How is π calculated to trillions of digits?

Рет қаралды 122,605

Күн бұрын

For record-breaking π (pi) computations, the Chudnovsky algorithm is used. In this video, I go over what affects the performance of algorithms for π computation. Then, how to apply "binary splitting" to the Chudnovsky algorithm. Then, implementing the Chudnovsky algorithm in Python. Then, why is π calculated to many digits. I may go a little quick at times, so I encourage you to pause the video to verify an understanding of each step.
00:00 Intro
00:11 What makes a formula better to calculate π?
03:26 Optimizing the Chudnovsky algorithm
12:00 Implementing the Chudnovsky algorithm in Python
13:57 What's the point of calculating π to many digits?
14:43 Outro
This video is for the #SoME3
References and further reading:
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Пікірлер: 195

@JoshuaRayton 6 ай бұрын

I am aware that the audio is quiet. It sounded fine to me in production and I only noticed once I had uploaded it.

@IRedBerryI 6 ай бұрын

On mobile, it sounds perfectly fine to me…

@nguyeninhhongphuc6168 6 ай бұрын

I thought that was my earphones problem but now I know the reason lol

@OwenGalaxy 6 ай бұрын

it sounds fine… but please not that S is pronounced ess not ass, ass means something else and everytime you said that letter I heard ass.

@JohnDlugosz 6 ай бұрын

You should look at the Level meters, and not rely on how it sounds during editing, as that is affected by the volume levels of your speakers, room conditions, etc. If you re-upload, I'll watch it. As it is, I can't make out what you're saying.

@AbbeyRoad69147 6 ай бұрын

Why is this video identical to the Wikipedia?

@scoutgaming737 6 ай бұрын

Fun fact: 63 digits of pi is enough to calculate anything in the universe (If you consider the plank length to be the smallest you can measure)

@Roi_Babar 6 ай бұрын

Look like you didn't watch the video until the end.

@alansmithee419 6 ай бұрын

@@Roi_Babar The video contains a similar, but different, fact. Not the same one.

@Roi_Babar 6 ай бұрын

@@alansmithee419 yeah technically true I guess. Though it's directly related I believe. I think it's not stupid to say if you can measure the size of the observable universe up to plank precision then you are able to measure anything with that same amount of precision.

@alansmithee419 6 ай бұрын

@@Roi_Babar The example in the video said the number of digits they specified could calculate the circumference down to the width of the "smallest atom", nothing about Planck lengths.

@PhysicsMathandMe 5 ай бұрын

Yeah, the Planck length is **twenty-five** orders of magnitude smaller than a hydrogen atom. It’s interesting to know how much more precision we need to measure circumferences down to this scale.

@kakyoindonut3213 5 ай бұрын

Chudnovsky: "math has advanced, billion must calculate pi"

@pay0t-pay0t 8 ай бұрын

Great video! I recently wrote a rust program with Chudnovsky's algorithm and this could've come in handy a week earlier so I could comprehend what I was doing

@ukissrulez 5 ай бұрын

nope your program sux and prolly doesnt work!!

@KewlKris 5 ай бұрын

@@ukissrulezrekt!!!!!!!’

@ZOCCOK 5 ай бұрын

As a person who just learnt print in python, your program prolly sux

@felixherz2864 6 ай бұрын

2:24 Quadratic convergence means that doubling the amount of iterations quadruples the amount of correct digits. In general, order of convergence n means multiplying the iterations by m multiplies the amount of correct digits by m^n. Doubling the amount of correct digits each Iteration would be exponential convergence.

@MrofficialC 5 ай бұрын

I'm inclined to believe that the inverse is true

@oberonpanopticon 5 ай бұрын

From what little I understand of that, that seems like it’d be extremely convenient The opposite of the usual effect of seeing the word “exponential” describing an algorithm in my amateur programmer experience

@SystemfehlerK 5 ай бұрын

@@oberonpanopticon It is extremely convenient. It's often called "spectral convergence", and it usually involves the construction of a series of polynomials of increasing order.

@raphaelsimonin9717 5 ай бұрын

No. Its a common missunderstanding cause thr word quadratic makes you think about the number 2. A quadratic convergences does mean doubling the amout of correct digits by 2 at every iterations. What happens is that the difference betwen the approximation and the correct answer is squared at every iteration, and when you square 1/10^n you get 1/10^2n. I hope this can help you !

@raphaelsimonin9717 5 ай бұрын

Quadratic convergence does not mean the order of the convergence is 2.

