To understand this proof it needs more. Showing f(0) is natural and the symmetry property is boring and more or less trivial. The difficult part of this proof is the idea to choose f(x) like it was done. Why this f(x) should help? If u figure out this, then one has understood this proof. :) But the big problem is following. Mathematicians play around with stuff. Try this. Try that. At the end u get a result if you are lucky. Problem? The following paper u write is scientific and not "My thouggt was this, and that." It's just: Theorem: blabla Proof. The specific thoughts of the author are not given. This makes it really hard to understand this proof. On the other hand: What makes this proof so difficult? Showing Pi is not a ratio is difficult because u can't use a explicit definition of Pi. Try to use the analytic definition: Pi is the double of the smallest positive zero of the cosine function. Good luck. That makes it difficult. So u have to be smart and find something what will help u. This is the chosen f(x). The choose is genius.
@@namangaur1551 Yes, you're precisely correct in a "whoosh!" kind of way. Pi is defined by a ratio, but that ratio doesn't fulfill the requirements to guarantee that it expresses a rational number and therefore there's no contradiction in the fact that pi is irrational. But the key detail you highlighted isn't explicit in the *construction* of the word "irrational", which in itself only says "not a ratio" - that discrepancy between construction/apparent meaning and formal/actual meaning is the source of the "irony".
@@StoicTheGeek Makes sense, but that's a pretty modern redefinition. Euler's number and the natural exponential function are much more recent discoveries than pi, and for the exponential function to have a periodicity the exponent must have an imaginary part, so it's not even just about using Euler's number as a base. It always seems odd to me when mathematicians claim the most basic definition of an ancient and simple concept is to derive it from much more recent and sophisticated ideas.
Throughout my life I have always known that π was irrational but have never seen a proof for it. And I ALWAYS tell my students to never just accept anything from their teachers - always ask them to prove or justify statements such as - Area of circle = π r² - Volume of sphere = 4/3 π r³ So finally I have seen the proof I should have looked for all of those years ago. It was well presented and easy to follow. It's also got a decent bit of mathematical meat to it. Thanks Presh! I really enjoyed this video.
That's a problem with maths. If you want to teach it rigorously then, e.g. in case of calculus, you need to start from the axioms for real numbers and work from there. After "some" time, you can derive those relations you mentioned (via integration).
@Suani Avila of course it does. Some things might be extremely hard to prove by yourself, but some genius mathematecian might have done a very creative solution which is easy to do if you know it, but veery hard to think on your own.
For those wondering how one would ever come up with this proof and how one would come up with the definition of, think of a function that is equal to 0 at x = 0 and x = a/b = π. With the fundamental theorem of algebra, one can easily construct the function x(x - π) = x(x - a/b) as satisfying this property. One can multiply by b to get x(bx - a), and it still satisfies this property. Finally, one can multiply by -1 to get x(a - bx), which satifies this property still. How does one transition from x(a - bx) to [x(a - bx)]^n/n! logically? Well, if you sum the latter expression over all natural n, you get e^[x(a - bx)]. And as this is a well known expression, you know the associated infinite series converges. This already lends itself to the proof as presented in the video. The rest can be figured out by taking these things and exploring further
You could also have started with x(π - x) = x(a/b - x), then multiply by b to get x(a - bx) The reason is because when 0 < x < π, then π-x > 0 and x(π-x) > 0. So better to start off with a function that is positive for all values of x in the interval (0, π).
@@Maxence1402a yeah, but you need integrals for the simple proof, and integrals were really hard to do before the 17th century (like it's so much easier now :D )
To understand this proof it needs more. Showing f(0) is natural and the symmetry property is boring and more or less trivial. The difficult part of this proof is the idea to choose f(x) like it was done. Why this f(x) should help? If u figure out this, then one has understood this proof. :) But the big problem is following. Mathematicians play around with stuff. Try this. Try that. At the end u get a result if you are lucky. Problem? The following paper u write is scientific and not "My thouggt was this, and that." It's just: Theorem: blabla Proof. The specific thoughts of the author are not given. This makes it really hard to understand this proof. On the other hand: What makes this proof so difficult? Showing Pi is not a ratio is difficult because u can't use a explicit definition of Pi. Try to use the analytic definition: Pi is the double of the smallest positive zero of the cosine function. Good luck. That makes it difficult. So u have to be smart and find something what will help u. This is the chosen f(x). The choose is genius.
TheSpecialistGamerX2 No, super pi day was 3/14/1592 Or alternatively it will be in 3141 on September 15, but I don't believe anyone will be celebrating it at that point
@@grottjam the goal of the proof was to create a function for the integers that make up rational pi that has nice cancellation and differention properties. Naturally the mathematician went with polynomial functions for the former and trigonometric for the latter. The choice for the polynomial wasn't exactly elegant, but probably done meticulously so through trial and error to make the cancellation work. When doing proofs for irrationality it's typically easiest to assume the opposite and arrive at a contradiction, because rational numbers have a very simple but essential rule baked into the definition, and irrational numbers being the compliment of rationals therefore have a very straightforward definition as well. It's likely that the proof writer realized that their chosen polynomial would be all integers, or all positive, or something. With that being the case, they simply have to find a property that shows that it, and it's contradiction, are both true, which is what they did by finding it's upper bound (a common tool in real analysis)
There's a much easier way to prove (pi^(n+1)*a^n)/n! goes to 0 as n approaches infinity. You're dividing an exponential function by a factorial. The factorial goes to infinity faster than the exponential. Note that pi^(n+1)*a^n = pi*(pi*a)^n = pi*Q^n. Once n becomes greater than Q, n! will increase faster. No need to go anywhere near Taylor series.
