A really interesting formula for pi

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Опубликовано:

9 ноя 2022

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Комментарии : 101

@MichaelPennMath Год назад

🌟To get started for free, visit brilliant.org/MichaelPenn/

@HershO. Год назад

Its fascinating how people come up with stuff like this which seems to come out of nowhere.

@guntherbeer8234 Год назад

These formulas did not come out of nowhere. Unfortunately, here and in classes, no one talks about how mathematicians go about solving a problem, or the many mis-steps. That's the most important part. The formulas are mostly irrelevant to the future of a typical student. What is important is that they learn to think mathematically.

@christianorlandosilvaforer3451 Год назад

thats was exactly what i was thinking after finish the vide

@christianorlandosilvaforer3451 Год назад

@@guntherbeer8234 he said "seems" but iam agree with u

@HershO. Год назад

@@guntherbeer8234 I agree with that completely. The school system (in most countries I suppose) doesn't promote creative, outside-the-box kind of thinking. What we are taught generally is to follow a particular monotonous approach and solve a bajillion problems of the same kind in the name of practice. Thankfully I had the resources(particularly the internet) to stave away from this path and explore more competitive examination sort of problems which are mostly quite thought provoking, but many people don't. We are never taught to journey out of the ordinary and I think we need that. The monotony may very well be the reason why so many people hate math.

@riccardofiori828 Год назад

In this specific case, the formula actually came out from a computer. If you search bbp formula on wiki you can read the details. The idea is to find some particular integer relation experimentally (with some search algorithm) and exploit them to construct this kind of sums. Sorry for my bad english, i hope this is understandable :)

@Alex_Deam Год назад

Apparently Plouffe has published a digit extraction formula for the nth *decimal* digit of pi this year

@dlevi67 Год назад

That would be interesting to see!

@etiennedeforas1566 Год назад

The paper is: arxiv.org/ftp/arxiv/papers/2201/2201.12601.pdf

@dlevi67 Год назад

@@etiennedeforas1566 Thank you!

@bjornfeuerbacher5514 Год назад

Really great stuff! I've seen this formula already several times, but I never would have thought that the proof could be so relatively easy... :)

@martinschulte3613 Год назад

Is there a plan when part 2 will be published? I would appreciate it!

@urgjendevetak3605 Год назад

me too!

@DynestiGTI 9 месяцев назад

Still waiting 10 months later... 😔

@donmoore7785 8 месяцев назад

Huh so I won' look real hard for it then. But I will try.

@SurfinScientist Год назад

I figured that the second part of this video would be kind of trivial, but then since Michael announced it as a separate video, I guess it is not trivial. But it is 5 months now, and I still did not see the second part. Did I miss something?

@Alex_Deam Год назад

11:57 In the partial fraction decomposition, there should be a minus sign that changes the first term when you multiply through, because the denominator on the LHS is 2-x^2, whereas on the RHS it's x^2 -2

@phiefer3 Год назад

I noticed that as well, but I don't think it ends up mattering. Because if he'd left the denominator as 2-x^2, it would still end up inside an absolute value after integration where the order of subtraction can be freely swapped.

@Alex_Deam Год назад

@@phiefer3 Tbf it wouldn't have mattered anyway because the values for the numerators Michael wrote down are correct, it's just the equation beforehand that was wrong

@knisleyjr Год назад

Very well done. Looking forward to part 2.

@gadxxxx Год назад

Interestingly, I read their book containing this formula and others over 20 years ago. A very entertaining book. Thanks for making a video about it.

@wolfmanjacksaid Год назад

Interestingly the formula evaluated at n=0 already gives a good first approximation of Pi with ~3.13333

@HoSza1 Год назад

This explanation is kind of going backwards, because they starts from the final formula and shows how its value is correctly equals to pi but it is more interesting how BBP came up with (i.e. derived) the formula originally.

@adamdickson608 Год назад

Did you ever complete the 2nd part?

@troydog3 Год назад

Amazing content. Thank you!

@kevinmartin7760 Год назад

If I had to guess, I would expect Plouffe to be pronounced to rhyme with proof, rather than like "plow-f"

@zathrasyes1287 Год назад

This is actually a proof, that the sum gives pi. But it is not a derivation of the formula itself. (Thx for the proof anyway.)

@fibbooo1123 Год назад

This formula was found by a big computer search- so this derivation is as good as you'll get

@christianmartin8751 Год назад

Very interesting formula, I am very impatient to see the second video on single digit application.

@oddlyspecificmath Год назад

This is wonderfully satisfying. Thank you.

@Jack_Callcott_AU Год назад

So satisfying, and beautiful. Thanks Prof. ! ✅🔊

@JavierSalcedoC Год назад

This is like cooking a meal but with numbers and you're cooking a pie

@CM63_France Год назад

Hi, I have been looking for this technique for at least twenty years, after having seeing this on a software in Python! Thanks! I can't wait for the second video! Is there a similar technique for the ten's digits?

@amaarquadri Год назад

I don't think there's an equivalent formula for base 10 😢

@shruggzdastr8-facedclown Год назад

"Borway, Blailey & Plouffe" reads like the name for a '70s Prog Rock band -- or that of a law firm! ;^}

@Unidentifying Год назад

7:51 how the 16 in denominator changed back to a 1 ?

