Ramanujan's Pi Formula

The second video in a series about Ramanujan. Continuing the biography and a look at another of Ramanujan's formulas. This one involves Ramanujan's pi formula.

Ramanujan was literally self-educated and he came up with some superb formulas and more. I would say that among the recent 500 years of geniuses Ramanujan definitely deserves a place. That said, computer science of today use many of Ramanujan's formulas because they not only are brilliant, but computers love them with only a few tweaks. What is that? A guy, from India, self-taught in math, provided results that benefits all of the globe today (with just some small tweaks), isn't that GREAT? I am amazed, to say the least.

[@piratesofphysics4100]

He should have lived 80 years. I'm from Bangladesh. My most respected man in the world. He is the most genius man from the big bang.

[@AdarshRajCR7]

If he would have lived till 80 , we would have met aliens already.

[@hemantkumar9303]

Indians are genius is every field they opt for especially MATHS.

[@sowmitriswamy6718]

@@hemantkumar9303 Stop your bragging, it doesn't look good. Geniuses - Newton, Galileo, Gauss, Euler, Einstein, Shakespeare, Darwin, -can be born anywhere.

[@MABfan11]

@@piratesofphysics4100 if he had lived to be 80, Fermat's Last Theorem would've been proved much earlier. hell, he would probably be the one to solve the Riemann Hypothesis

[@priyamshukla8670]

Ramanujan was genius. We Indian celebrate his birthday as a National Mathematics Day at 22 December.

[@leif1075]

I feel like I could do what he did..

[@harshitkumar4760]

@@leif1075 why not? Anyone can be a genius :)

[@niteshsrivastava3122]

@@harshitkumar4760 😂😂😂😂

[@_DD_15]

@@leif1075 I guess anyone can add all the positive numbers and come out with a negative one. 😂😂

[@leif1075]

@@_DD_15 Well I can do much more than that.

[@malcolmbyrne]

3:04 wow! Just look at that formula. How can someone "discover" that! That must be real genius. I'm really enjoying this series, thanks.

[@prajeeshprasannakumar1079]

When he died he was 32. & Hardy found him when he was at 26.

[@ddstar]

he used number theory based on a prime table of 25. I'm sure of it.

[@musik350]

@@ddstar could you elaborate?

[@adrianbiber5340]

*3:14

[@maloxi1472]

@@musik350 No, like many lunatics on the internet, he cannot

[@AlexeiRamotar]

Ramanujan's brain must have been wired differently. He saw math.

[@josed6739]

Yeah he a mathmatican he knows How to solve partitions

[@amalsrivastava6853]

Ramanujan said that goddess saraswati would whisper all this in his ears

[@niemandniemand2178]

dumbass

[@hidgik]

@@amalsrivastava6853 You got that wrong. It was the Mahalakshmi of Namakkal. Big deal!

[@stardustreverie6880]

@@niemandniemand2178 dumbass

[@leobekayombo8087]

I like it that you talk about Ramanujan's work. It seems that western mathematicians tend to minimize his contributions. That's not easy though.

[@RinkuYadav-pn4jo]

I doubt why he died so early in a era when birtish colonialism and rascism was at its peak

Definitely not -- but my knowledge about Ramanujan begun by a beginner's class of computer science, so I cannot 100% agree with you on the "western" statement ... except the class was about numerical computing and thus not mathematician-only

[@NetrocMagi]

They definitely do a lot. I've seen westerners straight up deny Ramanujan and said he made up stuff on his own this lacking proofs. Incredibly ignorant, but I'd have to say none of those westerners seem quite educated on higher mathematics let alone basic math. l@

[@pierrejeanes]

wow I wonder what Ramanuja would discover and prove if he wouldn't die so yung

[@user-og9nl5mt1b]

Time travel and , worm hole math

[@omkarpawar14]

The man who knew infinity.

[@bhavishyasharma7834]

@MATHE MATICS don't make gods, they already exist

[@sanketgore8915]

The pi formula that you mentioned which was discovered by Leibnitz was actually discovered by an Indian mathematician named Madhava about 300 years before Leibnitz was even born!

[@azzteke]

Leibniz please.