I thought that many people are lost from the beginning because of functions f(x) and G(x) that fall from the sky. So I rewrited it for the Applied Math type. (1) If π is rational, i.e. =a/b, then bπ is integer (=a) and so is any polynomial formula of the type b^n(c₀π^n+...+c_n) with integer coefficients c. So we are looking for a function that solves to this shape, and which can be proved to NOT being an integer. Best candidaites : functions greater than zero and less than one. (2) The use of an integral ∫f(x)sin x dx allows to work with derivatives instead of primitives (=antiderivatives). Trig function is also hinted at by the problem, which concerns π. (3) We want a function that zeroes on 0 and on π, since the sinuses are zero and the cosinuses are ±1. The first that comes to mind is f(x)=x(π - x). But its "sine integral" is not less than one (it is equal to 4). Reminding the series expansion of exponential (or equivalently, remembering the term P(n) in a Poisson probability distribution) we know that : z^n/n! tends to zero when n is large. So let us define our f(x) as [ x(π - x) ]^n/n!. (One should write f_sub_n, but here it would be too heavy.) (4) Integrating any f(x)sin x is easily done by parts (repetitively). It is actually F(x) = - f(x) cos x - f'(x) sin x + f''(x) cos x + and so on. Now as we integrate from 0 to π, the sines disappear (equal to zero) and the cosines alternate signs... hence in F(π) - F(0) they actually give the same sign to both terms. The integral from 0 to π is consequently : - [ f(π)+f(0) ] + [ f''(π) + f''(0) ] - [ f⁽⁴⁾(π)+ f⁽⁴⁾(0) ] + etc. (even derivatives) (5) What about those derivatives? Either you expand the [ x(π - x) ]^n terms and derivate the polynomials with binomial coefficients, like in the video. or you derivate the factorized form and get only [ x(π - x) ]^m terms for the n-1 first derivatives ; those same plus one [ (π - 2x) ]^m term for'the (n)th to (2n)th ; and zero afterwards. ALL HAVE INTEGER COEFFICIENTS. Meaning that at 0 and π , the first ones disappear and you are left with terms in [ (π - 2x) ]^m = π^m or (-π)^m. But remember, we were left with only the even derivatives! So both terms are equal and positive. (6) Result: the desired integral results in a polynomial in π with integer coefficients and only even powers from n (or n+1 if n is odd) to 2n. Please note that the coefficients can be positive or negative. (7) On the other hand it is easy to show that the integral must be larger than zero (all interior values of f and sin are positive) and, as we required, can be made arbitrarily small by increasing n. NOTE THAT THIS IS GENERAL RESULTS, VALID WHATEVER THE NATURE OF π. Now for the proof. Posit π = a / b, positive integers. By (1) we know that b^n times any integer-coefficient polynomial of degree n in π must be an integer. That is precisely the result of b^n times the integral. So it must be larger than zero and it can be made arbtrarily small (same Poisson argument). An integer between zero and epsilon ==> Contradiction.
Thank you for making this profound fact accessible to almost a third of a million of people! You did a fantastic job explaining this! I was happy after watching the video, thinking I understood it, until I realized that I had missed one detail: To be valid, the proof must actually use the assumption that a/b is the ratio of the circumference and the diameter of a circle. Without that, it would just prove that we made a mistake somewhere. It's not mentioned explicitly in the video where that happens. As far as I understand, the point where we use it is at 7:46 in the video, where we assume that sin(a/b) = 0 and cos(a/b) = -1.
Yes, I also noted that this was the only time in the argument when a/b is assumed to PI. It made me wonder how this could be generalized a bit farther.
A great and clear proof!!! I can understand 85% when I listen to it for the first time!!! and of course, I listen to it serval times!!! Please please please video proofs for (1) pi^2 is irrational (2) pi is transcendental (3) e is transcendental. You are an excellent professor!!!
My only comment is that I think the function f(x) should be denoted as f_n(x) {the n being a subscript), or f(n,x) to show that your f(x) is a function of both x and n. Other than that minor detail, great video.
You missed Bhaskaracharya Do you know the indian formula of calculation of pi It is (12^1/2)×[(1÷(3×3))×(1÷(3×5))] and so on If u can understand the later on series
I was randomly laying in my bed when the thought occurred to me “how the hell did we prove that pi is irrational” but I haven’t learned calculus yet and don’t understand any of this whatsoever
Wait, didn't pizza mean the area of a circle with diameter Z? (Unlike American deep-dish pizzas, real traditional pizzas are flat; they aren't supposed to have "height".) Pi*Z*Z/4
Lvl 1 guy with nothing special Gets killed by random cult for no reason lvl 100 mathematician Drowned in the sea by the Pythagoreans for demonstrating that √2 is not a rational number That's how mafia works
If he wanted to present the simple proof it could take 2 minutes. Most things that are difficult to prove don't have "simple proofs" that you can explain to a lay audience. Proving Fermat's Last theorem: (1) notice, if x^n+y^n=z^n for positive integers x,y,z and n>2, we can form an elliptic curve that is not modular (2) It also follows from the axioms that all elliptic curves are modular (3) this is a contradiction, proving Fermat's last theorem *details left to the reader
@@MK-13337 _Footnote: related to this problem is the Birch and Swinnerton-Dyer conjecture, proof of it is left to the reader, as it is considered trivial._
0 < integral x^8(1-x)^8*(25+816*x^2)/(3164*(1+x^2)) from x= 0 to 1 = 355/113 - pi, So 355/133 > pi and 355/133 is closer to pi than 22/7 is :-). In fact, if you replace 3164(1+x^2) with 3164(1+0^2) and again by 3164(1+1^2) and integrate between x= 0 to 1 and compare these results to the integral above, then we find that 3.14159274 > pi > 3.14159257. Note: pi = 3.141593 ( 6 decimal places).
i remember in algebra in 8th grade i got into an argument with the teacher when i claimed it was impossible for both the circumference and diameter of a circle to be rational numbers. same guy i forced to call the high school teacher when he claimed there was no such number as i when he asked us if we could use the same method we used to simplify x^2-1 to (x+1)(x-1) and i said yes.
Nice job, Presh. It was not too hard to stop the video a few times and fill in the proofs of the pieces, and it was fun. It left me wondering, though, just why it goes so wrong. You do a bunch of simple calculus stuff that you should be able to do with any old numbers and then voila--a contradiction! Where would the proof blow up if we tried to tun through these calculations with a rational number a/b which is extremely close to pi? Because of course it would blow up, that's the point of the proof. Perhaps someday I'll try to follow that through and see what happens.... MORAL: Proofs by contradiction are often a bit unsatisfying, because they don't always illuminate the underlying mathematical relationships. BUT, the video wasn't unsatisfying. The video was perfect. Thanks again.
The key is that sin(pi) = 0 and cos(pi) = -1. This was used in finding the definite integral to be an integer, but it wouldn't be true if we replaced pi with a rational number, not even one very close to pi such as 355/113.
Pi is irrational, because it is simply measuring ANY circle's diameter around its circumference. ANY circle, means ANY length of the diameter. This means ANY number, however big. The bigger the number, the more decimal places it will result for the pi. So the only way we can have a final value for pi, is if we figure out the biggest number possible, which of course is non-existent. Infinity exists, so a circle with an infinite diameter can exist, and putting that round it's circumference will give an answer with 3. + infinite values of decimals numbers. That is the reason why Pi is an irrational number.