@someone-wv3ds 6 ай бұрын

The video quality and montage is insane dude when I saw this video in my recommendation i thought that this channel is multi million subs but turns out its ≤ 200! You earned an instant sub I have 2 recommendations: 1- Like alot of people mentioned try to make the sound higher either by you talking loud or via sound editing softwares like audacity 2- I recommend using a black background instead of white for more comfort watching at night and it feels Comfortable on the eyes and personally I see it cooler and more professional Thats the 2 suggestion that I have other than that your contact is so fun to watch and professional

@someone-wv3ds 6 ай бұрын

Oh and one more recommendation try to make the video fill up the screen by making its height and width 1920x1080 because there are black spots up and down and a little on the right and left, the video isnt filling the screen

@ME0WMERE 6 ай бұрын

I disagree with 2, personally - it's a lot easier to read if it's black on white, and this is coming from someone who uses dark mode on everything. If you want to watch it at night, don't. Please sleep instead.

@someone-wv3ds 6 ай бұрын

@@ME0WMERE You are right :) But White color on black background gives the same result as black color on a white background, I think I should really sleep instead of watching youtube :)

@ME0WMERE 6 ай бұрын

@@someone-wv3ds maybe. I guess it could be more like white on grey, which would have the same effect but wouldn't be as harsh.

@convergence9990 5 ай бұрын

Billions must pi

@gobbleguk 5 ай бұрын

The algorithm has fallen

@warwolfsoad4270 5 ай бұрын

Incredible work, i dont have enough knowledge to understand everything but it was fun to watch, thanks for the video !

@ahsmeg4069 5 ай бұрын

I had a hard time following the math but got it as soon as you started implementing it in Python. Thanks for this video!

@obscurity3027 5 ай бұрын

I’ve been wondering about this for years! Awesome video!

@Magicraft13 6 ай бұрын

great video! can't wait to see more :)

@YouShisha1393 5 ай бұрын

But the sqr of 10005 is also an irration number and it should be calculated to the same precision as pi.

@alegitnolife 5 ай бұрын

man nobody cares abt the sqaure root of 10005

@Jo-rz6bs 5 ай бұрын

@@alegitnolifeit's one of the terms in the chudnowski one though

@shophaune2298 5 ай бұрын

I think the main difference is that the square root of 10005 is algebraic (by definition) so doesn't need such a fancy algorithm to find.

@zball_6396 4 ай бұрын

@@shophaune2298 You still need a fancy algorithm, just is actually very fast (square roots needs only newton iteration which can be computed in O(M(n)) time)

@dedu15 8 ай бұрын

Very well done and interesting video! 👌 One point You could improve upon is the audio volume, it’s a bit quiet..

@JoshuaRayton 8 ай бұрын

I agree, I only noticed the audio was quiet once I had uploaded it.

@newtzy16 8 ай бұрын

@@JoshuaRaytonskill issue

@vy_productions 6 ай бұрын

@@JoshuaRayton A good way to check the audio is okay in your editor is if you can see your volume mixer. Assuming you find it: if the average volume is between -6 dB and 0 dB then you're likely in a good place for audio, or at least those numbers are the range I've found to be good for KZfaq in the past.

@itsentdev 6 ай бұрын

shut up

@kenneth_romero 5 ай бұрын

when the audio quality is trash but the video is well formatted and edited, you know you're in for a doosy

@inspectormills3290 6 ай бұрын

Fascinating video. Thanks for sharing

@ZephyrysBaum 6 ай бұрын

43 secs ago lol! Lots of people seem to be watching this

@apotterlockian 5 ай бұрын

i loved it, and your animations! thank you!!

@guillermobarrio55 5 ай бұрын

Stunning vid. The optimization part was just hypnotic. Thanks so much.

@ZephyrysBaum 6 ай бұрын

This video is really good!

@girth8613 5 ай бұрын

As a current Calculus B student, this is fascinating and scary 😂

@1JWL 5 ай бұрын

Sometimes I find myself wondering, "Is there a mathematics model about any purely natural phenomena that stands without any weird magic in it?" I ask of this, concerning those "strange" constants embedded inside of the Chudnovsky Algorithm.. they make it look both obvious and mysterious at the same time. Cool lecture meanwhile 🤞😊

@lanye2708 6 ай бұрын

loved the first half of the video - you lost me about when you defined S. so many functions and variables that my eyes started glazing over. if you manage the spacing/timing a bit better i think you have great potential :)

@lexkoal8657 5 ай бұрын

Very good topic and presentation

@VarunGupta3009 5 ай бұрын

This was randomly recommended to me, and even though I didn't understand a whole lot, I watched the whole thing haha.