[@backpropagation8365]

Euler, Gauss, Ramanujan, Jacobi are the true geniuses who spit out concepts after concepts and formulas that are the basic of every math today. Black holes are now being explained by Ramanujan’s mock theta functions discovered by him 80+ years ago, while on his deathbed and wanted to share it right away via a letter to Hardy because he knew it was so important to share. This is quite unexplainable how he could have seen the importance of mock theta functions that early. I do not recall any science genius or even any prominent scientist of having written down a piece of complex math that was applicable to an application that would only be discovered so far into the future. Basically, Ramanujan had the math ready to be put to use to explain a physical discovery that happened way into his future? This is superhuman. No other mathematician ,past or present , was this superhuman. It is quite unexplainable and bizarre actually.

[@Narcissistic.nAz_i]

Gauss was a noob

[@chaoticdays]

At 03:14 you say "no one knows how he actually got the formula". GH Hardy has stated that when he asked the same question to Ramanujan, he replied "God talks to me". This dialogue is also shown in the movie 'The man who knew infinity'.

[@koushikmediga8397]

I think he is the one who does the math, but as he believes in god, he just thinks that god communicates it to him.

[@adityaagarwal636]

@@koushikmediga8397 Actually not. It did happen that her deity goddess came in his dreams and told him the formulas, and then his work was just to prove them!!

[@adithyadanaj9768]

@@adityaagarwal636 yeah that's that's he meant. Our mind works like that. Many scientists have had encounters like that. Where they get answers just after a sleep or having written something in their notebook during sleep. There are accounts for that. Ramanujan just attributed that to his godess but it was he himself who was actually doing it subconsciously. Our subconscious mind is surprisingly more productive when we focus on it

[@koushikmediga8397]

@@adithyadanaj9768 yeah that's my point

[@bv6168]

​​@@adithyadanaj9768 ramanujam is different than rest of the scientist or mathematician. He is uncomparable

[@DANGJOS]

That man was one of the greatest mathematical geniuses ever!

[@brian554xx]

A pie formula is called a recipe.

[@Tfin]

+Brian Schiefen Everyone loves pie, and I want some now. I have no way of accessing pie at this hour.

[@higherbeingX]

ReciPIE to be exact.

[@saw7191]

brian554xx ReciPI

[@YazFidz]

Hey james, i just wanna say thank you. You're doing a great job at teaching us viewers about math. Keep on keepin on.

[@vector8310]

As an inspiration, Ramanujan is irreplaceable. Euler and he have that effect on me. I'm compelled to take pencil to paper and explore mathematics

[@yavuz1779]

Evariste Galois too

[@zwwx2142]

@@yavuz1779 yes😎😎😎

[@piratesofphysics4100]

If God told me that ramanujan will come back but I have to die first. I would definitely die happily and instantly 😐

[@pratik-taank]

If ramanujan would have lived 10-15 years more, we would be having the technology of self replicating spacecrafts to go to other planets/galaxies.

[@Roshkin]

And as always, if you have been, thank you for watching.

[@U014B]

IT'S NOT THE SAME! 😣

[@joyjeetdas6821]

ramanujan the best mathematician ever came on earth...proud to be an indian

[@whendaybreaks4317]

2k views. 400 likes. 4:20 long video. my favorite mathematician of all time

[@dannygjk]

Here is a sobering thought I wonder how many 420 worshipers know that is also Hitler's birthday?

[@egenriether]

Or George Takei..... Oh My.

[@ITsmapleTimexD]

KEEP IT COMING! I'M LOVING IT!

[@DaC10101]

It's staggering how someone managed to find such an apparently bizarre formula for τ/2...

[@wyattarent156]

What I didn't know you had a personal channel! Instantly subscribed

[@VideoMathTutor]

I have a similar abacus as the one on his desk. Had it since the '80s. Love it!

[@explorewithcate3040]

God I love your explanation, I've been watching your videos all day long for goodness

[@stellagilbert]

Love these!!

[@RSLT]

Look forward to seeing a new Riemann Hypothesis video by you. I believe it will be fantastic. Your insights on such a profound and challenging topic would be incredibly valuable. I've always admired your ability to explain complex mathematical concepts clearly and engagingly. A new video on the Riemann Hypothesis would be an excellent update to your content, and I'm sure many others would benefit from and enjoy it as well. Please do it!

