How Some Formulas for π Are Faster Than Others
HTML-код
- Опубликовано: 8 фев 2025
- PDF link if you want a more detailed explanation:
drive.google.c...
Visit our site to access all the PDF's:
dibeos.wordpre...
Our goal is to be the #1 math channel in the world. Please, give us your feedback, and help us achieve this ambitious dream.
---
Some great books for learning math or physics
www.amazon.com...
Need a VPN?
go.nordvpn.net...
🐦 Follow me on X: x.com/dibeoluca
📸 Follow me on Instagram: / lucadibeo
🧵 Follow me on Threads: www.threads.ne...
😎 Become a member to have exclusive access:
/ @dibeos
📈 Check out my Udemy courses (you may find something that interests you 😉): www.udemy.com/...
📊 Do you need a consultation on Math & Physics, or do you know somebody who does? I might be helpful! Our email: dibeos.contact@gmail.com
🥹 Consider supporting us on Patreon:
www.patreon.co...
![I.N "HALLUCINATION" | [Stray Kids : SKZ-PLAYER]](http://i.ytimg.com/vi/n5B5q1Hwt_U/mqdefault.jpg)
PDF link if you want a more detailed explanation:
drive.google.com/file/d/1RaMVsYpmJANzJnvaNh6feCCsvhvfZ2_S/view?usp=sharing
Buen video 👍🏻
But you didn't answer why some formulas are faster at converging, just that some are faster
@@anderskallberg7969 well, we did show exactly how some are faster. The reason is because the terms you choose at each iteration (after multiplied by the previous one) produces an error (last formula of the video) that is less than another formula’s error
@ well, I’m sorry that your expectations were not met. We really tried and thought it was the answer you are asking for. Next time we will be more precise
@ thanks, well at least now we know something you guys actually want so we can make a video on that 😅
@@dibeos i thought it was cool thank you for the video
what a shame, feel like i lost my time
I’m so happy to have found your channel! Subscribed!
Somebody used three dots. Other mathematicians: woah! what an invention!
@DadundddaD 😂😂😂 yeah, it was exactly how I imagined it
but second formula doesn't use square root computations, which have higher computational costs. So, in practical applications, we should compare amount of basic arithmetic operations (which can be performed by the processor)
@@Vovik-fz4tx Hm… yeah, good point. It’s a balance between operation cost and convergence speed
How does a square root have higher computations
@@lavyagaur6293 In order to compute a square root, you have to approximate the solution of the equation x*x - 2 = 0, using algorithms such as Newton's method or the bisection method. Those algorithms can take many steps to reach a good approximation of a square root, so we say they have a higher computational complexity.
@DaedalusCommunity thanks
@@lavyagaur6293 Calculating a square root is more complex and computationally expensive than basic operations such as addition, subtraction or multiplication due to the iterative nature of the algorithms used. Calculating square roots requires a more elaborate sequence of steps, often involving successive approximations or numerical methods. The same idea used in the video to calculate PI has to be performed multiple times each time you want to calculate a square root, making it one of the most computationally expensive mathematical operations.
I loved it. Thanks for making these videos.
More generally one could say that the irrational and trancedental nature of π makes it easier to be approximated from irrationals (e.g sqrt(2) ) rather than rationals , that's why one formula is better than the other.However,in the more efficient formula one has to approximate irrational square roots ,which are taken for granted in the scope of calculating π and in practice they aren't and they need approximations themselves.Don't know which of these is more efficient after taking this into account
That was fun. Are there other approximations that are even more efficient?
What I would love to see is the thought process behind discovering these infinite series. Were they intending to find this? Were they just messing around and happened to notice the usefulness? Is there a best process or even a formula for finding these kinds of approximations? What do these approximations tell us about pi?
@@someguy-k2h The most efficient formula for approximating π is probably the Chudnovsky formula, but I’m not really sure. Historically, the first formula was discovered using nested radicals when exploring infinite series and geometric relationships in polygons inscribed in circles. The second formula was derived from insights in calculus and infinite series, starting with integrals for sine functions and their relationships. We have detailed proofs in the PDF link. If you have a good understanding of Calculus 1, it should be enough to study them. But yeah, both were methodical: Viete (first formula) worked from geometric principles, while Wallis looked for patterns in integrals. They didn’t directly aim for π but sort of “stumbled” upon it
@@dibeos Thank you so much for replying, and quickly. I know the Chudnovsky algorithm, from the Chudnovsky brothers. It's based on the famous Ramanujan-Sato series. I can easily see how a nested sum of increasing polygons can get you to π, but it's very interesting to me that this number which is used to describe the curvature of space-time pops up in so many strange places. I will read the PDF. Thank you for the response and the detailed content. You guys got my subscription.
@ thanks for the new videos ideas haha
Ramanujan tho...