@frozenpeak1524 5 ай бұрын

This is such an amazing video. My understanding of math is just not sound enough to understand anything past the 4 minute mark though

@ingiford175 8 ай бұрын

Really liked this. But I was wondering if there is a good video on how the algorithm itself was derived.

@JoshuaRayton 8 ай бұрын

There likely won't be any videos deriving the Chudnovsky algorithm. Here is a proof of the Chudnovsky algorithm: arxiv.org/abs/1809.00533

@sinobou 5 ай бұрын

this video is awesome

@poop1124 4 ай бұрын

Great video. I suggest 'cobsons theorem for your next video.

@ukissrulez 5 ай бұрын

I like how “b” is pronounced… beei

@justincase5228 5 ай бұрын

Pretty sure the author of the video is a Neimoidian diplomat.

@primenumberbuster404 5 ай бұрын

I had to turn off all the electronic appliances of my home and draw down curtains to able to listen anything. It's a cool video but the sound issue needs to be addressed. Goof luck!

@xozu 7 ай бұрын

wait but how do we even know what the correct digits are?

@JoshuaRayton 7 ай бұрын

Any digits of π can be computed in base 16 (and converted to base 10) with the Bailey-Borwein-Plouffe formula. This computation doesn't require all the digits before to be calculated, making it efficient for checking if arbitrary digits are correct. It's assumed (and highly likely) that if the last few digits of a π computation are correct (match what the BBP formula gives) then all the digits before that are also correct.

@fullfungo4476 6 ай бұрын

You just calculate pi up to N digits. Then for any K

@theforeskinsnatcher373 5 ай бұрын

great video

@ResidentofVirginia 5 ай бұрын

Billions must do calculate

@tessenary_ 6 ай бұрын

great video! just raise the audio next time please

@creatorofimages7925 6 ай бұрын

Is it only me or is there almost no difference in volume change when I trie to up the volume? It's like he is mumbling and has additional sound supression.

@picrust314 5 ай бұрын

How do you create your animation of formulas?

@JoshuaRayton 5 ай бұрын

I made the animations in Adobe After Effects.

@user-iy6dt4xp5o 5 ай бұрын

I saw the thumbnail and was like "????"

@petterlarsson7257 5 ай бұрын

good video

@MatthisDayer 5 ай бұрын

Is this revealjs with solarized theme?

@ferdynandkiepski5026 5 ай бұрын

The biggest optimization to the churnovsky algorithm here would be not to use python. These computationally heavy algorithms are best implemented in C or C++ and if you want peak performance and are an expert even assembly could be used.

@StoicTheGeek 5 ай бұрын

I did laugh after he spent 10 minutes optimising the calculation and then said “let’s use python”. But I understand the thinking - it’s educational and python is easy to understand and well known. I’m sure there’s lots of computational trickery you can do in C, but that’s a different video.

@ferdynandkiepski5026 5 ай бұрын

@@StoicTheGeek what he actually did here was just to optimize the equation. I agree that optimizing the implementation is a completely different topic.

@cbuchner1 5 ай бұрын

Python uses highly optimized numeric libraries. No need to implement the core algorithm in C/C++ as most of the time is spent in the numeric libraries anyways.

@ghotifish1838 4 ай бұрын

@@cbuchner1yeah most of numpy is implemented in c/c++ anyways so python would just act as a wrapper for these lower level implementations anywayz

@ReasoningWellspring 5 ай бұрын

Awesome explanation of the maths and its simplification for doing it in python. You didn't seem to show how to display the list of digits. To print the list, you just need to add this after import decimal at the top. import decimal as * getcontext().prec = 1000 Then you just need to choose the amount of iterations you want to do at the print function.

@mathisnotforthefaintofheart 5 ай бұрын

I am just wondering, how did the Chudnovsky algorithm come to be to begin with?

@JoshuaRayton 5 ай бұрын

It was based on Ramanujan's π formulas.

@c.almeida6864 5 ай бұрын

The next question would then be "how did Ramanujan came up with his series?" to which one can easily answer: we have no clue.

@rodrigolaparva 8 ай бұрын

I cannot hear anything.... did you make a video wispering?

@krunotomicic 6 ай бұрын

it sounds fine to me, hmm

@aceae4210 6 ай бұрын

if you check stat's for nerds in the right click menu you can see that the "content loudness" is "-25.5dB", the normal video volume is around -10dB to -5dB it can reach -1dB to 0dB if it's a loud video so yeah it's not just you

@primenumberbuster404 5 ай бұрын

Math ASMR

@user-xw6ky8ob4l 5 ай бұрын

Whispering?

