuming the DAC has 0 rise time). If the DAC has bad enough rise time, it could introduce the ring to the audible range, but that's so bad that it would also distort the audio significantly.
1. Start with a square 2. Continue with zigfinity till you reach the planck length. 3. It is physically impossible differentiate this circle and a "smooth" circle 4. Pi = 4 5. ??? 6. Profit!
You also have to remember that Planck Lengths are not an accurate representation of space, and that they are just how far it takes light to travel in a vacuum for 1 unit of Planck time, not simply a grid that the universe is built out of.
No, it's applied logic. A lot of Physics was predicted and discovered simply based on mathematical formulas and theorem, but it exists also for its own consistency. If you break one segment of Mathematics, you bring down the whole and the tool is completely pointless then.
Dorth Lous Logic is just corrective reasoning, of course a system with it's own made up abstract constants is going to interact and follow it's own abstract and made up rules. Essentially it's structured nothingness. To me, numbers are only useful if you're approximating the physical world, once you reach the realm of the irrational and the imaginary, you're pretty much speaking metaphysics.
And it's also wrong (or at least it would take a lot of effort to make it right). Basic issue is this: there is no curve with the properties of a zigfinigon that lives in the real plane - so if you want to talk about zigfinigons, you need to invent an entirely new space of curves, ones that don't correspond directly to any that live in the real plane (maybe you can define each new-curve as an equivalence class of ordinary-curves, idk). It's not clear at a glance that any choice of such a space will have nice, intuitive properties. So if you have the requirement that the limit of a sequence of curves (if it exists) is an ordinary curve, then the limit of those zigzag curves has to be the the circle (or hypotenuse) exactly. Does this pose a problem? No - we had no reason to expect in the first place, that if you have a bunch of curves converging to a limit, that their lengths will converge to the length of the limit. This assumption is equivalent to saying that the function that sends a curve to its length is continuous (for an appropriate topology on the space of curves). All our examples show is that this assumption is false.
Roman Chapkis We will start with the diameter of the circle, which is 1cm and splits the circle in half, i know what you're thinking: A circle is a circle; you can't say it's only a half.
The fundamental flaw in your friends first "proof" is that all the zigzags extend outside the circles circumferense, which means it will always be bigger than PI. If one instead did the reverse and begun the same procedure from a square fitted perfectly inside the circle, the same method would "prove" that Pi is smaller than it is. Only if the zigzags, regardless of number and size, actually cross between the interior and the exterior of a circle in equal proportions would such a zigzag-circle actually have the same length as the circle it's based on.
Skadu Skitai You could also make a zig zag pattern that stayed inside the circle yet had a larger perimeter... because the smooth circular curve is a short cut relative to all those turns.
It's interesting that for the perimeter of a "circle" to equal 4 it has to have infinite zig zags but the perimeter of an actual circle has infinite digits. You have to have a finite shape or a finite number.. you can't have both. Also that real circles don't really exist. The circles on my screen aren't really circles. Due to the pixels their perimeter is closer to 4 not pi. The pixels aren't perfect squares either. It's really only a circle from the conceptualization of it in my frame of reference. Math or whatever this is is interesting as fuck.
I believe that what you're talking about being really interesting is attempting to impose the infinitely precise nature of mathematics on our inherently finite and quantised world. Essentially, despite what we might think, mathematics does not perfectly represent our world has a concrete idea of "the smallest possible quantity", whereas mathematics does not.
I'm in seventh grade, but I'm in accelerated math, so I have just started geometry. I was lost about how I was going to get through the year without failing, but then I remembered Vihart! You've got an odd way of explaining things that makes math easier and dare I say it, fun.
kiwibirbz all of us in seventh grade have the same program like you...ALL OF US... even the little KU-KU students... but we are smart... well most of us
They're not fractals! Zigfinigons, if they exist in a meaningful sense, can't live as curves in the real plane, since no curves in the real plane have the required properties. Vi proceeds with the assumption that because the length of the apparent-limit is not the limit of the lengths, that the real-limit must be some new strange object whose length *is* the limit of the lengths. The actual underlying matter is much simpler and mundane - the function that sends a curve to its length is discontinuous. That doesn't mean zigfinigons can't exist, but to show they do (in any meaningful sense) requires a lot more legwork, since they can't just be sets in the plane.
I know right.. I feel so special while reading all these comments seeing that people thinking not in a correct way. I feel bad about feeling good about this :D
haha, when i first saw this "proof of pi = 4" i was like, what the?, then i sat down, took a sheet of paper, and drew a circle with the radius of 1/2 , and drew around it a square of length a = 1, then i started doing the same thing, but at the end, i recalled that a circle has a radius, so if this thing really is then its radius will be the same as the number of "zig zags" approaches infinity, but then the radius will be disturbed at each zig zag thus, R the radius will not be equal to the same number. i figured out at the end that this would lead to a contradiction (1/4 != 1/2), thus π != 4, :PPP
It's amazing though how the "zig zags" that we cannot tell apart causes pi= 4 ! :D That's how infinity works i guess (as there are infinity dot's in a circle.)
***** but its not pi !!!!, how can you calculate pi with a non-constant radius, think about it, pi = C/2r, now look at that zig-zagy circle, where is the radius?, there are infinility many radii (because you can draw line from center to the side, and draw another which will intersect with the zigzagy bit of the circle, thus u will get different values) but im sure you can calculate pi from that zigzagy circle, the formula may be like PI = 4 - Σ(something)
English my sec. language so i didn't notice i made a mistake in grammer there. Let me put it like this; It is amazing how the "zig zags" that we cannot tell apart COULD cause pi=4 . And as you and Vi' explained very well, the non-consant radius causes the mistake there. The mistake that gives pi=4 .
I love how UA-cam suggested me this video 9 years after I just watched it the first time. Love you Vihart, never stop amaze us to the pi-nfinity and beyond
You know, I'm just amazed by your videos. I have interest in math. But you turn my math into something that is never much cooler. You made me realize how doodle can be more fun. It's just awesome. My mind is already blown by your things. And yeah your drawing skill... omg!! that'a amazing... No one can say that was doodles. Wish I had some of 'em. Guess What? This is the first channel I subscribed to. I'm glad I made the right choice. The whole channel is full of awesomeness. I'm gonna watch all of 'em. wish me luck! Lol 😁😂 And Yeah Keep it up to the sky! 'Cause you're doing great. 👍
I remember that, when math was still relatively new to me I realised that, for positive numbers, the function y=x^2 could also be interpreted as x=sqrt(y). A few years later I learned the function notation where y=f(x), but other variables could aslo be used at the left or right side. From that moment I could see that all variables with a relationship had an equal relationship; it was two-way traffic and the concept of dependency was just a mere matter of perspective. So the way I see it y=2x is not an unhealthy relationship, but the way y and x work together in this particular situation; it could be formulated with equal correctness as x=y/2.