Hi Presh, I’m wondering, whilst the modern study of Pi has progressed somewhat beyond the Archimedes Method, does the Method actually quite neatly prove that Pi is irrational? If we think in terms of polygons with ever-increasing numbers of sides, we also polygons with ever-increasing (assuming inscribed polygons) perimeters, and thus an ever-changing precise ratio to the widest measurement of the polygon (or circle diameter, again assuming inscribed polygons) - I suggest that the simple fact that the ratio (specifically in relation to a polygon) will never exactly stabilise (but only ever approximate) is pretty good proof that the ratio is irrational, and thus that Pi (as the value of the precise ratio relating to a circle, or to a polygon of infinite sides) is irrational.
Interesting. But if I take a 90 degree step size I get an approximation of pi of 4*sqrt(1/2), which is already irrational (2.82...). As a counter example: the sum of an infinite amount of fractions can be fully integer too. Like 1/1 + 1/2 + 1/4 + 1/8 + .... == 2. So why can the sum of an infinite amount of smaller and smaller fractions not be a fraction as well?
You can get REALLY close to rational, though... If you have a circle with a diameter of 113 (whatever unit), the circumference will be 355 (whatever unit)... 355/113 = Pi (at least up to a few decimal places) - but, being more exact, the circumference of such a circle would actually be 354.99997.
This reminds me of how in high school I learned the difference between deduction and induction on accident. My one friend told me that you can't prove that pi repeats forever because you would have to actually find a last digit so you can only disprove it but never prove it (like how you can disprove a theory but can never prove a theory; inductive reasoning and the scientific method). I repeated that to my math teacher and she said no we know it's infinite because we can simply prove that pi is irrational ( deductive reasoning) My mind was blown
It wasnt him who named it simple, it was the author of the original paper. And indeed it is simple, if compared to other known proofs which generaly use several tools from mathematical analysis or abstract algebra. This one is simple in the sense that it is short and only uses basic calculus. It is however very hard to come up with, so much so that it was only found in the 40s.
For those who didn’t know what happened to the baby it got abducted by aliens and the parents had already bought the baby shoes, so they decided it would be worth selling the baby shoes, because there was other ways of remembering their child , and they needed some money. Honestly a complete tragedy… I am in tears rn
@@ErikBongers The guy who did the proof surely spent several weeks playing around with several functions to come up with that. This is the kind of proof that looks like magic upon completion because you are not following the full thought process of the creator. The guy no doubt made several falty attempts before arriving at that.
Erik Bongers The function has it so that if x = a/b = π, then f(x) = 0, and dividing by n! makes it suitable for taking derivatives because it cancels out the factor. Those are two very desirable properties, and it's easy to construct the function from those properties alone. Hence this gives you a very elementary reason to work with this function.
Completely lacking in the source of the equations. "if you just pull this equation out of your butt do some figuring about it then test it for assumed rational pi" was what I got from that.
@@HeyKevinYT I have seen complicated ones. And he doesn't deserve to present those complicated proofs if he can't solve that Einstier riddle or than Hardest Australian highschool prob I may sound rude here, but I used to love his videos and became a big fan of him. Now a days he is just using clickbaits to popularize his channel. His channel ain't anymore about strategical combinatorics and game theory😞
There are far more irrationals than rationals, but that doesn't make them "More important". In some very real, physical way, irrational numbers exist only as mathematical concepts.
@@johanliebert6734 Many more Indian scientists were there other than Ramanujan. Although Ramanujan's series are more famous. One was Madhava of Sangamagrama who proposed the value of π as: 4×{1 - 1/3 + 1/5 - 1/7 +...} He was of 14th century and Ramanujan was of 20th😱😱😱
@@AAAAAA-gj2di It's still an infinite series but Leibniz used the results of that Indian mathematician to approximate pi. That's the reason that both of their names are used when taking about it. But as with many discoveries, the result carries the name of the one that found it and in this particular case leibniz used the infinite series of Madhava for the inverse tangent to give us this beautiful result.
Reminds me of a joke: A math professor is teaching a class. He's in the middle of a proof, and, referring to a complicated expression, says, "It is intuitively obvious that this is an integer." Then he frowns, looks at his notes, looks at the board, looks back at his notes. He steps to the side, and starts scribbling unreadable shorthand equations in the corner of the board, scratching his head. After five minutes of this, he switches to writing in pencil on his notes. The class is mystified. Another ten minutes go by, with him alternating between writing furiously on the paper and staring intently at what he'd written. Shortly before the class is scheduled to end, the professor suddenly looks up and says, "Aha! Yes, I was right. It *is* intuitively obvious that this is an integer."
The joke is that if it is intuitively obvious, you wouldn't have to go through the process of figuring out the proof to feel confident enough to make that statement.
The argument that pi^(n+1) a^n / n! -> 0 was just awful. It's a circular argument. The domination of exponential growth by factorial growth is used to prove the convergence of the exponential Taylor series! Anyway, there are far more intuitive ways to explain this behavior!
We can take a unit vector from the origin (0,0) to (1,0) and we can rotate it 180 degrees or PI radians so that the point (1,0) is transformed through rotation to the point (-1,0). The length or magnitude of the vector is 1. The overall distance from (-1,0) to (1,0) is 2. The Arc length that is generated is PI radians. How many unit vectors or line segments of length 1 are there that make up this arc during the rotation between the points (1,0) and (-1,0)? There's an infinite amount. This is similar to asking the question: how many Real numbers are there between the interval [0,1]? We know that some of those values are rational but we also know that the vast majority of them are irrational. There is a very high probability due to the infinite complexity that it would highly suggest that PI is irrational. This is not meant to be a direct nor an exact proof. Yet, I would claim that this would be the simplest possible proof that there is to demonstrate the reasoning that PI is irrational. Well, it's more of a supposition, more of a conjecture than an actual proof. Here's an example to support this. The surface of the earth is approximately 70% water and 30% land mass. You have a much higher probability of landing in water than on the ground if you were randomly dropped from the sky. The ratio of Irrationals compared to that of rationals within the Real Number domain is much higher than this 7:3 ratio. So to randomly drop a constant value on the real number line has a much higher probability of being irrational than it does being rational. This is just a way of thinking outside of the box. What more proof do you need? If you keep trying to search and divide into an an infinite domain you will never stop dividing nor will you ever stop diving. Sometimes having a relatively close enough answer is good enough! And I think this was simple enough to explain. Just food for thought.
