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.
@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.
@@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.
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).
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).
@@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
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.
@@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)
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.
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
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.
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._
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.
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?
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!!!
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.
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
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.
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).
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
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.
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.
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
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
@@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😞
@@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.
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.
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.
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 am not a mathematician but I do believe that trigonomic functions are derived from circles. So Nivan used these to prove that pi is irrational. Does this mean the trigonomic functions are irrational too? Just correct me if I am wrong. I hate arguments.
I am teaching my son about rational numbers. We understand that Pi is an irrational number. But thr book we are using says that (Negative) -Pi is a rational number. Could you please help up understand how that is true?
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.
Why does Ivan Niven define f(x, n) in this way? As in, if you were challenged to prove pi is irrational, which logical steps would you go through to suspect that such an f(x,n) would be a useful function to play with? I understand why sin(x) comes up as a factor, because that is a function closely linked to pi, but I don't understand where we pulled f from.
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.
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
@@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.
I understood and proved all steps, but there is something I don't get, why is it important to consider pi as the number in the proof?, I mean, I can replace pi for any other number and the proof stays the same, where is it essential to that a/b = pi ? can you explain me please?
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.
Hi, for some strange reason, I have to calculate pi upto a millionth decimal places in the shortest possible time. As always, I will be using the Ruby programming language. Can you make a video about calculating the infinite series of pi in the most efficient way?
"Proving Pi is irrational - What you never learnd in school" The idea of your video is nice. U can make it. Why not? But it is neccessary to write in the title "what you never learnd in school"? Come on. At least be fair and come up with some facts. First: I can show you school books where the irrationality of pi is a small topic. Not rigorous but it is part of the book. The proof itself is quite easy to follow, I accept that. But why you don`t explain other very important points? There you find your answer why it is not part in school. You start your video with quotes like "it is not easy...". Then you came up with "an easy proof for pi is irrational" in 1 page. Following the proof is easy. I accept that. The reason why it is actually not easy is following. Being irrational is defined by not rational. So there is a "not". Being rational is defined by "q is rational if it exist integers m and n, n not zero, such that m / n = q. So in the definition is the word "exist". This is the reason. A number p is irrational is equivalent to "for all integer ratios m / n we have: m / n != q". This makes it difficult. How you can check all ratios? So the main problem of beeing irrational is. The definition doesn`t say how you show this property. It is not constructive. Another point what makes this proof difficult is the definition of pi. All definitions of pi are not so "nice" in some sense. So the reason what makes this proof very difficult is to understand why on earth such a f(x) like it was chosen should help? This is the difficult part.
We are also told that pi is transcendental. Is there any simple proof of this? If there is, it will transcends the proof given here that pi is irrational, because if a number is transcendental, it must also be irrational. Transcendental numbers form a subset off irrational numbers.
Sou pesquisador Matemático e trago para este singelo canal; MindyourDecisions com sua apreciação e reconhecimento a Racionalização de Pi, sendo Racional e Irreversível, provando Cientificamente e Matematicamente; Pi, usado para representar a constante matemática mais conhecida: a razão entre a circunferência de um círculo e o seu diâmetro, é objeto de estudo há muito tempo, e continua despertando a curiosidade matemática. O primeiro cálculo foi feito por um grande pensador e matemático, Arquimedes de Siracusa ( 287 a.C - 212 a.C ) que para aproximar a área de um círculo, utilizou o "Teorema de Pitágoras", para encontrar as áreas de dois polígonos regulares. essa teoria é o ponto de início da qual parte a presente pesquisa. que tem por objetivo gerar conhecimentos para aplicação prática, facilitando e melhorando o aprendizado dos(as)alunos(as). Ao longo do tempo, inúmeros estudos tentaram desvendar esse valor, tendo como base inúmeras hipóteses, a persistência e contínuos esforços. Em um breve relato histórico, conheceremos mais sobre Pi, seus pesquisadores, métodos de estudo e cálculos realizados até hoje, agora também com o auxílio da tecnologia que diminuiu significativamente o tempo gasto nos cálculos, culminando com a demonstração prática deste enigmático número, comprovadamente Racional e Irreversível(Com Três inteiros e quinze centésimos finito depois da vírgula)(3,15). o Autor da obra "A ousadia do pi ser racional" Sr Sidney Silva.
What I do find interesting is, that as an engineer, maths, physics, computers, nature all agree on some intuitive level. But as we take these disciplines to the extremes they agree less and less. When we take maths to the extreme where Pi is never fully resolved. Computers cannot follow that rule, and pi becomes finite in precision. So there math and computers no longer agree. Then we bring in physics where the smallest time interval is Planck time, and you have a ruler calibrated from 0 to 30. as you move along the ruler, you cross the point 3.14... with infinite precision, no matter how slow you go, there will never be a time interval in physics where you were at the value of Pi exactly to infinite precision as in any point of time, you are either before it or after it and never on it. Suddenly physics do not agree with Pi's existence as it can never be found. Then nature do not use maths at all and nor does it use physics. Then the theoretical circle has a minimum circumference any real circle would strive to but never achieve. as you increase your precision, so does the irregularities in a real circle become a problem that increases its circumference to the point of Planck length where no finer measurement is possible and the circumference becomes the maximum realistic value. Like a road can have a specific length in kilometer precision. but if you measure it in meter intervals it becomes longer, and in mm intervals much longer etc. The circumference of England is determined by the resolution of measurement, the finer resolution the larger the circumference. Hence a real circle circumference can be larger than a theoretical circle to any degree, but not equal or smaller. So what is the ratio between the circumference at Planck resolution and the average radius now? We can say the average ratio will strive to Pi. but the real circle will have a rational ratio as its building blocks are quantize in physics. Suddenly nature do not agree with the physics, the computers or the maths as nature refuses to cut particles in infinitely small sections to make our maths work. Hence the number of Planck lengths in a circumference and in a radius are always integers. All these disciplines and reality is running of into different directions as we head to the extremes. Hence the proof of pi being irrational is pointless. Excuse the pun. This makes me think... if our tools are used in our quest to compute reality as we move to the extremes we find ourselves increasing in error and paradoxes while natures knows no error nor paradox. The only time our tools could reasonably solve problems are when we stay reasonably away from the extremes. Because its at the extremes where we find mathematicians, physicists, computer science and nature are each in their own reality.