@bro_vega_1412 Год назад

Just a mistake

@mguzjebesku2591 Год назад

Just a mistake sponsored by brilliant 😇

@yvesdelombaerde5909 Год назад

Nice video, very interesting to see the demonstration using secundary school maths.

@mariooliveira5768 Год назад

Very nice!

@ZedaZ80 Год назад

This was so cool, such a snazzy trick!

@m9l0m6nmelkior7 Год назад

Luv your shirt

@yashvardhanbiyani7617 10 дней назад

Is there a part 2 for this?

@goodplacetostop2973 Год назад

16:15

@donmoore7785 8 месяцев назад

Phenomenal. As a digital design engineer, I am looking forward to part 2!

@DarinBrownSJDCMath Год назад

4:14 Thought I was losing my mind until I finally said to hell with it and unpaused the video, lol.

@michaelempeigne3519 Год назад

go to ?? for part II

@synterr 3 месяца назад

Where is video part 2?

@dilbertojunior Год назад

Indeed, very very interesting.

@dilbertojunior Год назад

I like your videos so much. Congrats from Brazil.

@abrahammekonnen Год назад

Finally got it. Thank you for the problem. It's always interesting to see the transition between discrete sums and definite integrals.

@richardheiville937 Год назад

too complicated. 1/(8n+k+1)=integral of x^{8n+k) for x=0 to 1 and sum(n=0,infinity, x^(8n+k)/16^n)=x^k/(1-x^8/16)

@The1RandomFool Год назад

I've seen this formula before, but had no idea where it came from.

@b33blebrox Год назад

I wonder is there something like this formula for base 10? I believe it should

@khatharrmalkavian3306 Год назад

Calculus is madness.

@dilbertojunior Год назад

Nice

@DynestiGTI 9 месяцев назад

15:38 it's been 10 months and still no part 2... 😔

@laux_dtm5027 Год назад

For the last equality, how did the (u+1/ u² +1) in the integration become (1/ u² + 1) ?

@TheEternalVortex42 Год назад

He kind of skipped over that part but he split it into two integrals u/(u^2+1) and 1/(u^2+1)

@laux_dtm5027 Год назад

@@TheEternalVortex42 oh ok thanks you

@alquinn8576 Год назад

when Al Quinn wants to find pi, he just goes to marie callender's

@yvesdelombaerde5909 Год назад

Voir le livre de Delahaye à propos de Pi, dans la collection Pour la science

@scullucs Год назад

Respectfully, Simon "Plouffe"'s name is pronounced more like "Ploof". (I actually spent an evening in his apartment in Montréal in the late 90s, (when he was away for some reason), watching "Jeopardy" with a friend who was taking care of the place. I'd never met him, nor did I know he was this important mathematician. Just a weird coincidence (the coincidence being me having lived that and being here now)).

@pedrofigueiredo7850 Год назад

The nice integrals of 16x^(k-1)/(1-x^8) do not converge.

@samueldevulder Год назад

Il faut savoir se jeter à l'eau. .oO(comprenne qui peut !)

@woodchuk1 Год назад

Does anyone have an idea on how fast that infinite sum would converge to the true value of pi?

@WhattheHectogon Год назад

Plug it into Desmos! It converges very very quickly :)

@ngtrxpwgdrk Год назад

Linearly. Each term gives you at least 1-2 correct digits, because the tail is bounded like in geometric series.

@idjles Год назад

It’s used to verify billions of digits of pi, by calculating the last few digits and see if they match.

@ojasdeshpande7296 Год назад

VERY FASSSSSSTTTTTT

@idjles Год назад

@@ojasdeshpande7296 it is actual really slow, that’s why its only used to check much faster algorithms.

@gregoryknapen9133 Год назад

Plouffe is pronounced ploof BTW.

@minwithoutintroduction Год назад

رائع جدا.أحسن مدرس و أفضل قناة . يعجبني طريقة ترتيبه للسبورة رغم صغرها

@alcodark Год назад

towards the end.. bit annoyed by the comeplete the square to (1-x)^2... only to change signs in the end..it's not wrong it's just a step towards confusion..xD

@schweinmachtbree1013 Год назад

He did it for a reason - the substitution u = 1−x is nicer than u = x−1 here because the mapping x↦1−x swaps 0 and 1 which are the bounds of integration. If he had used u = x−1 then x=0 and x=1 would have become u=−1 and u=0.

@hasanjakir360 Год назад

Hey Mike, would you make a video one the following question? Find y, if 2y=x²y".

@Noam_.Menashe Год назад

He already has. Search Euler Cauchy.

@keithmasumoto9698 Год назад

I don't get how the denominator becomes 16 - x^8. Anyone?

@colonelburak2906 Год назад

Check out the last line at 5:59. There you have the 16-x^8, but it seems he made an error in copying the integral to the next board. I was also confused for a moment!

@keithmasumoto9698 Год назад

@@colonelburak2906 Ah yes! That's it. Ty!