Ramanujan: Making sense of 1+2+3+... = -1/12 and Co.

[@tahmidt]

Imagine this; A guy comes up to you and offers you $1 on day 1, $2 on day 2, $3 on day 3... all the way to $n on day n and he keeps doing this for the rest of eternity. You might think that he's giving you free money but a mathematician will quickly realize that he is trying to con you and steal a twelfth of a dollar. But now you know.

[@ishwar8119]

+Tahmid Islam (Tanveer) Fine of $0.09Mathematician: That guy lost 2/3 of a cent because of this

[@rasowa2958]

+Tahmid Islam (Tanveer) That's just what Federal Reserve and other central banks do.

[@dixonbuttes]

I know I'm 9 months late, but I wanted to let you know this is the best comment I've ever seen on youtube.

[@ramirez9919]

Tah mid Tanveer, But why only 1/12 and why not any other figure

[@vijayakrishna07]

Tahmid Tanveer nice comment

[@rifkyalfarizy9945]

before i started watchibg mathologer i hate math and have bad grades after i found this channel, i love math, and still bad grades luv dis channel

[@nestorv7627]

hahahaha XDXD

[@adrianncoello7223]

LOL!

[@HimanshuSharma-eg5li]

Lul

[@andreherath5210]

Adriann Coello ñhk )0 Nike Air

[@adhidwiyana2723]

emejing

[@donmoore7785]

I like how Ramanujan left out his bold assumptions, but the recipient of the letter knew exactly what he meant.

[@maxwellsequation4887]

Recipient was Hardy tho...

[@abhaynatoo]

it was THE ramanujan talking to THE hardy. no commoners here. so its understandable.

[@vaibhavkrupakar240]

Not an assumption, he proved it himself

[@Alkis05]

It wasn't an assumption. If you pause the video, he says that he explain further in what conditions those divergent equations make sense. Most pointedly, he understood they were divergent and he made it clear he was not talking about standard summations.

[@stevethea5250]

@@Alkis05 timestamp?

[@kevnar]

An infinite number of mathematicians walk into a bar. The first guy orders a beer. The second guy orders half a beer. The third guy orders a quarter of a beer the fourth, an eighth. The bartender just pours two beers and says, "You guys need to know your limits." I finally get that joke. Thanks.

[@kalliboymusic]

Lmao

[@78anurag]

Lmao

[@Hyperdrive]

Lmao

[@alex.g7317]

lmaoo

[@literallyaperson7125]

Lmao

[@PickyMcCritical]

List of jumpscares: 4:02 9:28 11:53 22:50 30:12 31:57 34:14

[@ankitaaarya]

PickyMcCritical 😂😂😂😂

[@ekap1916]

PickyMcCritical You ignorant knob!!

[@biscuitsalive]

PickyMcCritical terrifying

[@vaishnavplays203]

Lol

[@Akshaylive]

Thanks muck. This helped me prep.

[@villagelog1966]

Ramanujan’s work in pure mathematics is now being used in engineering too. “In 1918, Srinivasa Ramanujan introduced a summation, known today as the Ramanujan-sum. He used this to express several arithmetic functions in the form of infinite series. For many years, this sum was used by other mathematicians to prove important results in number theory. In recent years, engineers and physicists have found applications of this sum in digital signal processing, especially in identifying periodic components of signals with integer periods,”

[@rustalisin8950]

Pls Help ..Any Polynomial may be completely determined by a part of it. Analytic function is uniquely determined by a small part ( sample ) of it. OK. How can a Eta / Zeta Analyt function be used to extract binary modulation from a high density set of timespaced sloping carriers (let all carriers be linear, thus parallel, with guard band) I can see this series extracting (defined) harmonic modulation using array summation to yield an entire constellation across a message timespan of say 12milliSec. given 512 symnbols in 12mS.

[@RecliningFurniture]

Great, but what on Earth does all that mean? It is really not difficult to see why 1+2+3...etc does NOT equal -1/12 [but, incidentally, that 0.999...(0.9 recurring) DOES equal 1 (which I mention only to show that I don't have an issue with infinite numbers in calculations, where the calculations are actually grounded in logic)] . As that is so (and all you really need to know is that a series of plus numbers added together cannot logically equal a minus number unless you introduce a minus figure into your calculation, in which case it is no longer a series of plus numbers) I am not sure where that leaves the 1+2+3....series in terms of any practical application. Can anyone enlighten me on this, please?