It's not always easy to travel in both directions. You may end up with something that is not a "function" (a function has only one value for each "x") To illustrate, you said you could interpret y = x^2 as x = sqrt(y), when it's actually x =±sqrt(y) which has more than 1 value of x for each y, tehrefore is not really a "function". But yeah, you are right about the perspective thing xD Maybe teacher should emphasize this, or maybe it would tend to confuse students more.
moraigna66 That's true: the functions that are not one-to-one are problematic in this context (a function is one-to-one if it never takes on the same value twice; that is f is one-to-one if f(a)≠f(b) when a≠b); but now that term makes even more sense; in a way y=x^2 is one-to-two as x=±sqrt(y) can actually be seen as two functions. In the same way, one could say y=tan(x) is one-to-many as x=tan^-1(y)±k*pi for k=±0;±1;±2;... can actually bee seen as infinetely many functions. So whenever something is not one-to-one it can become a set of multiple functions from the perspective of the other variable. It's kinda like a relationship between people where one of them only uses one means of communication while the other uses other means as well.
Starting at about 3:48 you are very close to providing an example of a function that is continuous everywhere but differntiable nowhere. Just add them all together.
What you've pointed out very well here (without telling it) is that when you make a perimeter aproach to a line, you always end up with something in the rational number set instead of ending in the real number set. Nice job!
I wish my high school classes of math were like your videos...i would have paid attention....though not keep up but still paid attention. Thank you for the education videos!
A Sloth Elementary math facts are basically the most important thing in math. In fact, if you pull out complicated math when simple math will do then you're just an idiot because you waste time. Didn't Einstein say that if you can't explain something to a child, then you don't understand it well enough?
aitor ormazabal It is not stupid. It is wise. Einstein really did say something like that. I don't think that he meant it to be taken too literally. I'm sure that he didn't mean that you should be able to explain it to a baby for instance. He also said, "everything should be made as simple as possible, but not simpler".
This just happened to pop into my head today. It's because the perimeter does not converge to any parameterization of the circle in W^{1, 1}. I was in grade school when I first saw this. Wild.
if a=b a*a=b*a a^2=ab a^2(π-1)=ab(π-1) a^2(π-1)+a^2=ab(π-1)+a^2 πa^2-a^2+a^2=πab+a^2-ab πa^2=πab+a^2-ab πa^2-πab=a^2-ab πa(a-b)=a(a-b) divide both sides by a(a-b) and you get π=1 problem solved... or is it?..
Matt Fellenz the area is the same but the perimeter is different. math gets weird at infinity. like she said you could crumple a line into a single point. you can make it whatever shape you want
In the video you said the peaks of the triangles would eventually reach zero, you said they must, that's not true, they never will they would reach numbers infinitely close to zero but would never reach zero because the base of the triangle is at zero, since you're halfing the hight of the triangle but doubling the amount of triangles each time, you're not actually doing anything to the base of the triangle, if the peaks reach zero they aren't considered triangles any more then it becomes a straight line
a circle isn't a real shape it doesn't exist in reality. all shapes are polygons with sides. a circle simply designed is a shape with all of it's points equal distance from a center. you can add what seems like an infinite number of points right down to the plank distance(smallest physically possible distance) but in the end each of those points will be connected with line segments. I think that's part of why pi exists as an irrational number. it's trying to describe something that doesn't really exist.
There isn't an object in real life that is perfectly circular, but that is far different from saying that circles don't exist. They are a concept. It would be like saying that meters don't exist because there is no object that is exactly a meter long down to the planck length. If you try to understand math exclusively in terms of real life, you won't be able to get much further than basic arithmetic.
okay. the circle doesn't even really exist in a mathematical sense because every point is connected with a straight line segment. there is no curved line connecting these points.
Infinitesimals explain all of this. I have fallen in love with infinitesimals. They explain so many questions!!! Many people use zero when they should use infinitesimals. Let me put it nice and simple. 1/infinity does not equal 0. It equals an infinitesimal. Which is basically zero but it has different properties when you multiply it by infinity. Sweet.
***** but its quit different because the lenght of the blows up to infinity by shortening the measuring "tape". here the perimeter is constant which is very important for the problem
i somehow manage to understand exactly what you're talking about while at the same time i have literally no idea what you're talking about and it's amazing
It's cool that her video piques interest, which is the point- but the entire reasoning is hinged on a flawed statement- Pi is not the circumference of a circle- it is the ratio of any circle's circumference to its radius. This is why the area of a circle is pi(r)^2.
I like your hands, your voice, your joy and your poetry, your doodles, your ideas and your musical ear. I like your texts, I like the way you make me think. Bravo, keep doing your videos. They're clever, deep yet light and enjoyable.
1 is sum of all parts 2 is just double 1 3 4 is just double 2 5 6 is just double 3 7 8 is just double 4 9 is just triple 3 10 is just double 5 11 12 is just double 6, which was double of 3 13 14 is just double 7 15 16 is just double 8, which was double 4, also 4 was double of 2, and 2 was double of 1 (16 has alot going on here) 17 18 is just double 9' which was triple of 3 19 20 is just double 10, which was double of 5 There is a pattern here, i will find it
+Mukine I'm in eighth grade and had never heard of parabolas until I started watching Vi. I have not actually been taught in school what a parabola is.
Lucy Niemann Oh, I'm in Algebra I now, (eighth grade advanced math). I guess it's because of different education in different countries. Or maybe it's a difference between the fifty states because I think I heard something about that. I hope parabolas aren't next. My grade is already terrible. But the quarter (yes we use quarters of the year up until high school, then its semesters) is almost ended thank god for that. But, i assume it's where we live of why we learn things at different times.
is it odd that someone Who hates math Just watched this vid on their own time... and loved every second of it. let me explain, I work with my hands and can build(given the resources to do so) just about anything (it may not be efficient) so numbers have always stumped me, teachers try to teach me but I never seem to understand, need less to say I kinda understood what You said and that is why I loved this vid. keep up the good work :D
eon star I am in 11th grade and thank You for the offer but Im afraid I must respctfuly decline your offer. My math teacher watched the videos this UA-camr and found out how to teach me. Thank you for the offer. Have a nice day Signed Treeless
Treeless Phantom my ghad I thought you were like an elementary schooler XD. You probably know a lot more than me XD(I'm only in trig/algebra II). Have a nice day also :D.
You can draw diagonal lines between the corners that aren't touching the circle. This will produce a path that is shorter then 4 (since it takes the shortest paths between non touching corners instead of diverting to the touching ones), but longer then pi. This path is also close to the perimeter of a circle since it simulates the constant turning by equal amounts that the circle's perimeter has, instead of awkwardly turning 90 degrees in one direction and the 90 in the perpendicular one like the weird square does.
This obviously does't make sense at all. If this is true, we can say "Anything in the end is 0, so teacher, give me the full score because I answered everything 0". This is why I think 0.999... does not equal 1. The idea of this whole video is based on that false (and widely accpeted) assumption. We need to accpet the fact that there are infinitely small numbers. Acutally, it's strange that we haven't accpeted this fact. We accepted the existence of i ( square root of -1 ) which defies all of the "Mathematical proving process", and we denied the existence of 0.000...1? It sounds stupid, isn't it?