[@MOHNAKHAN]

I am proud to say that I am from that Great Land India from where Ramanujan was...

[@TheVicky3030]

Mohna Khan He was a "Bootparast".

[@yahikouzumaki2767]

You people always need a reason to take pride in something with almost zero effort . Why not do something great and take pride in yourself ?

[@prateekdalal1659]

@naz de even i dnt know from where are you from ? and you are just talking about india, try to make your country proud , dnt take any credit from other counrty . you are so negative like a nil/zero.

[@prateekdalal1659]

@@yahikouzumaki2767 with zero effort LOL, you are the biggest joke , can you count me your contribution regards to your country which provide some impact on other countries ?

[@prateekdalal1659]

@naz de if you cant respect your own country , so its very sad to say you have done nothing in your life and also for others its like a , as you can so you become, now come to the point of cleaner things , you have fully filled with self ego and a uncertainty , i am also lived in Paris but i cant figure out any cleaner things compare to urban city like in india. India is much cleaner than your thoughts .|| Respect from Indian guy ||

[@madhukaraphatak]

Loving this series.. Keep it up

[@_jenaissante_]

Always love Jame Grime's videos!!

[@TheMaplestrip]

So when are you starting on Crash Course Maths with Matt Parker?

[@joshnigam7733]

Yea, Parker squares in general

[@MrHSX]

Oh that'd be so awesome Hank recruiting the entire Numberphile team to star on their Maths videos...

[@TheMaplestrip]

Matt said he'd be in for it if James would do it, for what it's worth ;p

[@onafehts]

oh, that reference though haha

[@ramanparashar1]

Ramanujan was on the mathematical scene for only 4-6 years doing Maths along with managing his livelihood in a foreign country, fighting ill-health & prejudices and scores of other issues.. Researchers are still discovering new things in those 4 years of Mathematics that he has done..

[@shashanksistla5400]

Another great video!

[@petermerrotsy1011]

Well, this result was actually published by Ramanujan in 2014 (QJM, 65, 350-372), hence accomplished before he went to Cambridge, and "values" such as 1103 and 9801 were not "deduced from numerical examples", but arise from his deeper, more genralised work on modular functions.

[@petermerrotsy1011]

@rf4life Yes, of course, 1914, thank you for the correction.

[@DarkNomadness]

I appreciated the energy and content

[@georgemissailidis7581]

Or for a better approximation, 9089/(747√15)...Or for a better approximaion, 116881/(7932√22)...Or for a better approximation, 58651979/(3466840√29)...You see, π ≈ A/(B√C) and there seems to always exist two distinct integers A and B such that A is odd, A > B, and C = 7n + 1, ∀n ∈ ℕ.Now you might say there always exists two integers X and Y to satisfy the equation π ≈ X/Y but if it was instead approximately of the form A/(B√C), then it seems like there exists two integers A and B where this form is accurate to π by at least 6 decimal places ⇔ C = 7n + 1.

[@georgemissailidis7581]

And did you know, 9801 = 99²

[@ericvilas]

controversial maths? Would that happen to mean Ramanujan sums of divergent infinite series?

[@Mrius86]

How do I calculate the first series described in the video?

[@jackwright2495]

FYI, the stress is on the second syllable of "Ra-MA-nu-jan." BTW, 355/113 = 3.141 592 920 = pi * 1.000 000 085!

[@jonathanschossig1276]

Is that the prime number generator in the background?

[@idjles]

I've loved this formula for 30 years. Please show us a few terms or how it was derived, or how you get root(8) to X digits

[@zwwx2142]

What formula?

[@idjles]

@@zwwx2142 Ramanujan‘s pi formula!

[@ernst9100]

Ramanujan himself did not derive it from first principles..it was given to him in a dream by the God he believed in. The structure of Benzene which had evaded scientists was revealed to Kekule in a dream. God does miracles thru his chosen subjects . No one can explain how a simple Indian from those days who had no mathematical background in university could solve so many theorems, even attempted to solve Fermat's last theorem, thus came up with the magic number 1729 as the smallest number that can be expressed as sum of two cubes in two different ways...simply amazing

[@AdarshRajCR7]

Ramanujan didn't knew a lot more than any other person. The fact that he studied all the mathematics on his own using a book gifted by his friend is Insane.