[@achyuththouta6957]

@@RecliningFurniture 1+2+3+4..... is indeed equal to -1/12 although the explanation given by Ramanujan is completely wrong but this is used in string theory and has merit. Ramanujan gave insane explanations to this theorems and equations but most of them are correct

[@RecliningFurniture]

@@achyuththouta6957 Have to say I'm surprised people are still taking the whole -1/2 thing seriously. Mathologer has done another video about this (I'm not sure if it was before or after this one) where he completely debunks it. I don't know which proof you favour; the one I followed introduced plus and minus 1s and tried to make out these implied an average, when this was very obviously a completely different sequence - a mathematical sleight of hand. But, for the sake of argument, I would be interested to know where the logical fault lies in positing that adding only positive numbers to positive numbers must result in a positive number and cannot result in a negative one. I cannot personally fault the logic in that, but, as I say if the -1/2 sum is correct, there HAS to be a logical flaw in what I just said. Can you identify what it is, please? I would be extremely impressed (enlightened, even, in the truest sense) if you can.

[@RecliningFurniture]

@jamesk479 Thanks for this. You will need a specific rule to assign a solution to sqrt (-2). Such a rule would not accord with any maths I know, but that's OK, because it would be a rule specifically dealing with minus numbers, and the point of the rule I referred to above is that it deals only with positive numbers. To test the validity of the c = 1+2+3....etc here it should help to first consider a much simpler sum. So let's look at c* = 1+2+3 (which seems as reasonable a sum to consider as anything not containing an infinite series). So, c in this case equals 1+2+3. Which equals 6. Now, going on to use Ramanujan's methodology, we multiply by this 4 and subtract the x4 number from the original number. According to the demonstration in this video, this should result in -3c = 1-2 +3. But it doesn't. What we actually get is -3c = -3-6-9 (which is -18, so c =6, as we already knew). The practice of missing out every other number in the demonstration when multiplying by 4, and then including numbers missed out when subtracting the bottom number from the top, is going to give a bogus result. What we can actually say about the 'c' in = 1+2+3....etc is that c must >6, because the infinite series following 3 contains only additional positive numbers. Similarly, we can say with absolute confidence that c >10, because we know the next number in the sequence is a 4, and 1+2+3+4 =10, and there are more positive numbers to follow (an infinite number of them). Taking into account the number after that, we have 1+2+3+4+5 = 15, so c must be greater than 15. And so on. Forever. It turns out there are an infinite number of things we can say about c. We know exactly what types of things they are, and none of them are that c is equal to - 1/12. *c here obviously does not stand for the constant in an infinite series as Ramanujan used it, it is just c.

[@parinitachakma765]

A simple stack overflow bug. God will patch it in the next update.

[@CapWilder2504]

Skylark Phoenix 😂😂😂😂

[@rohitupadhyay1116]

There is no bug under his rule my friend

[@andrewzhang8512]

'stack overflow'. EVERY PROGRAMMERS LOVE!!!!

[@wliaputs]

Rohit Upadhyay but i see bugs everyday

[@rohitupadhyay1116]

@@wliaputs 🤣🤣

[@sjegannath6295]

It's important to note that this doesn't mean that the sum of all natural numbers is -1/12 but rather it's the solution to the riemann zeta function where this series occurs.

[@binayakthakur5122]

It is like the software running the universe got overwhelmed and gave error - 1/12 Integer Overflow

[@waywardson4472]

Binayak Thakur I was thinking along similar lines when he had all powers of 2 summed up to -1

[@thesunexpress]

Actually probably the Universe spat out "Cannot compute! Error: 1/137 "

[@bee_irl]

-1/12, my favourite integer

[@aklipsigman3451]

It's real number not integer

[@vivanmakwana2253]

it is not error it is a smarter idea of an indian and i am proud to be an indian

[@kevintan9483]

Every time I make a mistake on a maths paper, I'm going to hear that sound and be scolded and scared out of my skin by Ganesha descending from the ceiling.

[@pranavchandrav6071]

Jumpscare

[@yedosti3092]

This comment made me jump out of my bed laughing😂😂. Truly insane😂😂

[@Mathologer]

This week’s video is in response to a large number of you and my students at university asking me for my take on the whole 1+2+3+... =-1/12 business. Initially, the plan was to come up with one of my usual 15 minute long videos. However, after several unsuccessful attempts at not exceeding the magic Mathologer time limit, I realised that any “short” video like this would just be a clone of one of those wrong/misleading accounts of this topic that KZbin is full of. So, rather than just give up completely on this project I decided to do what you should never do if you actually want people to watch your videos, namely simply go for it and not look at the clock. The result - a video that is an insane 35 minutes long in which I say all the things that I think need to be said and can be said only using elementary math to (just barely) do this amazing topic justice.