The answer is simple in pne sense. And sounds silly on the other hand. This f(x) is chosen for purpose. Problem: How to find this? This proof is just genius.
Humans tangibly deciphered how to actualize, mathematically, the seemingly perfect thing that is a circle. A perfect circle has no angles or bases or true shape. Its 'infinite'. As humans under God, we already knew that. But they are trying to show the math behind it. A circle is Gods shape of perfection. We repeat this kind of symbol with our recycling symbol. A circle is the symbol of infinite. Explaining it mathematically it technically impossible. What your witnessing is humans attempting, and making good sense of it mathematically. Humans have limits. Pi being an irrational (infinite) number will forever remain a proposal or theory.
Why did it take so long to prove that pi is irrational?
ANS: It is not simple to prove that pi is irrational.
To understand this proof it needs more. Showing f(0) is natural and the symmetry property is boring and more or less trivial.
The difficult part of this proof is the idea to choose f(x) like it was done.
Why this f(x) should help?
If u figure out this, then one has understood this proof. :)
But the big problem is following.
Mathematicians play around with stuff. Try this. Try that. At the end u get a result if you are lucky.
Problem?
The following paper u write is scientific and not "My thouggt was this, and that."
It's just:
Theorem: blabla
Proof.
The specific thoughts of the author are not given.
This makes it really hard to understand this proof.
On the other hand:
What makes this proof so difficult?
Showing Pi is not a ratio is difficult because u can't use a explicit definition of Pi.
Try to use the analytic definition:
Pi is the double of the smallest positive zero of the cosine function.
Good luck. That makes it difficult.
So u have to be smart and find something what will help u.
This is the chosen f(x).
The choose is genius.
@@easymathematik r/whoooosh
@@avikdas4055 What?
Yui
@@avikdas4055 You don't know the meaning of r/whoosh.
“It is not easy to prove that pi is irrational”
Thank you for your profound insight!
Last video : 2+2=4
Today video : prove that pi is irrational
Tomorrow. Proof for the Riemann hypothesis.
Tomorrow. Proof for pi is transcendental
Tomorrow: still 1+0 = 1
Day after tomorrow: Riemann's hypothesis proof
Lol 🤣
Then, wau
I love the irony that pi, a number that is most basically defined by a ratio, is irrational.
But isn't there an irrational number present in the ratio?
So it's not really in the p/q rational form...
@@namangaur1551 Yes, you're precisely correct in a "whoosh!" kind of way. Pi is defined by a ratio, but that ratio doesn't fulfill the requirements to guarantee that it expresses a rational number and therefore there's no contradiction in the fact that pi is irrational. But the key detail you highlighted isn't explicit in the *construction* of the word "irrational", which in itself only says "not a ratio" - that discrepancy between construction/apparent meaning and formal/actual meaning is the source of the "irony".
The same could be said of φ, the Golden Ratio. Its construction by line segments used to go by the name, "mean and extreme ratio."
Fred
Generally mathematicians defined pi as the periodicity of the exponential function, but I take your point.
@@StoicTheGeek Makes sense, but that's a pretty modern redefinition. Euler's number and the natural exponential function are much more recent discoveries than pi, and for the exponential function to have a periodicity the exponent must have an imaginary part, so it's not even just about using Euler's number as a base. It always seems odd to me when mathematicians claim the most basic definition of an ancient and simple concept is to derive it from much more recent and sophisticated ideas.
Yes, we get it... you went to Stanford
Yeah but, did you know about this textbook that i took in Stanford?
😂😂😂
I too went to Stanford... taqueria.
Let him, he never boasted about it in his earlier videos. Man deserves a self plug.
@@NorthernlionLP dude, it's so obvious that he is proud of himself in almost every video
Throughout my life I have always known that π was irrational but have never seen a proof for it. And I ALWAYS tell my students to never just accept anything from their teachers - always ask them to prove or justify statements such as
- Area of circle = π r²
- Volume of sphere = 4/3 π r³
So finally I have seen the proof I should have looked for all of those years ago.
It was well presented and easy to follow. It's also got a decent bit of mathematical meat to it.
Thanks Presh! I really enjoyed this video.
How come your comment is 2 days old but the video came 5 hours ago...?
The vid was unlisted I think
That's a problem with maths. If you want to teach it rigorously then, e.g. in case of calculus, you need to start from the axioms for real numbers and work from there. After "some" time, you can derive those relations you mentioned (via integration).
Do u teach 5th graders or what? A teacher is expected to know this proof . I'd be really disappointed if my teacher didn't know this.
I can bet that merely 1 % teachers in my city knows why π is irrational
They run over marks and exams
2:22 I just want to say that many things are not easy to prove, even though they might have a simple proof.
Yeah
@Suani Avila of course it does. Some things might be extremely hard to prove by yourself, but some genius mathematecian might have done a very creative solution which is easy to do if you know it, but veery hard to think on your own.
there is a difference between hard and complex after all
For those wondering how one would ever come up with this proof and how one would come up with the definition of, think of a function that is equal to 0 at x = 0 and x = a/b = π. With the fundamental theorem of algebra, one can easily construct the function x(x - π) = x(x - a/b) as satisfying this property. One can multiply by b to get x(bx - a), and it still satisfies this property. Finally, one can multiply by -1 to get x(a - bx), which satifies this property still. How does one transition from x(a - bx) to [x(a - bx)]^n/n! logically? Well, if you sum the latter expression over all natural n, you get e^[x(a - bx)]. And as this is a well known expression, you know the associated infinite series converges. This already lends itself to the proof as presented in the video. The rest can be figured out by taking these things and exploring further
You could also have started with x(π - x) = x(a/b - x), then multiply by b to get x(a - bx)
The reason is because when 0 < x < π, then π-x > 0 and x(π-x) > 0.
So better to start off with a function that is positive for all values of x in the interval (0, π).
this is the part I needed explained
Why did it take so long to prove π is irrational?
*It is not easy to prove that π is irrational*
Thanks
Yet, the proof is simple.
Vedanth Mohan it is if you know what irrational means
@@Maxence1402a yeah, but you need integrals for the simple proof, and integrals were really hard to do before the 17th century (like it's so much easier now :D )
@Work is worship put * at the beginning and end of your sentence
To understand this proof it needs more. Showing f(0) is natural and the symmetry property is boring and more or less trivial.
The difficult part of this proof is the idea to choose f(x) like it was done.
Why this f(x) should help?