Well... it is you versus all qualified mathematicians in the planet, because 0.999... does equal 1. But obviously everyone but you is wrong, of course.
(1/3)*3 = 1 0.333...*3 = 0.999... 1/3 = 0.333... Therefore 1 = 0.999... Also the video never claimed that all numbers are zero... in any sense. Somewhat ironically the main point I took from this is not to trust your intuitions when dealing with this sort of thing. Which is what you're probably doing. Or you're right and your the greatest mathematician of the century. Or you're a troll.
Ah, THINK!!! Throw all of the "Qualification" or "Reputation" away. Let's just play with numbers and logic Let's think about square root of 2 which this video also covered. You can play with pi, but it is not for you who are obsessed about "Great mathematicians said so" Ok, the basic idea this video is saying is "The height of each peak will be zero because "Mathematica said so", and if that happens, square root of 2 will be 2 because the sum of the two other lines is 2" The point is the operation here is exactly same with 0.999.... argument. The number of peaks after 1 zig operation is 2, 4 after another round, 8 after another round, 16 after another... and goes infinitely. 1/2+1/4/+1/8... equals 1 is what mathematicians including her said. and this time, she said it is less than 1, so it can NEVER CONVERGE OR DIVERGE. It is just a move to defend the fact that pi is not 4 and square root of 2 is not 2. Ultimate self-contradiction in mathematics EVER If 0.999... equals 1, pi is indeed 4 and square root of 2 is indeed 2. However, all of us know that does not make sense at all. If this is true, circle must be just square, and triangle can never exist at all. and if we follow this logic, we can push any number to 0, and claim "Teacher give me full score because I answered everything 0. Mathematica said so" Stupid. Just stupid. Why mathematicians use i (square root of -1) in real calculations, and not this one??? It is just Ultimate stupidity.
OK Lets begin Sometimes you sounds like you understand some of the video sometimes you don't. Just to clarify: I think you understand that this video is an argument *against* pi=4 and root2=2 but you think that this is contradiction with 1 = 0.999... The two things have little to nothing to do with one another. I think your problem is you understand just enough to feel you understand but not quite enough to understand quite how complicated these things are. When you're accusing thousands of people who have devoted their lives to mathematics of being stupid in the subject you should at least consider that they might know something you don't. And that thing might not be easy to understand. But the 1 = 0.999... thing is easy. See my above proof. Or this one: 1/9 = 0.111... 9*(1/9) = 1 9*(0.111...) = 0.999... 9*(0.111...) = 9*(1/9) Therefore: 0.999... = 1 If you could point out were either of those is wrong please?
OK Lets begin if your having trouble with a mathematical argument, I was able to convert my tok teacher using a philosophical argument. If two real numbers are distinct, there must be a number between them. A foolproof way to find said number is to average, take a + b then divide by 2(this is not important it is just here to make it un falisifiable) There exists no real number x such that 1 > x > 0.999___, therefore by contradiction 1 = 0.999___. We know this number cannot exist again by contradiction Assume x exists and x = a.bcdefg... where the letters are digits. Lemma 1: If a > b then a1 >= b1(first digit of a is less than or equal to first digit of b) and if a1 = b1, then a2 >= b2 so on and so forth. Therefore 1 >= a >= 0 Case 1 1 > a > 0. If this is true than a must be a natural number between 1 and 0, which doesnt exist by definition. Therefore 1 > a > 0 is not true, x does not exist. Case 2 1 >= a > 0. If this is true than Case 1 can arrise(which we know a doesnt exist for) or 1 = a > 0. Otherwise this means a = 1. But if a = 1 then 1.0000___ > 1.bcdefg... > 0.99999___, which means if a = 1 then 0 > b > 9 -> 0 > 9 (not true) from Lemma 1. Therefore x does not exist Case 3 1 > a >= 0. If this is true than Case 1 can arrise. Otherwise a = 0. Though if a = 0 than 1.000____ > 0.bcdefg... > 0.9999___ and from Lemma 1 0 > b > 9 which is not true. Therefore x does not exist Case 4 1 = a = 0 Direct contradiction because if this is true then 1 = 0. Therefore by contradiction x cannot exist and by extension 1 = 0.999___ I made this proof so it only required the fundementals of mathematics(no algebra or analysis involved) and uses barley any arithemtic and has no use of infinity(we are not manipulating it in any way).
This post has a few years but heres what people mix up, because screens use pixels they will always be squared, that way the process of making a complete round circle is by making smaller squares. For optimization, and its not a coincidence, any circle and round shape calculated by a computer with Math's Pi has a finalising process of 4 (2² = 100(binary)) which preserves memory when calculating visual/graphic content.
I havent got to all this smart and advanced stuff in math yet but I'm honestly loving your videos so much. I love the drawings and the smartness😂 Keep doing what you do, it's great.
I really like your brain. Basically you conclude at the end that R is waaay bigger than Q (rational numbers), and that's why a line cannot be achieved by those zig zags. And that's why a circle doesn't have a perimeter of 4, etc. Really nice.
By the way, that way of approximating a circle actually does approach a circle, and the limit of the length really is 4. The issue is this in no way implies the length of the limit is 4, order matters.
at 2:13 Vi says square root 2 is the ratio of the diagonal of a square to it's perimeter. I think she meant to say that square root 2 is the ratio of the diagonal of a square to its side.
Here's an explaintion I saw a while ago: the proof of making a circle-like infinigon from a square does not work because not only the shape has to approach the circonference, but the tangent vector has too. I'll explain better: If you imagine an arrow tangent to the circle, and you spin it around, the arrow will never point inside of the circle. If we instead make this arrow walk through all the sides of the square infinigon, it will point in and out infinite times, so the circle approximation is incorrect. If we try this with a hexagon or an octagon, or some regular shape with I-don't-know-how-many sides, the arrow will never point inside, mimicking the behaviour of the circle, making the approximation correct. Sorry if it sounds like a "I know everything" kinda thing, but I just wanted to share this more rigorous explanation with you
Commenting before continuing to guess that the main thing that’s happening is ignoring the actual definition of distance in 2D space. As in Pythagoras’ theorem. The perimeter of moving along x and then y is always a + b but moving directly from one point to the next is always sqrt(a^2 + b^2). The idea of repeatedly refining the square works asymptotically for area, but for length/perimeter the limit only approaches a circle if you’re repeatedly refining triangles that have a proper 2D length .
She asked what circle means a round plane figure whose boundary (the circumference) consists of points equidistant from a fixed point (the centre). draw a circle with a compass. • something in the shape of a circle: the lamp spread a circle of light | they all sat round in a circle. • a dark circular mark below each eye caused by illness or tiredness. she was pale and rather beautiful, with dark circles around deep, exhausted eyes. • Brit. a curved upper tier of seats in a theatre or cinema. she sat in the front row of the circle. • Hockey short for striking circle. 2 a group of people with a shared profession, interests, or acquaintances: she did not normally move in such exalted circles.