[@GyanTvAmit]

thats why he ruled brits

[@jamespklett]

What is that device on your desk behind you? And are those markers or tapes to the right? Can't make them out, just curious - thanks for your videos !!!!

[@sidhambaramaadhimurugan5475]

One of the reasons for genius of ramanujan was his food.. What food we eat determines what we are... Food influences our mind , body even our behaviour , our character... Our inteligence , our thinking .... Animal flesh , egg will have negative influence on our thinking, (which means not giving high level thinking which means we cannot think broader as well as deeper , if we eat non veg we can able to think little deeper but not clearer ) Gives Negative to our intelligence growth, (lower intelligence) Gives negative to our behaviour and character( anger, hatred, lust , desire, ego, selfish which are all negative or dark) Gives the bad karma which is action/reaction which means you do killing and cause pain , then you will get pain frim others Ramanujan was vegetarian.... Thats why he doesnt have the bad karma of killing and eating animals.... Thats why gods love energy/light can easily getdown in his consciousness... If we eat non veg , we are making our karma darker and thicker that it becomes very very difficult for god energy to enter into us.. If you are vegetarian its enough...... dont need to be vegan ...

[@roberthoffenheim7861]

Bullshit. Most great mathematicians in history ate meat. Not saying that eating meat is a good to thing, in fact factory farming is very cruel and the meat industry is destroying the planet, but eating meat is not detrimental to mathematical ability.

[@Fiyaaaahh]

Will you talk about Ramanujan's almost integer numbers at some point this week? For me that's his works that appeals the most to my imagination. Especially considering he derived them way before any type of serious (digital) calculator was available.

[@frankharr9466]

Blech, meters. Does the sq.root of 8 have anything to do with the eight decimal places thing?

[@clashott4372]

The man really knew infinity ♾

[@jaggubhai44]

You could be a good teacher.you explains complex things very understandable and easily comprehensible

[@ayasaki.pb_787]

I am interested in it and looking forward to watching the movie. =D

[@kyanite1164]

Can you please post your resources!! I would love to read more and where you got your info from!!

[@geoffklassen9402]

And we have been, so you're welcome for watching. ;)

[@earlystrings1]

I have a dim sense that Leibnitz's infinite sum to approximate pi is doing, in essence, what Archimedes was doing with his calculation of ever-more-sided inscribed and circumscribed polygons. But I am as far from being a mathematician as it is possible to be. Perhaps someone can explain if this in any way correct?

[@Dave-lr2wo]

What's the electronic thing in the corner?

[@monosizroy7017]

Ramanujan formulae joins the chat My brain : Alright imma head right out

[@pnachtwey]

I used 355/113 as an estimate for PI when I was programming 16 bit micro controllers without floating point. Try it.

[@ragnkja]

Accurate enough for the vast majority of practical applications on Earth.

[@egenriether]

without floating point isn't it rounded by the processor to 3? I use the Parallax Propeller (an integer-based controller). If you put in 355/113 for example, I think you just get 3. If there's a workaround I'd like to have it

[@stevenvanhulle7242]

egenriether - I guess he means a FPU, i.e. floating point done in hardware.

[@zakkenwasser65]

egenriether, It is a scaling thing I think. For instance you can map the 16bit integer range -32768 to 32767 on the real range -32.768 to 32.767. Then you can use 1000*355/113 in your formulas or calculations to represent PI and there will be no loss of precision because 355/113 approximates PI with about 7 digits precision.

[@egenriether]

Pedro Pendejo Amigo oh yeah... that seems like it would work. I use floating point libraries now that do the job but I’ll try that sometime. Thanks

[@Zedoy]

You are the best James you are awesome! :-)

[@sidorak12814]

Love that the prime number generator is still on the desk

[@jesusthroughmary]

Why isn't it 2206 root 2?

[@SuperYoonHo]

awesome!