If u figure out this, then one has understood this proof. :)
But the big problem is following.
Mathematicians play around with stuff. Try this. Try that. At the end u get a result if you are lucky.
Problem?
The following paper u write is scientific and not "My thouggt was this, and that."
It's just:
Theorem: blabla
Proof.
The specific thoughts of the author are not given.
This makes it really hard to understand this proof.
On the other hand:
What makes this proof so difficult?
Showing Pi is not a ratio is difficult because u can't use a explicit definition of Pi.
Try to use the analytic definition:
Pi is the double of the smallest positive zero of the cosine function.
Good luck. That makes it difficult.
So u have to be smart and find something what will help u.
This is the chosen f(x).
The choose is genius.
So next year (2020) the symbol for pi will be 314 years old!
TheSpecialistGamerX2 No, super pi day was 3/14/1592
Or alternatively it will be in 3141 on September 15, but I don't believe anyone will be celebrating it at that point
@@TheRealFlenuan there are only 12 months bruh.....3/14 aint possible
@@xcarnage8632 the dates are in MM/DD/YYYY
@@HarshSharma-wj8mc well the conventional format is DD/MM/YYYY
@@TheRealFlenuan pi wasn't discovered until 1706...
As Boromir once said, "One does not simply prove that PI is irrational."
Karl Speth As Euler once said: "One does not simply walk into Mordor eating pie."
@@cmarley314 LOL that too.
@@cmarley314 Or was that "You'd better not walk into Mordor with the One, sweetie pie"?
The difficult thing here is not the proof itself but where did it come from and why does it work.
That is not answering the question.@@grottjam
@@grottjam the goal of the proof was to create a function for the integers that make up rational pi that has nice cancellation and differention properties. Naturally the mathematician went with polynomial functions for the former and trigonometric for the latter. The choice for the polynomial wasn't exactly elegant, but probably done meticulously so through trial and error to make the cancellation work.
When doing proofs for irrationality it's typically easiest to assume the opposite and arrive at a contradiction, because rational numbers have a very simple but essential rule baked into the definition, and irrational numbers being the compliment of rationals therefore have a very straightforward definition as well. It's likely that the proof writer realized that their chosen polynomial would be all integers, or all positive, or something. With that being the case, they simply have to find a property that shows that it, and it's contradiction, are both true, which is what they did by finding it's upper bound (a common tool in real analysis)
This is slander, Pi is the most rational guy I know.
Underrated comment imo
Pi is pretty unstable though
I loved his stripes! 💕
On another note, can a troll have a TH-cam channel? 🤔
Sometime is chill like a CIRCLE ... sometimes act like a SERIES killer
@@henryalexander9152 Also bipolar, he can be Pi+ and Pi-.
There's a much easier way to prove (pi^(n+1)*a^n)/n! goes to 0 as n approaches infinity. You're dividing an exponential function by a factorial. The factorial goes to infinity faster than the exponential. Note that pi^(n+1)*a^n = pi*(pi*a)^n = pi*Q^n. Once n becomes greater than Q, n! will increase faster.
No need to go anywhere near Taylor series.
It was a brilliant exposition of the proof! Thanks for your dedication.
Happy Pi's Day :D
Damn...
The "for sale: baby shoes, never worn" hit me hard....
Then try this one: Six Zombies, Five Bullets, Two Zombies.
Zombies lives don't matter, don't care
@@MK-wm9zi I believe you missed the main point of the story, it is not about the zombies
@@BitcoinMotorist I didn't get it either. Would you mind telling me?
@@yourlordandsaviouryeesusbe2998 The guy with the bullets becomes a zombie...
Too high mathematics for me once again lol. Maybe in few years...
I am gonna learn these this year..am so excited!!
Geometry dash player on a math video... Suiting...
-Fellow GD player
Don't worry you'll get there just stick to it and keep learning
few years? what age/grade are you now then?
Achelois Nonce
Engineer be like
π=3
π^2=g
lol
Pi squared is suspiciously close to g tho
g = 9,81 m/s^-2
pi^2 around 9,86
π = π / 1 rational
In engineering π≈e≈3.
I'm having an irrational impulse to eat some pi.
I thought that many people are lost from the beginning because of functions f(x) and G(x) that fall from the sky. So I rewrited it for the Applied Math type.
(1) If π is rational, i.e. =a/b, then bπ is integer (=a) and so is any polynomial formula of the type b^n(c₀π^n+...+c_n) with integer coefficients c. So we are looking for a function that solves to this shape, and which can be proved to NOT being an integer. Best candidaites : functions greater than zero and less than one.
(2) The use of an integral ∫f(x)sin x dx allows to work with derivatives instead of primitives (=antiderivatives). Trig function is also hinted at by the problem, which concerns π.
(3) We want a function that zeroes on 0 and on π, since the sinuses are zero and the cosinuses are ±1. The first that comes to mind is f(x)=x(π - x). But its "sine integral" is not less than one (it is equal to 4). Reminding the series expansion of exponential (or equivalently, remembering the term P(n) in a Poisson probability distribution) we know that : z^n/n! tends to zero when n is large. So let us define our f(x) as [ x(π - x) ]^n/n!. (One should write f_sub_n, but here it would be too heavy.)
(4) Integrating any f(x)sin x is easily done by parts (repetitively). It is actually F(x) =
- f(x) cos x - f'(x) sin x + f''(x) cos x + and so on.
Now as we integrate from 0 to π, the sines disappear (equal to zero) and the cosines alternate signs... hence in F(π) - F(0) they actually give the same sign to both terms.
The integral from 0 to π is consequently :
- [ f(π)+f(0) ] + [ f''(π) + f''(0) ] - [ f⁽⁴⁾(π)+ f⁽⁴⁾(0) ] + etc. (even derivatives)
(5) What about those derivatives? Either you expand the [ x(π - x) ]^n terms and derivate the polynomials with binomial coefficients, like in the video. or you derivate the factorized form and get only [ x(π - x) ]^m terms for the n-1 first derivatives ; those same plus one [ (π - 2x) ]^m term for'the (n)th to (2n)th ; and zero afterwards. ALL HAVE INTEGER COEFFICIENTS.
Meaning that at 0 and π , the first ones disappear and you are left with terms in [ (π - 2x) ]^m = π^m or (-π)^m.
But remember, we were left with only the even derivatives! So both terms are equal and positive.
(6) Result: the desired integral results in a polynomial in π with integer coefficients and only even powers from n (or n+1 if n is odd) to 2n.
Please note that the coefficients can be positive or negative.