@Jonas-gm4my 5 ай бұрын

How do you know whether all the digits are correct.

@fullfungo4476 6 ай бұрын

Your video is super quiet.

@FINXainarskrastins 5 ай бұрын

Bro your audio is so low I got jumpscared by the midroll ad

@cobalius 5 ай бұрын

and instead of going for recursion straight away, we could use a generator. that would do the same but would probably more readable and controllable

@GeorgeGeorgalis 5 ай бұрын

Why do those constants have so few (and finite) significant digits?

@DonjiKong 5 ай бұрын

Wow, just wow… I’m speechless, this video was so beautiful. In the back of my head while watching, I was thinking, “this guy has millions of followers, this video is so fucking cool, I can’t wait to subscribe and check out all the other ones he has made now.” I’m absolutely shocked this is your first video and the video doesn’t have a million views, and you don’t have a million followers. Please make more!!! You are incredible. Keep it up 🦾

@warrendargusch5873 5 ай бұрын

Do you have a sound engineer?

@AlFasGD 5 ай бұрын

y-cruncher has not been explicitly mentioned anywhere and it really bothers me

@kkklkkl 6 ай бұрын

Can't hear you

@ShivamYadav-jo5bl 3 ай бұрын

can anyone tell me why did he add that at 7:35?? plz

@pppkenken6610 6 ай бұрын

ありがとうございます。√π

@joshmckinney6034 5 ай бұрын

Can someone please explain how do they know it gave correct digits? What are they comparing it to? Is there a formula or algorithm that is the "gold standard " for pi that is used to compare other possible formulas or algorithms?

@Banana5320 5 ай бұрын

So the reason why these give the correct digits is because they appear in equations which have pi as a solution one example of this is the leibnitz algorithm which comes from the Taylor series (a way of representing certain functions as an infinitely long polynomial) of arctan (Since arctan(1) =π/4 from geometry) this means that by performing an infinite amount of additions for arctan's Taylor series will yield you the exact value for pi you know that a correct digit once after a certain number of additions the digit stays the same and all later terms are too small to change it (since each term is successively smaller)

@zachansen8293 4 ай бұрын

1:00 no, computational complexity doesn't determine the speed of computation -- until you get to a large enough N. What "large enough N is" is variable. There are lots of times were higher computational complexity algorithms are MUCH faster than lower ones for seemingly "large" Ns

@columbus8myhw 5 ай бұрын

I think you misspoke when talking about quadratic order of convergence. It means the number of correct digits goes like n^2, not 2^n like you said. EDIT: Never mind - you were right, I'm wrong.

@zball_6396 5 ай бұрын

No, this is correct. Order of convergence is not the asymptotic power of digits after n steps, but the limit of next step digits divide last step digits.

@alegitnolife 5 ай бұрын

the chudnovsky algorithm looks like it was made by people who knew the real value of pi and badly reverse engineered it

@hoggif 6 ай бұрын

Very very low volume. :(

@fuxtnegrx 6 ай бұрын

Can't hear a thing!

@gregorymorse8423 6 ай бұрын

There is a famous spiggot function which can generate the nth digit of Pi in any number base. Not mentioning it or showing it is a shame. Using the size of the universe which is extraordinarily speculative isnt necessarily a good upper bound for practical accuracy. That estimate changes every few years and is subject to much debate.

@davidaugustofc2574 5 ай бұрын

The minimum size of the universe isn't subject to much debate, couldn't care less about the upper limit tho

@aurabozzi228 5 ай бұрын

Why does the Liebnitz formula get so many digits right after that erroneous 0?

@JoshuaRayton 5 ай бұрын

en.wikipedia.org/wiki/Leibniz_formula_for_%CF%80#Unusual_behaviour

@aryansinha4992 5 ай бұрын

My doubt is that how do they check if the digits are actually correct if nobody has ever calculated that far yet

@cbuchner1 5 ай бұрын

There are other algorithms to compute specific individual digits of Pi at arbitrary locations. One can use these to cross check digits at various positions. Not all of them, but enough to be fairly confident.

@aryansinha4992 5 ай бұрын

@@cbuchner1 ohh thanks for letting me know I'll see how they work

@davejoseph5615 5 ай бұрын

Now I even I would celebrate in rhymes unapt...

@devth125 5 ай бұрын

the sound is fine its the stupid commercials that blast in with full volume, blasting you out.