[@jayrajganatra8782]

Can anyone please explain how Ramanujan derived the formula. I know he addressed it but I didn't fully understand.

[@megawattapps]

Leibniz did not actually come up with that formula you talk about at 1:41, it is miscredited to him. Check out the documentary "The History of Maths" on Netflix it talks about it

[@bemusedindian8571]

Its called the Madhava-Leibniz series. Madhava of Sangamagrama is still not recognized for his work. In a way, rightly so, because of very little historical evidence.

[@marlenepoblet555]

A new subscriber! Thanks

[@Earwaxfire909]

Has AI been applied to this type of series calculation?

[@skarumuru]

The formula for pi you attribute to leibnitz was originally discovered by Sankara , an Indian mathematician some 200 years earlier, leibnitz name still figures in math history books ‘cos - white man invented everything.. great ..

[@uselessperson1731]

hey do you clear maths queries ? queries in particular problems?

[@mridul303]

Formula for calculating value of PI up to 100 digits 31415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679/10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

[@yusufqaddura7103]

lol

[@ThePerfect1077]

+Mridul Tiwary isn't that "evaluates" not "calculates" ?

[@mridul303]

+ThePerfect1077 well someone has to do the division, either your brain or a calculator, which will actually "calculate" the final result.

[@ThePerfect1077]

Mridul Tiwary hmmm i wouldn't call it a formula though cos there aren't any variables, nothing u can sub values into. like volume of a cube: v=h^3, u can sub in different values for h into the formula to find the volume

[@mridul303]

ThePerfect1077 that's true. Technically, it's a function.

[@lindy7985]

Math geeks complaining about too much mail?

[@seemlyme]

Thank you so much

[@shahbazsheikh3545]

So what is applications of these pi formulas? Don't we already know pi to be 22/7 (or is that also an approximation formula)?

[@astrovert.ed2321]

Hi, this is very informative. I have seen formulae by Ramanujan and the Chudnovsky brothers, but how do these compare in accuracy with the actual value of pi?

[@ernst9100]

There is no actual value of pi.....it goes on and on to infinity, just like the number e

[@astrovert.ed2321]

@@ernst9100 Noted Sir. But generally these two would align with the actual irrational value to what extent?

[@Eric-im6eq]

How about dividing 16 by 113 plus 3, that will also give Pi with six correct digits. With help of my HP-35s calculator.

[@Ace1King1]

There is only one number that consecutively repeats itself nine times in the first one and a half billion digits of pi. What is the 10 digit number and where does it begin in the sequence?

[@gurnihalsingh6252]

Bro I have tried the second fomula and it is same as first and we get only get first 6 digits of pi and not 8

[@marcschmidtpujol550]

Do you know if he really did write sqrt(8) instead of 2sqrt(2)? Would really like to know!! Good video :)

[@DRD363]

how about a video on the roots of 200, 64, 90, and 30 and their relationship to each other.

[@DRD363]

finally!!!! The Pi equation is shown!!

[@icecubegaming3661]

sir I have a pi formula of myself which can calculate upto 7 decimal places

[@jhonyangarita9961]

son muy buenos tus vídeos

[@ParticleJesus]

Get your calculators ready, the _-1/12_ series is coming.

[@leif1075]

How did Ramanujan and Leibniz come up with those formulae?

[@Yakushii]

This might be a hard habit to kick, but Ramanujan was Indian, and the "-man" syllable is the one that should be emphasized. Not the "-u".

[@EmperorZelos]

+Yakushii and worship the cow, don't forget the cow worshipping

[@KevinSolway]

+Yakushii "the "-man" syllable is the one that should be emphasized" Not very politically correct in the modern climate!

[@Yakushii]

Kevin Solway - Haha, sorry. I should have said Ramanujan/Womanujan.

[@EmperorZelos]

Kevin Solway BURN THE FEMINIST!

[@AnotherCuber5050]

+Yakushii You're right. It's pronounced as RAA-MAAN-u_Jen

[@Jerome...]

love how the shirt matches the blinds

[@wiredog771]

can someone please explain something to me. One of the flashy points of the Ramanujan story is that he had no formal training. well, even I can tell that his letter to Hardy was filled with calculus notation. do they mean just that he didn't study math on the college level?