[@MK-13337]

I just finished watching this because I'm a time traveller. Great video!

[@wkwmenen]

Jesus, that Indian God works as almost a screamer

[@icefreezer7]

+Hououin Kyouma lol. me too. and nice name

[@Mathologer]

+Matti Kauppinen Fill me in on the secret of time travelling sometime. Would make for a great video I am sure :)

[@Mindsway]

great video ! the cesaro methode is cool :) But can we.. take the average of the average of the average... and so on to the infinite ?

[@aadityasingh5575]

I love how whenever Ganesha comes up we get a death threat

[@xendu-d9v]

Jay shree Ganesh

[@yashkrishnatery9082]

@@stratonov +1

[@Kevbotomy]

On a scale of 1 to -1/12, how would you rate this convergent series?

[@teamplayer2helper]

Kevbotomy infinity

[@-_Nuke_-]

XD

[@AshishPandey-uq1uw]

0

[@rayoflight3224]

i

[@ir2001]

@@teamplayer2helper Lol amazing pun

[@aniketthakor]

Happy Ganesh Chaturthi :) Love and respect form India

[@MikeJohnMentzer]

@danny supersell Bruh

[@felipelopes3171]

I think Ramanujan was quite aware of what his sum meant. He mentions it's the "center of balance" of the series. You can get the -1/12 value using the Euler-Maclaurin formula with an expansion around 1/n -> 0. This is a pretty novel way to interpret the value of the zeta function. Anyway, about this confusing people in calculus, I think people learn much more math by analyzing where the -1/12 comes from than when they dismiss the formula.

[@akanksha8311]

well said

[@Entropy3ko]

The problem is that people do not understand how limits work, often Functions like F(x)= Sin(x)/x has a limit equal to 1 at x-->0, but the function itself does not exit at 0. So you can pin a value to F(0), but this does not mean the function at that point actually exists.

[@Wallofdenial50]

Kind of a divide by zero accident

[@general_paul]

​@@Entropy3ko people conveniently skip the word "approach" when studying limits and then complain when shit blows up

[@guillermoh.k.2967]

Remember: Ganesha knows when you stop paying attention...

[@sanky1211]

😂😂

[@danielharris9403]

Someone had to address the Elephant in the room 👍

[@096astiksharma8]

@@danielharris9403 😂😂😂

[@IbrahimKhan-gv6cw]

No, it does not. Just stop lying.

[@nithishnithish504]

@@IbrahimKhan-gv6cw we have a religious fanatic here

[@techsandbulls1869]

man who knew infinity now i understand why the title of his biography makes sense

[@rajendramisir3530]

Srinavaasa Ramanujan said every equation was a prayer for him. Ever since I learned about him in my Calculus 2 class 30 years ago, he has been my Mathematics role model. He approximated the value of 1/Pi to a few decimal places using infinite series. Srinavaasa Ramanujan gave our world his gift of genius and departed peacefully from this material world. It feels sad to loose men of great minds like him. Although he was not a trained Mathematician, he was able to enlighten our world of Mathematics with his genius of Mathematical deductions and profound insights. I am impressed at how he caught the attention of Professor GH Hardy of Cambridge University by his(Ramanujan) letter of theorems on infinite series that diverge( 1+2+3+ ... = -1/12.

[@samtgamee3202]

thanks to Ganesha I will never forget infinity can't be subtracted from itself.

[@gulshansharma2131]

Why u can't subtract infinity from infinity ? Becoz u can't think up to infinity? Or u can't make sense about infinity ? Or u want to just follow others' work?

[@gulshansharma2131]

@Sam please elaborate a little bit😀

[@rishisingh1035]

@@gulshansharma2131 u can't subtract infinity from infinity , but u can subtract 2 numbers tending to infinity and get 0

[@beastgamer4932]

@@rishisingh1035 we’re not talking about limits here ok?