(7) On the other hand it is easy to show that the integral must be larger than zero (all interior values of f and sin are positive) and, as we required, can be made arbitrarily small by increasing n.
NOTE THAT THIS IS GENERAL RESULTS, VALID WHATEVER THE NATURE OF π.
Now for the proof.
Posit π = a / b, positive integers.
By (1) we know that b^n times any integer-coefficient polynomial of degree n in π must be an integer. That is precisely the result of b^n times the integral.
So it must be larger than zero and it can be made arbtrarily small (same Poisson argument).
An integer between zero and epsilon ==> Contradiction.
No offense, but pi=22/7
(This comment was brought to you by the engineering gang)
But it doesn't
OMG, I haven't seen that ratio in a very long time. Brings back old memories. These days, all I see is how much digits of Pi you can regurgitate.
Its an approximation
335/113
π = √g
Reads Liu Hui's estimate
Me: carries on with the 100 digits of pi song
Thank you for making this profound fact accessible to almost a third of a million of people! You did a fantastic job explaining this! I was happy after watching the video, thinking I understood it, until I realized that I had missed one detail: To be valid, the proof must actually use the assumption that a/b is the ratio of the circumference and the diameter of a circle. Without that, it would just prove that we made a mistake somewhere. It's not mentioned explicitly in the video where that happens. As far as I understand, the point where we use it is at 7:46 in the video, where we assume that sin(a/b) = 0 and cos(a/b) = -1.
Yes, I also noted that this was the only time in the argument when a/b is assumed to PI. It made me wonder how this could be generalized a bit farther.
okay that is strange how I don't even remember writing this a year ago.
This was not the simple proof I was looking for 😅
A great and clear proof!!! I can understand 85% when I listen to it for the first time!!! and of course, I listen to it serval times!!!
Please please please video proofs for (1) pi^2 is irrational (2) pi is transcendental (3) e is transcendental.
You are an excellent professor!!!
Mathlogger has a video on it th-cam.com/video/WyoH_vgiqXM/w-d-xo.html
My only comment is that I think the function f(x) should be denoted as f_n(x) {the n being a subscript), or f(n,x) to show that your f(x) is a function of both x and n. Other than that minor detail, great video.
You missed Bhaskaracharya
Do you know the indian formula of calculation of pi
It is
(12^1/2)×[(1÷(3×3))×(1÷(3×5))] and so on
If u can understand the later on series
@Samurai Jack right bro
HIMESH VIEWS Correct formula is
12^(1/2)×(1-1/(3×3)+1/(3^2×5)-1/(3^3×7)+...)
@@dudz1978 yes you are right too
Well, Indians already estimated it in a 700 pages book
Called Salbatroos
Thanks for the correctuon of tge fault
@@Tianzii2k4 subscribe to pewdiepie though
When you realise that John Courant's Introduction to Calculus and Analysis proved this on page 29/30 with simple algebra
Lies
I was randomly laying in my bed when the thought occurred to me “how the hell did we prove that pi is irrational” but I haven’t learned calculus yet and don’t understand any of this whatsoever
Did you figure it out? 🤔
Did YOU figure it out?
@@NorthernlionLP did YOU figure it out?
@@dissmo706 Did you figure it out?
@I love Gaming Did you figure it out?
Did you figure it out ?
I like your timing. After you say something that might warrant some additional thought, you give that pause to compensate
You know what pizza exactly means.
The volume of a solid cylinder having radius z and height a
pi×z×z×h
Wait, didn't pizza mean the area of a circle with diameter Z? (Unlike American deep-dish pizzas, real traditional pizzas are flat; they aren't supposed to have "height".)
Pi*Z*Z/4
I wonder who would dislike this video.
Maybe it was the Pythagoreans
i might as i don't understand all this and he says it's not hard too often
Lvl 1 guy with nothing special
Gets killed by random cult for no reason
lvl 100 mathematician
Drowned in the sea by the Pythagoreans for demonstrating that √2 is not a rational number
That's how mafia works
Dude went to Stanford and started a TH-cam channel. I love this community.
My boy be flexing that he from Stanford!
1:45 We do what we say , we really summarised it ☠️
"Simple Proof" _11 minutes_
And leaves a lot of details for us to complete.
If he wanted to present the simple proof it could take 2 minutes. Most things that are difficult to prove don't have "simple proofs" that you can explain to a lay audience.
Proving Fermat's Last theorem:
(1) notice, if x^n+y^n=z^n for positive integers x,y,z and n>2, we can form an elliptic curve that is not modular
(2) It also follows from the axioms that all elliptic curves are modular
(3) this is a contradiction, proving Fermat's last theorem
*details left to the reader
@@MK-13337 _Footnote: related to this problem is the Birch and Swinnerton-Dyer conjecture, proof of it is left to the reader, as it is considered trivial._
Well I think that's short. You must not have a good attention span.
@@gogl0l386 Footnote 2: the Taniyama-Shimura conjecture is a trivial corollary of the above.
0 < integral x^8(1-x)^8*(25+816*x^2)/(3164*(1+x^2)) from x= 0 to 1 = 355/113 - pi, So 355/133 > pi and 355/133 is closer to pi than 22/7 is :-).
In fact, if you replace 3164(1+x^2) with 3164(1+0^2) and again by 3164(1+1^2) and integrate between x= 0 to 1 and compare these results to the integral above, then we find that 3.14159274 > pi > 3.14159257. Note: pi = 3.141593 ( 6 decimal places).
Or 223/71
But Irrelevant...
This is Because In the Riemann Paradox and Sphere Geometry System Incorporated π = 2 (Whole Number)
Correction: proving that π is irrational its highly not trivial
You forgot to mention Zu chongzhi who proved π is between 3.1415926 and 3.1415927
This proof is majestic. Thank you Ivan Niven. ^^
i remember in algebra in 8th grade i got into an argument with the teacher when i claimed it was impossible for both the circumference and diameter of a circle to be rational numbers. same guy i forced to call the high school teacher when he claimed there was no such number as i when he asked us if we could use the same method we used to simplify x^2-1 to (x+1)(x-1) and i said yes.
Happy π day to everyone
Proof Kit for Pi Irrationality. Some assembly required.
Archimedes and Pythagoras have left the chat
Btw im Greek and im proud of them
Good. But don't let that pride grow into arrogance. Learn, grow and become something your ancestors would be proud of in return.
But in india they r not proud of their ancestors
@@technicalgamers7324 why not
Nice job, Presh. It was not too hard to stop the video a few times and fill in the proofs of the pieces, and it was fun.