Interesting video that deserves more views
@@ssdegfteghytr166 thanks!!! Please let us know what kind of videos you’d like to see in the channel
Good video, however I want to make a clarification, as a student in computer science, I can say that the conclusion you reached is quite incorrect, requiring fewer iterations does not imply that it is more efficient, since calculating square roots is much more computationally expensive than multiplying. So although it seems like many more operations, for a computer probably (the experiment has to be performed, so I cannot fully affirm it) the second formula is faster and more precise.
Except for that small detail, the rest of the video seems extremely interesting to me.
@Druxorey thanks for the clarification :)
I been using the Leibnitz formula which was actually discovered by someone else. If you take the products of each iteration and add them together in triangle fashion you will get pi. I have calculated pi to 26 digits using this method. You can start at any number and end at any number. The father in you start the more powerful your formula will be. If you add them together in triangle fashion you will get a number very close to pi. But no matter how far you calculate you will never reach the end. It's fun to do in your spare time. I would like to know more about the new method discovered by the string theorists. I see the formula but I don't know how to work it yet.
@@joshuawhitworth6456 wow, it sounds really cool. I’ll search more about it
Second comment, and I am thankful for your efforts in making these great videos, and I highly love pi and transcendental numbers.
@@mahmoudalbahar1641 that’s awesome!!! 😎 there is more coming in the next week…
Merci pour cette excellente vidéo j'ai bien compris les formules vous avez bien expliqué vous êtes en anglais et c'est traduit en français et je suis content de l'exploitant de l'exponentielle
@@felixbouvet1746 Merci beaucoup ! Nous sommes ravis que la vidéo vous ait plu. Pourriez-vous nous dire quel type de vidéo vous aimeriez voir sur la chaîne, s’il vous plaît ?
@@dibeos p parler de l'équation de schrodinger's
I stumbled upon this channel purely by accident. Amazing videos! The PDF file you provided is really nice. May I ask, where are you guys from? Keep up the good work.
@@simondobes8570 we are glad that you enjoy our content! Well, this is always a hard question for us, but here we go: I (Luca) was born and grew up in Brazil. Since my family, from both father’s side and mother’s side, are of Italian origin, I’m Italian and Brazilian, and learned Italian from a very young age. Sofia was born in Ukraine, but spent most of her life in LA (in the US) and Moscow (in an American High School). For a few years now we have been living in Italy 😎
@dibeos Ohh that's wonderful! Thank you for your reply and I wish you all the best :D
Cheers
What I want to know is what these sequences have in relation to the ratio between the length of the circumference and the diameter.
They seem like totally different and independent things.
@@Mariosergio61 yeah, we honestly don’t have the answer for this question. But let me think about it… maybe there is a cool geometrical connection between them and a circle somehow… 🤔
@@Mariosergio61 well, if you see the formula written in the PDF (involving cosines) of the first one (Viete), maybe we can relate each iteration to the lengths of the sides of a polygon inscribed in a circle. The more iterations, the better is the approximation to perimeter of the circle. I don’t know… it is just an idea, but we need to check whether it works or not… 🤔
Use an irational number to calculate another irational number, got it. Regards from Mexico
@@carbajalromerofernandoulis1053 yep, not the best approximation in the world
Nice video! Loved the explanation and animations.
(EDIT: the rest of the comment is no longer relevant, as the name was changed)
I have just one objection, I think the title is a bit misleading. In the video, you only compare the convergence speed of the two algorithms empirically, by showing that one of them can achieve a certain precision faster than the other one. You did not quite go into **why** it is that one is faster than the other, you simply performed an experiment that shows that, indeed, one does converge at a faster rate. Showing "why" one algorithm converges faster than another would involve providing a proof, or at least an intuition, of why one iteration of the better algorithm is able to improve an intermediate solution more than one iteration of the worse algorithm would, or something along these lines.
I suppose a more accurate title would be "Some Formulas for π Are Faster Than Others", as it is indeed what you brilliantly discuss in the video.
@@DaedalusCommunity hi! Thanks for the tip, we really appreciate it. Many people said that. We just changed from “why” to “how”, but please tell us: do you think that “some” would still be more appropriate? 🤔
@@dibeos Hi again! I think with "how" it's much better!
Considering other formulas, which is the one that requires less iterations? Please somebody tell me
How to graph the Riemann zeta function on the complex plane? I can only find graphing calculators that include the real parts
Uh did I miss it or is that a horrible conclusion.
If the "faster" way required 30000 calculations each iteration while the "slower" way only 3, a radically different conclusion would be reached.
Where's the flop count?
That's because he mistakenly assumed that fewer iterations means greater efficiency, which is incorrect.
Aproximation... That's the thing, are all the digits exact? Clearly you need many factors to even get close, so when it gets relevant? This looks more theoretic and dangerously innacurate, than any good pi generator 🤔
'Coz you can approximate infinitelly and never get to the value, or oscilate around it ot other kinds of not-getting-there apptoximations 😮
Lovely 🌹
I want a pin just pin my comment
@Speed85 hahahah sorry but we can pin only one comment and we need to pin the comment with the pdf link 😬
Sorry but I want to learn more In mathematics
1st comment
Very well done 👍🏻