@cleitonfelipe2092 5 ай бұрын

My key takeaway here is that this video shows a lot of numbers

@Wilhelm-mg1jf 5 ай бұрын

Name of the piece?

@justincase5228 5 ай бұрын

J.S. Bach, Well-Tempered Clavier, Book 1, Prelude C Maj

@aeryluna 5 ай бұрын

Peak 3am content

@BigBoyAdvance 5 ай бұрын

The universe works in weird ways. Pi lies in many fundamental equations of physics. It's a number that cannot be defined in a finite amount of time. Yet the universe is able to do that instantly. Actions that would take forever happen instantly from the universe's perspective. It's a mystery that we will never understand.

@davidaugustofc2574 5 ай бұрын

I'm fairly sure the universe doesn't look at the entirety of Pi before deciding to do anything, since time and space have a resolution that's not infinite, then all circles and circular movements are gross approximations at the plank scale.

@vikenjl6257 5 ай бұрын

The way you say "b"

@oaardvqrk5965 5 ай бұрын

one day i will understand this... one day...

@heco. 5 ай бұрын

Absolutely!

@apotterlockian 5 ай бұрын

you got this!

@VirtualBrushStudio 5 ай бұрын

π² = ?

@christianmartin8751 5 ай бұрын

Excellent content, thank you. For what is of the quality of the container, for your next video, I would advise you several things : 1- Forget about music. Music does not add anything to math. Especially that some people cannot concentrate with music, and everyone has his personnal music that suits him. And in your case your voice is difficult to understand, therefore why add hurdles ? 2- Use a voice generator with a basic voice that has no accent. You can be understood with pain by English natives, but you just loose all non English natives. And there are plenty. 3- When you notice there is a problem with sound not being clear or loud enough, you can still delete, adjust, and then reload...

@user-xw6ky8ob4l 5 ай бұрын

Music for the Ears shuts the mind and its neurons . It is impossible to ascertain who, and how, type of Laptop used, by these series arrived at. How can we be sure that the trillion-digit number of Pi has been discovered? No way to test it. You have mentioned four types of series but what computer language has been used. Your editing skills are wanting.

@davidaugustofc2574 5 ай бұрын

I find it funny you felt the necessity to argue non-natives would have a harder time understanding his voice, and didn't even bother to ask any of them.

@ambiguousheadline8263 5 ай бұрын

1: music was fine, not too loud 2: his voice is clear 3: nothing is wrong with the volume

@garfungled7093 5 ай бұрын

@@ambiguousheadline8263 What? His voice was so quiet I had the volume on max in order to hear him correctly. Did you?

@ambiguousheadline8263 5 ай бұрын

@@garfungled7093 i could hear him just fine on 20% volume

@gabrielss_8913 5 ай бұрын

tinha como deixar a voz mais baixa n, ta alta dms

@user-nn9gl6rv3h 5 ай бұрын

Chud

@user-xw6ky8ob4l 5 ай бұрын

Chud or Chodo, which one ?

@gobbleguk 5 ай бұрын

Billions

@rebusd 5 ай бұрын

Your audio is too quiet fyi

@DeSlagen8 5 ай бұрын

You sound like you voiced Salad Fingers

@ensiehsafary7633 5 ай бұрын

Playing Bach's C major prelude over and over isn't a good idea .I couldn't watch to the end

@o_2731 6 ай бұрын

You start the video by optimising an algorithm you haven’t explained using a method you do not explain :/

@procactus9109 5 ай бұрын

I can't hear this, what's the point.... You ruined the video

@AlanTheBeast100 5 ай бұрын

Please work on your audio track before posting

@lisaalisa127 5 ай бұрын

LMAO 22/7

@anthony-wy1qc 5 ай бұрын

chud?

@yesmangaming9284 5 ай бұрын

It’s over

@fakkmorradi 5 ай бұрын

whhhhhhhhhhhhat

@MissiFull 5 ай бұрын

My question is how the fuck does someone create that formula (0:55)?

@dickdoesbackflip 5 ай бұрын

it's based on Ramanujan's π approximation, which he saw in his dreams (literally)

@augnix888 5 ай бұрын

@wagonfixer16 6 ай бұрын

After like 10 digits they just make them up at random anyways

@arthurcampos7290 5 ай бұрын

chud

@debblez 5 ай бұрын

why you talk like that

@robert-skibelo 5 ай бұрын

Playing music in the background is a very bad idea. For musically literate viewers it's an intolerable distraction from your interesting content. Sorry, unwatchable.