[@rishisingh1035]

@@beastgamer4932 then u cannot subtract infinity from infinity

[@yashjobanputra3832]

When he pauses for Hardy and then said "He is not dummy either", lol

[@paulczar]

If this guy had been by math teacher is high school, I’d probably be a mathematician by now. Amazing what a good, passionate teacher can do for you.

[@jacksummer4041]

😂 yeah a mathematician who's trying hard to find contradictions to proof that originally do not have any contradiction

[@angelc9818]

he has been my teacher

[@DocBree13]

I feel the same

[@ripjawsquad]

@@jacksummer4041 isn't that what good mathematicians do

[@AlexanderSuraphel]

@@angelc9818 I've read someone commenting the same. Which school was it?

[@chrisdavis5698]

who else was baffled by how breath-taking gorgeous that handwriting was?? jeez.

[@kcwidman]

Oh my freaking goodness, I jumped right out of my seat with my headphones in at 4:00.

[@Mathologer]

:)

[@Sudhirmishra30]

Kai Widman 7

[@JoshuaCastnerTrio]

me too 😅

[@eat_ze_bugs]

I jumped twice, second time at 9:29

[@twincatemy8997]

great video but this fuckin elephant freaked me out

[@Nero-ox5tw]

This is why KZbin is leagues better than Facebook, Instagram, Snapchat, etc.

[@drbhaviify]

Yes agree

[@ign_angry]

i use all

[@Subs1338]

Why?

[@thatkindcoder7510]

@@Subs1338 Because KZbin mainly teaches the objective truth, while other social media sites degrade your brain with attractive but biased false facts.

[@dr.palsonp.h.d815]

@@thatkindcoder7510 "objective" "youtube" HAHAHAHAHAHAHAHAH

[@heyitsalex99]

i havent watched the video yet, but thankyou for 3 reasons, 1. ramanujan 2. infinite sum of natural numbers 3. 30 minutes

[@Mathologer]

+heyits- alex Well, if you like Ramanujan, I'll probably do at least two more videos this year featuring his math :)

[@heyitsalex99]

+Mathologer yassss

[@Mathologer]

+Appleguysfriend A video on Galois theory would be nice (probably a 60 minute one though :)

[@funcaogamma7663]

+Mathologer yes

[@4Pssf2w]

+Mathologer yes

[@slappy8941]

LOL I like the shirt; "JUST OVER DO IT". This describes the German approach to engineering.

[@Mal-Function4]

we aren't that crazy... look at the russian uragaan-class guard ships... THEY overdid it on the weight distribution

[@gladtobeangry]

True. I used to live close to the german border near Aachen. I remember walking through a park there one time and thinking to myself "what are all these bulky stone monoliths lining the paths?". So I checked. Turns out that it were little lamp posts. If a dutchman had built those lamp posts, they'd have been made out of plexiglass, be covered in birdshit after day one, and be kicked into pieces by vandals after day two, after which we'd decide the park probably doesn't need lights. In Germany, they'll still be attached to the debris of earth when we get finally sucked into the supermassive blackhole.

[@marlobardo4274]

@@gladtobeangry love your comment about German Massivbauweise

[@iaroslavakornach]

hahah just noticed it! I want that t-shirt now

[@srinathtankasala]

@@iaroslavakornach did you find it? could you send me a link to buy?

[@jaimeriveras]

As in the case of quantum mechanics, I understand very little, but it’s enough to appreciate the beauty of it all. Amazing stuff.

[@devlsi9715]

The way lord Ganesha pops up with that sound.... It feels like he's giving some sort of mythological warning Edit: I forgot to mention while writing this comment that I was watching the video on Ganesh Chaturthi(weird YT algo right, for most Indians) and that my half yearly exams were a week away starting with MATHS as the first exam

[@sergiin3]

A man who knew infinity. Good book about Ramanujan's life.

[@michaelwinter742]

Interestingly, this video is the zeta function output of all the videos on KZbin being added together.

[@beta5770]

rofl, I agree

[@ashharhasan3120]

Clever beaver.

[@tanmayakaushik]

Wicked

[@jedus007]

Ramanujan was sheer poetry in Mathematics.

[@zainulabideen5895]

One Word for Ramanujan, "Genius of Genuis", He is a idol of motivation for math students, #Respect

[@robsonishere9062]

that is three words

[@hackermango]

@@robsonishere9062 lmao

[@WolfPack-gi7br]

4:02 that scared the hell out of me lol

[@darkdelphin834]

Same here xd

[@himanv]

Ganesh is the remover of obstacles and the lord of knowledge and light-- so yeah, makes sense he would scare the literal hell out of you! lol

[@sarojnayak8350]

@@himanv the sound was scary

[@franklee8032]

And he never answers the question posed.. what's the point of even showing it?