I never would have known about you if I was never incarcerated. I discovered Khan Academy on the tablets in jail and they had you under math lessons or whatever. Your mind is the most amazing, loveable, beautiful blessing. I discovered you a year ago, but I couldn't tell you until now. It really made everything worth it. Thank you for your music and the captivating topics and your work! I love you! You are so wonderful.... Thank you! Thank you! Thank you with a function that approaches infinity as an exponent to already numerous thanks!
All circles on computer screens must have a perimeter of 4 because circles on screens technically have zig zags.
True
***** Yeah, the computer probably still uses pi to calculate how the circle will look.
Depends how you consider the perimeter of an antialiased pixel. Which is a bizarre and interesting question.
but computers show not zigzags but diagonal transitions between pixels
DHREAVER well an antialiased pixel is still one pixel.
Is zig the real life
Is zag just fantasy
Caught in a zig slide,
No escape from zig zagity
customer sevice open your zigs
look up to the zags and see
+fourisamagicnumber
I'm just a poor zig
I need no zagithy
Zigeanian Zagathy by who
+Arthur Moreira
Because I'm easy zig, easy zag
Little zig, little zag
Zigfinigons remind me a bit of how audio is encoded digitally. The higher the sample rate, the closer you can get to something that is LIKE the original waveform, but is actually a shape with a number of sides determined by the sample rate.
Kinda. When it's represented digitally sure, but when it's being played the sound passes through an antialiasing filter that smooths everything out. In theory, with a sample rate of 40,000kHz you can *perfectly* recreate any signal audible to the human ear (See: Nyquist sampling theorem). If you *did* try to play the weird polygon shape on a speaker, even with a high sampling rate, you'd get fantastically weird retro-sounding aliasing artifacts.
(Note: this is ignoring the issue of sample depth)
@@antonliakhovitch8306 Aliasing only occurs during sampling. Because of this, playing back a sampled signal won't actually introduce anything in the audible range. This weird polygon shape would even have it's high-frequency components attenuated by the speaker itself, so it would be difficult to even tell that the signal is being played by a DAC without removing the speaker from the equation. The artifacts that do make it through would only be at multiples of the fundamental frequency of the sampling rate, so at worst, a 40 kHz sampled signal would have a 40 kHz ring on top of it (assuming the DAC has 0 rise time). If the DAC has bad enough rise time, it could introduce the ring to the audible range, but that's so bad that it would also distort the audio significantly.
@@STRIKEcorperation Dammit you're right - I was thinking of anti-imaging, not anti-aliasing.
sound has an interesting property though where every wave is represented as a sum of sine waves, even the sharpest of square waves!
what's your sharpie budget?
UA-cam money
+Jess Smith money money money
I think it's closer to over 8000 tho
NightStriderXP 9000 what?
units
1. Start with a square
2. Continue with zigfinity till you reach the planck length.
3. It is physically impossible differentiate this circle and a "smooth" circle
4. Pi = 4
5. ???
6. Profit!
Hello, Physicist! xkcd.com/435/
You also have to remember that Planck Lengths are not an accurate representation of space, and that they are just how far it takes light to travel in a vacuum for 1 unit of Planck time, not simply a grid that the universe is built out of.
Dorth Lous
The thing about Mathematics is when it's no longer tied down to the physical world, it's just applied philosophy.
No, it's applied logic. A lot of Physics was predicted and discovered simply based on mathematical formulas and theorem, but it exists also for its own consistency. If you break one segment of Mathematics, you bring down the whole and the tool is completely pointless then.
Dorth Lous
Logic is just corrective reasoning, of course a system with it's own made up abstract constants is going to interact and follow it's own abstract and made up rules. Essentially it's structured nothingness.
To me, numbers are only useful if you're approximating the physical world, once you reach the realm of the irrational and the imaginary, you're pretty much speaking metaphysics.
it’s been 9 years since this came out and it’s still better than most content on the app
And it's also wrong (or at least it would take a lot of effort to make it right).
Basic issue is this: there is no curve with the properties of a zigfinigon that lives in the real plane - so if you want to talk about zigfinigons, you need to invent an entirely new space of curves, ones that don't correspond directly to any that live in the real plane (maybe you can define each new-curve as an equivalence class of ordinary-curves, idk). It's not clear at a glance that any choice of such a space will have nice, intuitive properties.
So if you have the requirement that the limit of a sequence of curves (if it exists) is an ordinary curve, then the limit of those zigzag curves has to be the the circle (or hypotenuse) exactly. Does this pose a problem? No - we had no reason to expect in the first place, that if you have a bunch of curves converging to a limit, that their lengths will converge to the length of the limit. This assumption is equivalent to saying that the function that sends a curve to its length is continuous (for an appropriate topology on the space of curves). All our examples show is that this assumption is false.
She should be sponsored by sharpie
True
lol
Because she's sharp?
Get it?
@@Nadie47 yes
In this video I explain why pi doesn't equal four
But first, let's talk about parallel universes.
Roman Chapkis sick reference bro
I didn't get it, is it Vsauce?
Xxx360NoScOpErgetrektSONXxx 666 nope its super mario 64 memes
Roman Chapkis We will start with the diameter of the circle, which is 1cm and splits the circle in half, i know what you're thinking: A circle is a circle; you can't say it's only a half.
UPG10 YES
So basically Pi isn't 4, but it might as well be 4, or anything else, if you're a topologist.
Or if you're an engineer
@@fetsexe2274 No, then Pi is 3.
@@ZenoDovahkiin 3,14 i think
@@bugandoamente251 engineers use 3
@@sawc.ma.bals.well, it's more easier
I never understand half of what she is saying but I still watch her because I love her and she makes me feel smart
*feel
You know you can edit your comments so you dont have to do the *feel thing? I guess that's why she only makes you feel smarter.
wow ur so smrt
smart*
Wow your so amrt
"Uh-oh, teacher's walking around. Better draw some axes, and pretend to be doing math." XD
Grant Fikes I was scrolling through the comments and she said this at the exact same time I read this. Super creepy.
Axis’s
@@chizkin3799 axes is the plural of a singular axis
Every time I watch these I feel smarter
Ellie April but pi doesn't equal 4
I know but just all these words I don't understand it just makes me feel smort
Ellie April yah cuz i already new
Sp00ky Jim me 2
ikr, I know its wrong but hearing this smart words make me feel like Einstein
Teacher: WHY ARE YOU TALKING TO YOURSELF!?
Edit: Why is this top comment?;;
Eraser Undyne Student: Why aren't you teaching?
OOOOOOOOOOOOOOOOOOOOOHHHHHHHH
Teacher: ...
Why aren't you listening? I asked a question.
Student: I asked a question, why aren't YOU answering
Teacher: I asked first, and I have THE POWER OF (F)
Her mind is super interesting and I love it, but sadly I don't get it.