It left me wondering, though, just why it goes so wrong. You do a bunch of simple calculus stuff that you should be able to do with any old numbers and then voila--a contradiction! Where would the proof blow up if we tried to tun through these calculations with a rational number a/b which is extremely close to pi? Because of course it would blow up, that's the point of the proof. Perhaps someday I'll try to follow that through and see what happens.... MORAL: Proofs by contradiction are often a bit unsatisfying, because they don't always illuminate the underlying mathematical relationships.
BUT, the video wasn't unsatisfying. The video was perfect. Thanks again.
The key is that sin(pi) = 0 and cos(pi) = -1. This was used in finding the definite integral to be an integer, but it wouldn't be true if we replaced pi with a rational number, not even one very close to pi such as 355/113.
The proof actually starts at 3:54
Pi is irrational, because it is simply measuring ANY circle's diameter around its circumference. ANY circle, means ANY length of the diameter. This means ANY number, however big. The bigger the number, the more decimal places it will result for the pi. So the only way we can have a final value for pi, is if we figure out the biggest number possible, which of course is non-existent. Infinity exists, so a circle with an infinite diameter can exist, and putting that round it's circumference will give an answer with 3. + infinite values of decimals numbers. That is the reason why Pi is an irrational number.
dude what
Non sequitur.
Fred
Hi Presh, I’m wondering, whilst the modern study of Pi has progressed somewhat beyond the Archimedes Method, does the Method actually quite neatly prove that Pi is irrational? If we think in terms of polygons with ever-increasing numbers of sides, we also polygons with ever-increasing (assuming inscribed polygons) perimeters, and thus an ever-changing precise ratio to the widest measurement of the polygon (or circle diameter, again assuming inscribed polygons) - I suggest that the simple fact that the ratio (specifically in relation to a polygon) will never exactly stabilise (but only ever approximate) is pretty good proof that the ratio is irrational, and thus that Pi (as the value of the precise ratio relating to a circle, or to a polygon of infinite sides) is irrational.
Interesting. But if I take a 90 degree step size I get an approximation of pi of 4*sqrt(1/2), which is already irrational (2.82...). As a counter example: the sum of an infinite amount of fractions can be fully integer too. Like 1/1 + 1/2 + 1/4 + 1/8 + .... == 2. So why can the sum of an infinite amount of smaller and smaller fractions not be a fraction as well?
You can get REALLY close to rational, though... If you have a circle with a diameter of 113 (whatever unit), the circumference will be 355 (whatever unit)... 355/113 = Pi (at least up to a few decimal places) - but, being more exact, the circumference of such a circle would actually be 354.99997.
I think you used another definition for the term "simple", not the one most of us are accustomed to...
Thank you for not keeping this in my school syllabus
It's spelled syllabalubulus
Very nice! One of the best videos on this channel from which I learned something I had never seen a proof of until now.
This reminds me of how in high school I learned the difference between deduction and induction on accident. My one friend told me that you can't prove that pi repeats forever because you would have to actually find a last digit so you can only disprove it but never prove it (like how you can disprove a theory but can never prove a theory; inductive reasoning and the scientific method). I repeated that to my math teacher and she said no we know it's infinite because we can simply prove that pi is irrational ( deductive reasoning)
My mind was blown
I wonder why you named this proof simple but you didn't name your : 6/3×(5-2) simple
It wasnt him who named it simple, it was the author of the original paper. And indeed it is simple, if compared to other known proofs which generaly use several tools from mathematical analysis or abstract algebra. This one is simple in the sense that it is short and only uses basic calculus. It is however very hard to come up with, so much so that it was only found in the 40s.
I thought you meant it when you said it would be *simple*
For those who didn’t know what happened to the baby it got abducted by aliens and the parents had already bought the baby shoes, so they decided it would be worth selling the baby shoes, because there was other ways of remembering their child , and they needed some money. Honestly a complete tragedy… I am in tears rn
Zu Chongzhi's best approximation of pi was between 3.1415926 and 3.1415927, which is accurate to 7 decimal places.
The result is better than Liu Hui
π=π/1
It's written in a/b form
**Mind blown**
Then you have to prove that numerator ( here π) is also an integer.
We want integers
@@arghadubey9509 I know it was a joke... 🙂
A very high level of degree level maths understanding is required to under this video...✌️
3:46 Started
>Simple
"Oh cool, this should be good... looks simple so far..."
4:13
>:|
That wasn't even the hard part
Yeah...where on earth did that first function come from???
@@ErikBongers The guy who did the proof surely spent several weeks playing around with several functions to come up with that. This is the kind of proof that looks like magic upon completion because you are not following the full thought process of the creator. The guy no doubt made several falty attempts before arriving at that.
Erik Bongers The function has it so that if x = a/b = π, then f(x) = 0, and dividing by n! makes it suitable for taking derivatives because it cancels out the factor. Those are two very desirable properties, and it's easy to construct the function from those properties alone. Hence this gives you a very elementary reason to work with this function.
Surprising simple proof for a problem most think is not easy
Why everyone pronounce it pi (πάι)? In Greece, we pronounce it pe (πι)
Most european languages pronounce the letter “i”, i.
The english say it’s the letter aï.
All comes down to their alphabet.
@@vincemarenger7122 Yes, but that doesn't change the way it should be pronounced.
I learnt in school in 9th grade. But I have always found it difficult to agree because pi by defination is a ratio
Pretty tough... but I guess I can understand most of it... faintly... I'm sure I'll forget this proof after I sleep :(
Completely lacking in the source of the equations. "if you just pull this equation out of your butt do some figuring about it then test it for assumed rational pi" was what I got from that.
Absolutely. A prove must be complete or is not a prove at all.
For those who say that presh's videos aren't tough.😁😁
Till not tough.
Imagine if he presents complicated proofs of pi instead. Remember, this is just a simple proof!
@@HeyKevinYT I have seen complicated ones. And he doesn't deserve to present those complicated proofs if he can't solve that Einstier riddle or than Hardest Australian highschool prob
I may sound rude here, but I used to love his videos and became a big fan of him. Now a days he is just using clickbaits to popularize his channel. His channel ain't anymore about strategical combinatorics and game theory😞
Look for 3 blue 1 brown
And the π definition stated there will blow your miiiinnnnddddddd
Presh: Did you understand?
Me: 😕
Presh: which part don't you understand?
Me: From last semester
Irrational numbers have far more real life importance than rational numbers..