[@abhilashavulla2813]

WireDog it is said that he didn’t had formal education.. he proposed many theorems which already existed at that time. He also proposed many new ones which proved his merit.. directly approaching the solution is mysterious..

[@emiligiltomas1998]

Hey, can anyone solve me this doubt? When we're looking for pi digits how do we ensure that the digits that we get from the formulas are right? Thanks.

[@manpreet9766]

Emili Gil Tomas eg. The Leibniz formula is derived from the expansion of inverse tangent function which is an identity. Putting 45 degree(Pi/4 in radians) angle on one side we get 1-1/3+1/5.. on the other side. In other words the formulas are exact and are proven to be correct.

[@sirfermainclancharlie1018]

Gosh such a smart fellow

[@randomreviews5693]

My brother who is engineering topper says physicists are afraid of none but Mathematicians.

[@alexandterfst6532]

Excellent

[@peterlindner3283]

What is the third video, since this is "The second video in a series about Ramanujan "? Thanks for your spirited and often enlightening series.

[@ankitdhage8748]

From his behaviour I can say that he is one of those genius away from world affairs.🙁

[@Creativity_sharma]

Absolutely right man

[@satyamjnu]

Love from India

[@adityakhanna113]

1:59, Don't we get that formula by putting in "1" in euler's sine formula (derived from expansion of e^ix), shouldn't it be sin(1)??

[@FFA1996]

+Aditya Khanna The infinite sum at 1:59 does follow from a series expansion of a function, but it's actually the series for the arctangent (or inverse tangent) evaluated at 1 (since sin(pi/4)=cos(pi/4)=sqrt(2)/2 so tan(pi/4)=1).

[@tigerdalandan]

Yay! When I saw your face, I immediately remembered Numberphile.

[@logaandm]

11-33-55 easy to remember. Last three divided by first three is 355/113 = 3.1415929.... and is pi to 6 decimal places or 27 parts in 100 million. Not as accurate as Ramanujan's formula which is 7.6 parts in 100 million, but easier to remember. So only estimate the circumference of the earth to about 3.5 meters. e can be estimate with 335588/123456 but this is only about 160 parts in 100 million Just in case you forget your slide rule.

[@narmathasri9633]

why the pi value is given in its inverse form

[@salo2132]

That formula looks like an easy for me. How I gonna remember

[@Kram1032]

Nice formula indeed.

[@alexandterfst6532]

Brilliant

[@twertygo]

looking forward to the next video since I don't know how math is going to be controversial :)

[@WatchingTokyo]

Maybe something related to 1 + 2 + 3 + ... = - 1/12

[@hadensnodgrass3472]

+WatchingTokyo No, the document he sent to Cambridge had already deduced that. So it unlikely Grime would use that, plus it has already been done a thousand and one times.

[@inkolore2]

+twertygo If you want to see controversy you just need to check the comment section of some of Numberphile's videos, like 1+2+3+ ... = -1/12, problems with zero, infinity paradoxes.

[@twertygo]

+IceNoob88 But it's all just logical and makes sense!

[@bonbonpony]

+twertygo Then jump into some math forum and ask if √4 = +2 or -2, or just the +2 ;) (On second thought: _don't_ :P )

[@TimJSwan]

Hilbert was a normal human who made sense. Godel was a guy who was rational. Einstein was a regular man who understood logic. Euler was a reasonable husband and father. Galois was a talented bachelor. Ramanujan.. Compared to them, he was like an insane alien genius who must have obtained non-finite intuition from spiritual beings. Screw what Hardy said about him being 100 and Hilbert being an 80, they are all 1's and he's still at the limit of 100 on that scale. We can't figure out where he was because nothing he figured out was anywhere near normal. Go look at his work for yourself. It was strange and different. It wasn't even proofs or restricted to logic systems. He had crazy intuition results and approximations which are insanely accurate yet not precisely the function, so all the more curious how he developed them.

[@a6hiji7]

He was a genius with a great intuition similar to the likes of Gauss but I think it was the lack of formal training and the lack of guidance in formal research which contributed towards his early work being more based on intuitions.

[@saintcelab3451]

I love Nottingham accent