Andrea Johnson I understand but I'm dead inside
@@lolkillme3998 same.
She's not telling the truth watch the odd1s out
The fundamental flaw in your friends first "proof" is that all the zigzags extend outside the circles circumferense, which means it will always be bigger than PI. If one instead did the reverse and begun the same procedure from a square fitted perfectly inside the circle, the same method would "prove" that Pi is smaller than it is. Only if the zigzags, regardless of number and size, actually cross between the interior and the exterior of a circle in equal proportions would such a zigzag-circle actually have the same length as the circle it's based on.
exactly.
Skadu Skitai You could also make a zig zag pattern that stayed inside the circle yet had a larger perimeter... because the smooth circular curve is a short cut relative to all those turns.
Reminds me of the Trapezium Rule and Squeeze Theorem
Whut...
Drawing the square inside the circle would give you sides of length (.5)^.5 ≈ .707 and a perimeter of 4(.707) ≈ 2.83 instead of 4.
Love to see your introduction to infinity *0. Very clever. Absolutely love it.
It's interesting that for the perimeter of a "circle" to equal 4 it has to have infinite zig zags but the perimeter of an actual circle has infinite digits. You have to have a finite shape or a finite number.. you can't have both. Also that real circles don't really exist. The circles on my screen aren't really circles. Due to the pixels their perimeter is closer to 4 not pi. The pixels aren't perfect squares either. It's really only a circle from the conceptualization of it in my frame of reference. Math or whatever this is is interesting as fuck.
I believe that what you're talking about being really interesting is attempting to impose the infinitely precise nature of mathematics on our inherently finite and quantised world. Essentially, despite what we might think, mathematics does not perfectly represent our world has a concrete idea of "the smallest possible quantity", whereas mathematics does not.
the view count is 1,234,567
coincidence?
absolutely
I THINK NOT
+Heron Myer nope
DAM DAM DAAAAAAAM
+I need A Social Life Illuminati
I'm in seventh grade, but I'm in accelerated math, so I have just started geometry. I was lost about how I was going to get through the year without failing, but then I remembered Vihart! You've got an odd way of explaining things that makes math easier and dare I say it, fun.
im in 7th grade accellerated math too
Same!
kiwibirbz
oh, I *love* geometry. as for *algebra* .... ugh😷
kiwibirbz all of us in seventh grade have the same program like you...ALL OF US... even the little KU-KU students... but we are smart... well most of us
7th grade? Geometry? Where tf do u live
Video is 5:55 long. "Rhapsody". Well played Vi, well played.
+Qwerky_Syntax I don't get it.
Pacvalham same here
Wvt Bohemian Rhapsody is 5:55 as well.
mine is 5:54 long. Maybe you're watching it on a phone or something because in those youtube adds a minute for some weird reason
Diana Escobedo Luna minute?
The fact that she has never misspelled a word in any of her videos is just incredible
Ummm... should that be the norm?
But yeah it’s nice to see that
@@chasejohnson8326 well I think what they're saying is that it is well made, doing all of this usually in one take with little error
I think there's a point where she writes steak as "stiek" unless I misread it
I have no idea what your saying, but I love it!
Same😂😂😂👌
Same😂😂😂
really? I am 13 that gets Cs in Math and I still understand exactly what she's saying.
wtf im 13 and in my school we are doing fractions :/
so am i
I wish I had some amazingly insightful comment about pi. But no, just wanted to ask where you bought that sweater; it's awesome.
This is way more interesting that I thought it was going to be, I mean fractals with finite length. Cool.
They're not fractals! Zigfinigons, if they exist in a meaningful sense, can't live as curves in the real plane, since no curves in the real plane have the required properties. Vi proceeds with the assumption that because the length of the apparent-limit is not the limit of the lengths, that the real-limit must be some new strange object whose length *is* the limit of the lengths. The actual underlying matter is much simpler and mundane - the function that sends a curve to its length is discontinuous.
That doesn't mean zigfinigons can't exist, but to show they do (in any meaningful sense) requires a lot more legwork, since they can't just be sets in the plane.
@@coolkusti u broke my brain
This video makes me smile, knowing that only a select few individuals really know what is going on here.
I know right.. I feel so special while reading all these comments seeing that people thinking not in a correct way. I feel bad about feeling good about this :D
haha, when i first saw this "proof of pi = 4" i was like, what the?, then i sat down, took a sheet of paper, and drew a circle with the radius of 1/2 , and drew around it a square of length a = 1, then i started doing the same thing, but at the end, i recalled that a circle has a radius, so if this thing really is then its radius will be the same as the number of "zig zags" approaches infinity, but then the radius will be disturbed at each zig zag thus, R the radius will not be equal to the same number. i figured out at the end that this would lead to a contradiction (1/4 != 1/2), thus π != 4, :PPP
It's amazing though how the "zig zags" that we cannot tell apart causes pi= 4 ! :D That's how infinity works i guess (as there are infinity dot's in a circle.)
***** but its not pi !!!!, how can you calculate pi with a non-constant radius, think about it, pi = C/2r, now look at that zig-zagy circle, where is the radius?, there are infinility many radii (because you can draw line from center to the side, and draw another which will intersect with the zigzagy bit of the circle, thus u will get different values)
but im sure you can calculate pi from that zigzagy circle, the formula may be like PI = 4 - Σ(something)
English my sec. language so i didn't notice i made a mistake in grammer there. Let me put it like this;
It is amazing how the "zig zags" that we cannot tell apart COULD cause pi=4 .
And as you and Vi' explained very well, the non-consant radius causes the mistake there. The mistake that gives pi=4 .
I love how UA-cam suggested me this video 9 years after I just watched it the first time. Love you Vihart, never stop amaze us to the pi-nfinity and beyond
You know, I'm just amazed by your videos. I have interest in math. But you turn my math into something that is never much cooler. You made me realize how doodle can be more fun. It's just awesome. My mind is already blown by your things. And yeah your drawing skill... omg!! that'a amazing... No one can say that was doodles. Wish I had some of 'em. Guess What? This is the first channel I subscribed to. I'm glad I made the right choice. The whole channel is full of awesomeness. I'm gonna watch all of 'em. wish me luck! Lol 😁😂
And Yeah Keep it up to the sky! 'Cause you're doing great. 👍
congratulation, you invented fractal geometry, miss Mandelbrot :)
I remember that, when math was still relatively new to me I realised that, for positive numbers, the function y=x^2 could also be interpreted as x=sqrt(y). A few years later I learned the function notation where y=f(x), but other variables could aslo be used at the left or right side. From that moment I could see that all variables with a relationship had an equal relationship; it was two-way traffic and the concept of dependency was just a mere matter of perspective. So the way I see it y=2x is not an unhealthy relationship, but the way y and x work together in this particular situation; it could be formulated with equal correctness as x=y/2.