3.14
There are far more irrationals than rationals, but that doesn't make them "More important". In some very real, physical way, irrational numbers exist only as mathematical concepts.
I am a math teacher from palestine i am fond of pi & e since long time thanks a lot for nice proof
I actually understood nothing 😅
“It is not simple to prove that Pi is irrational.”
TI-84: “You underestimate my power.”
That Indian infinite series to represent pi. That was the exact value of it. That series is used to estimate the value of pi. I still remember it.
you mean ramanujan's series?
@@johanliebert6734 Many more Indian scientists were there other than Ramanujan. Although Ramanujan's series are more famous.
One was Madhava of Sangamagrama who proposed the value of π as:
4×{1 - 1/3 + 1/5 - 1/7 +...}
He was of 14th century and Ramanujan was of 20th😱😱😱
@@AAAAAA-gj2di That's the leibnitz series for pi.
@@gregorykafanelis5093 So isn't it infinite. 😆😆😆
@@AAAAAA-gj2di It's still an infinite series but Leibniz used the results of that Indian mathematician to approximate pi. That's the reason that both of their names are used when taking about it. But as with many discoveries, the result carries the name of the one that found it and in this particular case leibniz used the infinite series of Madhava for the inverse tangent to give us this beautiful result.
Thank you so much. Im studying number theory, and this is gonna be very important for me.
Couldn't be simpler... :p
we did this in ninth grade and it was really simple back then but now when i see this i think i might have to spend months just to get a taste of it.
The product of three prime numbers is equal to 11 times their sum.
What are these three numbers?
(11,13,2) and (11,7,3) with all permutations.
5,2,2.
Also, if you'll allow numbers that aren't prime, then 14, 1, and 2 are also acceptable. No other solutions exist.
david21686 5 * 2 * 2 = 20, but 11 * (5 + 2 + 2) = 99
Xall Universe missing (11,4,5)
Paul Gagneur 4 is not a prime number
me: Now I will prove pi!
others: That's irrational!
Happy pi day 😘😘😘🎊🎊🎊🎊🎊
Pi is irrational ... the proof is left to the reader.
Reminds me of a joke:
A math professor is teaching a class. He's in the middle of a proof, and, referring to a complicated expression, says, "It is intuitively obvious that this is an integer." Then he frowns, looks at his notes, looks at the board, looks back at his notes. He steps to the side, and starts scribbling unreadable shorthand equations in the corner of the board, scratching his head. After five minutes of this, he switches to writing in pencil on his notes. The class is mystified. Another ten minutes go by, with him alternating between writing furiously on the paper and staring intently at what he'd written. Shortly before the class is scheduled to end, the professor suddenly looks up and says, "Aha! Yes, I was right. It *is* intuitively obvious that this is an integer."
Joke???!!!???
The joke is that if it is intuitively obvious, you wouldn't have to go through the process of figuring out the proof to feel confident enough to make that statement.
The argument that pi^(n+1) a^n / n! -> 0 was just awful. It's a circular argument. The domination of exponential growth by factorial growth is used to prove the convergence of the exponential Taylor series! Anyway, there are far more intuitive ways to explain this behavior!
toay is π day 3.14 the 14th of march ,My maths teacher told me this 5 sec ago
Oh yess
So, you thought to write this comment, took out your phone, opened TH-cam, opened this video and wrote this comment. All in 5 seconds
@@luigiboy72 Maybe they watched the video with the math teacher together! Awesome
We can take a unit vector from the origin (0,0) to (1,0) and we can rotate it 180 degrees or PI radians so that the point (1,0) is transformed through rotation to the point (-1,0). The length or magnitude of the vector is 1. The overall distance from (-1,0) to (1,0) is 2. The Arc length that is generated is PI radians. How many unit vectors or line segments of length 1 are there that make up this arc during the rotation between the points (1,0) and (-1,0)? There's an infinite amount. This is similar to asking the question: how many Real numbers are there between the interval [0,1]? We know that some of those values are rational but we also know that the vast majority of them are irrational. There is a very high probability due to the infinite complexity that it would highly suggest that PI is irrational. This is not meant to be a direct nor an exact proof. Yet, I would claim that this would be the simplest possible proof that there is to demonstrate the reasoning that PI is irrational. Well, it's more of a supposition, more of a conjecture than an actual proof.
Here's an example to support this. The surface of the earth is approximately 70% water and 30% land mass. You have a much higher probability of landing in water than on the ground if you were randomly dropped from the sky. The ratio of Irrationals compared to that of rationals within the Real Number domain is much higher than this 7:3 ratio. So to randomly drop a constant value on the real number line has a much higher probability of being irrational than it does being rational. This is just a way of thinking outside of the box. What more proof do you need? If you keep trying to search and divide into an an infinite domain you will never stop dividing nor will you ever stop diving. Sometimes having a relatively close enough answer is good enough! And I think this was simple enough to explain. Just food for thought.
where did you get ther original f(x) function from? seems like pulling thngs out of thing air!
you derive it by looking at what properties you want it to have
The answer is simple in pne sense. And sounds silly on the other hand.
This f(x) is chosen for purpose.
Problem: How to find this?
This proof is just genius.
I can see why you published this video now. :D best day of the year
I can not understand anything.......... Maybe in future i would.!!😌😌😌😌😌👍
I thought you were going to tell us that Chinese third graders had to understand this proof.
Wow! A video kept secret woah
Well from whatever u said I only understood that π is irrational and nothing else
Damn, if this is simple, I can't even imagine what's complicated for you.
Anyone else notice what day this was uploaded?
Sòoooooo easy.....wait.What was the video
Again??????
Just saved it until I'm mature enough to understand what happened.
Humans tangibly deciphered how to actualize, mathematically, the seemingly perfect thing that is a circle. A perfect circle has no angles or bases or true shape. Its 'infinite'. As humans under God, we already knew that. But they are trying to show the math behind it. A circle is Gods shape of perfection. We repeat this kind of symbol with our recycling symbol. A circle is the symbol of infinite. Explaining it mathematically it technically impossible. What your witnessing is humans attempting, and making good sense of it mathematically. Humans have limits. Pi being an irrational (infinite) number will forever remain a proposal or theory.
But you should know he went to Stanford cuz wow it’s in his bio. Congrats STANFORD GRAD. Enjoy your debt.
Pi is irrational but it is defined as a ratio of the circumference and diameter.
Yes, so? At least one of the two is irrational...
Not so simple?😉
That's why Pi is still single.
Yeah real simple, lol