It's not always easy to travel in both directions. You may end up with something that is not a "function" (a function has only one value for each "x")
To illustrate, you said you could interpret y = x^2 as x = sqrt(y), when it's actually x =±sqrt(y) which has more than 1 value of x for each y, tehrefore is not really a "function".
But yeah, you are right about the perspective thing xD Maybe teacher should emphasize this, or maybe it would tend to confuse students more.
moraigna66
That's true: the functions that are not one-to-one are problematic in this context (a function is one-to-one if it never takes on the same value twice; that is f is one-to-one if f(a)≠f(b) when a≠b); but now that term makes even more sense; in a way y=x^2 is one-to-two as x=±sqrt(y) can actually be seen as two functions. In the same way, one could say y=tan(x) is one-to-many as x=tan^-1(y)±k*pi for k=±0;±1;±2;... can actually bee seen as infinetely many functions. So whenever something is not one-to-one it can become a set of multiple functions from the perspective of the other variable. It's kinda like a relationship between people where one of them only uses one means of communication while the other uses other means as well.
TheSenator007 what have u done 2 ma brayne
TheSenator007 yes
God I miss old UA-cam I'm so glad new UA-cam algorithm brought me this channel
Pi = 8 cause that's how many pieces I cut it in
Pi = 8 cause I 8 it.
I thought pi =3.14
@@ultimateweeb2374 r/woooosh
@@kd013n what is woooosh?
@@ultimateweeb2374 just look it up
Starting at about 3:48 you are very close to providing an example of a function that is continuous everywhere but differntiable nowhere. Just add them all together.
You opened up my mind some more this morning. Thank you.
What you've pointed out very well here (without telling it) is that when you make a perimeter aproach to a line, you always end up with something in the rational number set instead of ending in the real number set. Nice job!
Circles are one of the simplest things yet one of the most complex
Entropy?
Order, Entropy, a never ending cycle
It was 42! I knew it!
Dr Öge ah, the League reference
This is beautifully well-made. I already love math, but this could make just about anyone see the beauty in it.
I wish my high school classes of math were like your videos...i would have paid attention....though not keep up but still paid attention. Thank you for the education videos!
Curse you UA-cam recommendations. Tricking me into learning middle school maths during my semester break.
Dear people in the comments, pointing out elementary school math facts does not make you smart
A Sloth Elementary math facts are basically the most important thing in math. In fact, if you pull out complicated math when simple math will do then you're just an idiot because you waste time.
Didn't Einstein say that if you can't explain something to a child, then you don't understand it well enough?
Connor Skudlarek Sure but what does that have to do with what I just said
Connor Skudlarek I don't know if he did but that is just stupid
aitor ormazabal It is not stupid. It is wise. Einstein really did say something like that. I don't think that he meant it to be taken too literally. I'm sure that he didn't mean that you should be able to explain it to a baby for instance. He also said, "everything should be made as simple as possible, but not simpler".
A Sloth 2+2=5
I don't think we do infinity well.
We keep getting weird answers.
This just happened to pop into my head today. It's because the perimeter does not converge to any parameterization of the circle in W^{1, 1}. I was in grade school when I first saw this. Wild.
Ah yes! Making a fake proof for my friend... oh wait... I don't have any
Make up a sqrt(-1) friend.
Pacvalham “i” “i” captain
Pactain*
if a=b
a*a=b*a
a^2=ab
a^2(π-1)=ab(π-1)
a^2(π-1)+a^2=ab(π-1)+a^2
πa^2-a^2+a^2=πab+a^2-ab
πa^2=πab+a^2-ab
πa^2-πab=a^2-ab
πa(a-b)=a(a-b)
divide both sides by a(a-b) and you get
π=1
problem solved... or is it?..
+theFox25 Stop dividing by zero :P
+Derek Lee aaaaw... he saw it...
Step five is wrong; pi + 1 does not equal pi
+Liberī Philosophus I just removed the brackets... a^2(π-1)=πa^2-a^2
and ab(π-1)= πab-ab
I don't see the mistake there
+theFox25 you assume a=b so (a-b)=0 and a(a-b)=0 . it's not possible to divide by zero. so πa(a-b)=a(a-b) can not divide both sides by a(a-b)
I need to stop watching your videos and start doing my real math homework but it's so hard :(
If only I had friends that randomly want to talk about proofs, Vi. Life would be better.
And since 0.999 = 1, pi = 4
Matt Fellenz the area is the same but the perimeter is different. math gets weird at infinity. like she said you could crumple a line into a single point. you can make it whatever shape you want
Matt 'Bacon' Fellenz it is impossible to square a circle
Ari Björn Össurarson I'm not. Vi said that the perimeter (circumference) is 3.999 and 0.999 = 1 so 3.999 = 4
that's wrong
No, she didn't. The perimiter never changes, it is always exactly 4.
I showed this to my math teacher and he shot himself.
In the video you said the peaks of the triangles would eventually reach zero, you said they must, that's not true, they never will they would reach numbers infinitely close to zero but would never reach zero because the base of the triangle is at zero, since you're halfing the hight of the triangle but doubling the amount of triangles each time, you're not actually doing anything to the base of the triangle, if the peaks reach zero they aren't considered triangles any more then it becomes a straight line
She says that the peaks reach zero at infinity, which is true.
leena kiyumi limits dude. its limits..
Watching this during online math class really hits different
a circle isn't a real shape it doesn't exist in reality. all shapes are polygons with sides. a circle simply designed is a shape with all of it's points equal distance from a center. you can add what seems like an infinite number of points right down to the plank distance(smallest physically possible distance) but in the end each of those points will be connected with line segments. I think that's part of why pi exists as an irrational number. it's trying to describe something that doesn't really exist.
Seta-San fuck you
Dieeye when and where?
There isn't an object in real life that is perfectly circular, but that is far different from saying that circles don't exist. They are a concept. It would be like saying that meters don't exist because there is no object that is exactly a meter long down to the planck length. If you try to understand math exclusively in terms of real life, you won't be able to get much further than basic arithmetic.
Spencer Wadsworth ... which was his point ;) math is entirely made up.
okay. the circle doesn't even really exist in a mathematical sense because every point is connected with a straight line segment. there is no curved line connecting these points.
y says, "Hey! x is only half of me!!"
+Enid Radaviq "x is only half of my power."
Y: WHAT THE X IS HE DOING HERE!
X: HUH? Y DID YOU SAY THAT TO ME!??! D;;;
*_B A D U M T S S_*
Getting infinitely close is not close enough; not the perimeter
Infinitesimals explain all of this. I have fallen in love with infinitesimals. They explain so many questions!!! Many people use zero when they should use infinitesimals. Let me put it nice and simple. 1/infinity does not equal 0. It equals an infinitesimal. Which is basically zero but it has different properties when you multiply it by infinity. Sweet.
Alright youtube I finally watched it
2nd video of yours I'm watching and I still find myself saying "what the fuck did i just watch?!" But I love it!
doesnt that have to do with the coastline paradox?
I was thinking this the entire video
yes it does^^
***** but its quit different because the lenght of the blows up to infinity by shortening the measuring "tape". here the perimeter is constant which is very important for the problem
Phraser But this is how the coast line paradox works....
***** The answer to all of your questions if Fractal cosmology.
i somehow manage to understand exactly what you're talking about while at the same time i have literally no idea what you're talking about and it's amazing
It's cool that her video piques interest, which is the point- but the entire reasoning is hinged on a flawed statement- Pi is not the circumference of a circle- it is the ratio of any circle's circumference to its radius. This is why the area of a circle is pi(r)^2.
Girl, you lost me at "Say you're me and you're in math class..." hahaa, wah wahh :S
1:54 OH MY GOD ITS THE ILLUMINATI AAAAARGH
i know its hrd 2 fce but 1 liek=i ipod
Moral of the story: Infinity is a freaking troll, don’t mess with it.
There are infinite segments infinitely small. You can't make them disappear
lmao "AXES YEAH"
So let me get this straight. She ditches math, to do math...-_-
ya me and friends do not pass notes on this level...
I love her videos
thats the joke
She ditches math class not Mathematics, she promotes Mathematics
IDWpresents Bashing Functions sounds very much like promoting mathematics.
I like your hands, your voice, your joy and your poetry, your doodles, your ideas and your musical ear. I like your texts, I like the way you make me think. Bravo, keep doing your videos. They're clever, deep yet light and enjoyable.
1 is sum of all parts
2 is just double 1
3
4 is just double 2
5
6 is just double 3
7
8 is just double 4
9 is just triple 3
10 is just double 5
11
12 is just double 6, which was double of 3
13
14 is just double 7
15
16 is just double 8, which was double 4, also 4 was double of 2, and 2 was double of 1 (16 has alot going on here)
17
18 is just double 9' which was triple of 3
19
20 is just double 10, which was double of 5
There is a pattern here, i will find it
You think 16 has a lot going on, just wait 'til you hit 32!
(psst, nobody tell him about 64)
Azukar (Pssssssssssssssssssssssst. Don't tell him about 1073741824.)
The numbers keep going and going, but something is up with the numbers that dont double or triple from the beginning
1,3,5,7,11,13,15,17,19, something different about 8 and 9 being missing, obvoius why others are missing
They not prime numbers, i would have to do a bit of geometry to find out the form of these numbers, 1,3,5,7,11,13,15,17,19
It might make a nice spiral
i'm in sixth grade and learning about graphing and parabolas and even crazier stuff... i realized this the sixth time i watched this video xD
+Mukine Those are freaky! In addition, try hyperbolas! O.o They're even freakier!! O_o
+Mukine I'm in eighth grade and had never heard of parabolas until I started watching Vi. I have not actually been taught in school what a parabola is.
+cutiecreeper66 Really? I learned about them in Algebra One (Seventh grade for me)
Lucy Niemann Oh, I'm in Algebra I now, (eighth grade advanced math). I guess it's because of different education in different countries. Or maybe it's a difference between the fifty states because I think I heard something about that. I hope parabolas aren't next. My grade is already terrible. But the quarter (yes we use quarters of the year up until high school, then its semesters) is almost ended thank god for that. But, i assume it's where we live of why we learn things at different times.
cutiecreeper66 Yeah, that makes sense haha
@-@ I think my brain exploded
is it odd that someone Who hates math Just watched this vid on their own time... and loved every second of it. let me explain, I work with my hands and can build(given the resources to do so) just about anything (it may not be efficient) so numbers have always stumped me, teachers try to teach me but I never seem to understand, need less to say I kinda understood what You said and that is why I loved this vid. keep up the good work :D
+Treeless Phantom to get good at math just practice it :P. What grade are you in >.
eon star I am in 11th grade and thank You for the offer but Im afraid I must respctfuly decline your offer. My math teacher watched the videos this UA-camr and found out how to teach me. Thank you for the offer. Have a nice day
Signed
Treeless
Treeless Phantom my ghad I thought you were like an elementary schooler XD. You probably know a lot more than me XD(I'm only in trig/algebra II). Have a nice day also :D.
+eon star thank you, you to
Treeless Phantom np
pi is equal to 3.14159.... just face it vi hart! you want to deny it but it's true that pi is 3.14159!!!
Dang calm down okay
I'm not mad, just some friendly advice,
Thx btw I'm 12/gifted program
Not really, are you considering the complete decimal expansion?
when you cant sleep!
So it's true: 0×∞=IR()
You can draw diagonal lines between the corners that aren't touching the circle.
This will produce a path that is shorter then 4 (since it takes the shortest paths between non touching corners instead of diverting to the touching ones), but longer then pi.
This path is also close to the perimeter of a circle since it simulates the constant turning by equal amounts that the circle's perimeter has, instead of awkwardly turning 90 degrees in one direction and the 90 in the perpendicular one like the weird square does.
This is awesome
i like turtles
Austin Page hahahahahaAhahahahahooooohehehehahaHaaa.
Turtles all the way down
me too. so tasty!
I like trains.
Mine turtle
Omg! im the 10000th Like! (lol no one cares :P) Great video Vi!
I love that you use graph paper. I'm the only one in class. I try to share it but people are like 🤔"what do I do this?" 🤷🏻♂️
Me: "Make life easy." 😎
MinuteMaths.
My brain hurts...
I came here from TheOdd1sOut, anyone else?
IVmAsTeRiEsXI
Same
IVmAsTeRiEsXI I did
same
Dracokitten Firetunez, Same
EᏉᏇ BᎾMЅᏇᎯᎶme
Huh 9 years...great time to recommend this 😂
How about a bit slower for a change
Watch at 50/25% if u dont like it
;3
Pass a note to explain who is dating who. not why some stupid irrational is 4.
Must be honors algebra :/
Really y, get a life
Really, y get a life
That was beautifully cute...
And made a whole lot more sense than any book ever read or teacher ever taught...
This obviously does't make sense at all. If this is true, we can say "Anything in the end is 0, so teacher, give me the full score because I answered everything 0".
This is why I think 0.999... does not equal 1. The idea of this whole video is based on that false (and widely accpeted) assumption. We need to accpet the fact that there are infinitely small numbers.
Acutally, it's strange that we haven't accpeted this fact. We accepted the existence of i ( square root of -1 ) which defies all of the "Mathematical proving process", and we denied the existence of 0.000...1?
It sounds stupid, isn't it?
Well... it is you versus all qualified mathematicians in the planet, because 0.999... does equal 1. But obviously everyone but you is wrong, of course.
(1/3)*3 = 1
0.333...*3 = 0.999...
1/3 = 0.333...
Therefore 1 = 0.999...
Also the video never claimed that all numbers are zero... in any sense. Somewhat ironically the main point I took from this is not to trust your intuitions when dealing with this sort of thing. Which is what you're probably doing. Or you're right and your the greatest mathematician of the century. Or you're a troll.
Ah, THINK!!! Throw all of the "Qualification" or "Reputation" away. Let's just play with numbers and logic
Let's think about square root of 2 which this video also covered. You can play with pi, but it is not for you who are obsessed about "Great mathematicians said so"
Ok, the basic idea this video is saying is
"The height of each peak will be zero because "Mathematica said so", and if that happens, square root of 2 will be 2 because the sum of the two other lines is 2"
The point is the operation here is exactly same with 0.999.... argument. The number of peaks after 1 zig operation is 2, 4 after another round, 8 after another round, 16 after another... and goes infinitely.
1/2+1/4/+1/8... equals 1 is what mathematicians including her said.
and this time, she said it is less than 1, so it can NEVER CONVERGE OR DIVERGE. It is just a move to defend the fact that pi is not 4 and square root of 2 is not 2.
Ultimate self-contradiction in mathematics EVER
If 0.999... equals 1, pi is indeed 4 and square root of 2 is indeed 2.
However, all of us know that does not make sense at all. If this is true, circle must be just square, and triangle can never exist at all.
and if we follow this logic, we can push any number to 0, and claim "Teacher give me full score because I answered everything 0. Mathematica said so"
Stupid. Just stupid.
Why mathematicians use i (square root of -1) in real calculations, and not this one??? It is just Ultimate stupidity.
OK Lets begin Sometimes you sounds like you understand some of the video sometimes you don't. Just to clarify: I think you understand that this video is an argument *against* pi=4 and root2=2 but you think that this is contradiction with 1 = 0.999...
The two things have little to nothing to do with one another. I think your problem is you understand just enough to feel you understand but not quite enough to understand quite how complicated these things are. When you're accusing thousands of people who have devoted their lives to mathematics of being stupid in the subject you should at least consider that they might know something you don't. And that thing might not be easy to understand.
But the 1 = 0.999... thing is easy. See my above proof. Or this one:
1/9 = 0.111...
9*(1/9) = 1
9*(0.111...) = 0.999...
9*(0.111...) = 9*(1/9)
Therefore: 0.999... = 1
If you could point out were either of those is wrong please?
OK Lets begin if your having trouble with a mathematical argument, I was able to convert my tok teacher using a philosophical argument.
If two real numbers are distinct, there must be a number between them.
A foolproof way to find said number is to average, take a + b then divide by 2(this is not important it is just here to make it un falisifiable)
There exists no real number x such that 1 > x > 0.999___, therefore by contradiction 1 = 0.999___.
We know this number cannot exist again by contradiction
Assume x exists and x = a.bcdefg... where the letters are digits.
Lemma 1: If a > b then a1 >= b1(first digit of a is less than or equal to first digit of b) and if a1 = b1, then a2 >= b2 so on and so forth.
Therefore 1 >= a >= 0
Case 1
1 > a > 0. If this is true than a must be a natural number between 1 and 0, which doesnt exist by definition. Therefore 1 > a > 0 is not true, x does not exist.
Case 2
1 >= a > 0. If this is true than Case 1 can arrise(which we know a doesnt exist for) or 1 = a > 0. Otherwise this means a = 1. But if a = 1 then 1.0000___ > 1.bcdefg... > 0.99999___, which means if a = 1 then 0 > b > 9 -> 0 > 9 (not true) from Lemma 1. Therefore x does not exist
Case 3
1 > a >= 0. If this is true than Case 1 can arrise. Otherwise a = 0. Though if a = 0 than 1.000____ > 0.bcdefg... > 0.9999___ and from Lemma 1 0 > b > 9 which is not true. Therefore x does not exist
Case 4
1 = a = 0 Direct contradiction because if this is true then 1 = 0.
Therefore by contradiction x cannot exist and by extension 1 = 0.999___
I made this proof so it only required the fundementals of mathematics(no algebra or analysis involved) and uses barley any arithemtic and has no use of infinity(we are not manipulating it in any way).
This is my favorite YT video so far and the only one that started comments that were interesting and not horrifying
This post has a few years but heres what people mix up, because screens use pixels they will always be squared, that way the process of making a complete round circle is by making smaller squares. For optimization, and its not a coincidence, any circle and round shape calculated by a computer with Math's Pi has a finalising process of 4 (2² = 100(binary)) which preserves memory when calculating visual/graphic content.
I havent got to all this smart and advanced stuff in math yet but I'm honestly loving your videos so much. I love the drawings and the smartness😂 Keep doing what you do, it's great.
This and undertale shorts are the only things that I watch right now
when she scribbles in an outline of a shape and then instantly cuts to when she completes it, like the square root 2 triangle, it's so satisfying
I really like your brain. Basically you conclude at the end that R is waaay bigger than Q (rational numbers), and that's why a line cannot be achieved by those zig zags. And that's why a circle doesn't have a perimeter of 4, etc. Really nice.
By the way, that way of approximating a circle actually does approach a circle, and the limit of the length really is 4.
The issue is this in no way implies the length of the limit is 4, order matters.
Basically, if we turn math into pixels, and zoom all the way in, pi = 4
instantly when I saw the "proof" of π=4, I was like,"no, π makes a circle, and 4 makes a really squiggly circle!"
at 2:13 Vi says square root 2 is the ratio of the diagonal of a square to it's perimeter. I think she meant to say that square root 2 is the ratio of the diagonal of a square to its side.
Here's an explaintion I saw a while ago: the proof of making a circle-like infinigon from a square does not work because not only the shape has to approach the circonference, but the tangent vector has too. I'll explain better: If you imagine an arrow tangent to the circle, and you spin it around, the arrow will never point inside of the circle. If we instead make this arrow walk through all the sides of the square infinigon, it will point in and out infinite times, so the circle approximation is incorrect. If we try this with a hexagon or an octagon, or some regular shape with I-don't-know-how-many sides, the arrow will never point inside, mimicking the behaviour of the circle, making the approximation correct. Sorry if it sounds like a "I know everything" kinda thing, but I just wanted to share this more rigorous explanation with you
this is very interesting, thanks for sharing!
Commenting before continuing to guess that the main thing that’s happening is ignoring the actual definition of distance in 2D space. As in Pythagoras’ theorem. The perimeter of moving along x and then y is always a + b but moving directly from one point to the next is always sqrt(a^2 + b^2). The idea of repeatedly refining the square works asymptotically for area, but for length/perimeter the limit only approaches a circle if you’re repeatedly refining triangles that have a proper 2D length .
She asked what circle means
a round plane figure whose boundary (the circumference) consists of points equidistant from a fixed point (the centre). draw a circle with a compass.
• something in the shape of a circle: the lamp spread a circle of light | they all sat round in a circle.
• a dark circular mark below each eye caused by illness or tiredness. she was pale and rather beautiful, with dark circles around deep, exhausted eyes.
• Brit. a curved upper tier of seats in a theatre or cinema. she sat in the front row of the circle.
• Hockey short for striking circle.
2 a group of people with a shared profession, interests, or acquaintances: she did not normally move in such exalted circles.