[@jacksummer4041]

I'm watching at 3am 😂 and i almost died

[@NoobMasterX1]

Don't wear headphone when watching this video, have a nice day

[@navdeeprahiya1964]

🤣 🤣 I get it now

[@Mpasayat4690]

At 4:02 especially 🤣🤣

[@gabrielkellar1935]

9:29

[@ralphmuller5664]

@@Mpasayat4690 d

[@shajuvv]

Too late man,too late😱

[@94D33M]

Next time when you get wrong in ur maths question, remember to tell your teacher about maths in another world, your answer is definitely right in atleast one planet in the universe

[@wav372]

If universe is infinite then your answer is right on the infinite amount of planets

[@shimpiyaa]

1 = 2 is right in our planet. Assume 1= 2 1/0=2/0 1×0=2×0 0=0 LHS and RHS are equal Hence proved. Infact any number is equal to any number.

[@shoumikchakraborty3604]

@@shimpiyaa Adding one more proof d/dx (X ^2) = d/dx(x) + d/dx(x) +d/dx(x) ... x times ....+d/dx(x) 2X= 1+1+...X times +1 2X=X Dividing X both the sides 2=1

[@hamiltonianpathondodecahed5236]

the teacher would be like : then go on dat planet

[@drbhaviify]

@@hamiltonianpathondodecahed5236 correct 😅😂😂😂😂

[@TommyAmbersonRD]

2:54 "keep it completely elementary" . My definition of elementary is clearly not the same as his.

[@aaronleperspicace1704]

😂

[@andreguimaraes9347]

The loud noise of the elephant thing coming up scares the crap outta me every time xD

[@acediamond5399]

+André Reis Yes, me too.

[@horselookinghumanheadass]

+André Reis That's Ganesh (Hindu God) XD

[@andreguimaraes9347]

Elephant thing is more accurate

[@horselookinghumanheadass]

lmao

[@nikhilwardrobe]

+André Reis I may not agree with you, but I will defend to the death your right to make an ass of yourself

[@DustinRodriguez1_0]

If given access to a time machine, my first stop would be to visit Hardy when he firsts open and reads that letter from Ramanujan. I can't even imagine what the next few hours and days of his life must have been like. He probably thought he was losing his mind.

[@Metalhammer1993]

Dustin Rodriguez nope he actually read it, showed it colleagues and set everything into motion Ramanujan got financial support and transport to England. Harvey was baffled someone with no formal training could deduce all that. but the content of the letter at best slightly surprised him. the shocker for Hardy was Ramanujans talent

[@dscruiter]

I would have made certain Ramanujan's health would be okay. Then we could have enjoyed his brilliance so much more and witness more math miracles.

[@tsresc]

@Tangent of circle. Thanks for giving me the idea. 😉 I'll teach him a few things in exchange of being my mathematician.

[@zoetropo1]

Dustin Rodriguez: if given access to a time machine, your first stop would be the explosion when you turned it on. That’s what happens when a backward-going time machine occupies the same space-time as a forward-going one.

[@chandrasinghrawat622]

Certainly

[@Poor_Vampire]

I never knew Ramanujan was SUCH A GENIUS ! Thanx to mathologer.

[@u.v.s.5583]

He actually could do much much more than this! But don't read this comment any further or you will regret it. Remember, I warned you. Stop reading this comment. Really. Ramanujan was such savage genius that he even caught in London Tbc instead of Covid.

[@gumnaam4302]

@@u.v.s.5583 Exactly, the man who knew infinity catches TBC in the heart of the nation that looted and pillaged more than half the world and was still grappling with TBC!! So much for being a pomp regime. Such a shame!!

[@shubhama891]

He actually wrote many formulas and assumptions in last year of his life. Researches are still going on. Many of his formulas are still used to find the black hole😲😲

[@user-gc6my9jg2c]

I like your videos! Several times I've had a technical comment that you specifically addressed at some point later in your video. I'm reviewing topics that I already know. Your explanations are pretty good! I also appreciate that your animations are simple and enough.

[@DaUsher]

Therefore, 1 > 1+2+3+4+5+...

[@chillphil967]

You jerk! lol

[@jimbean4751]

and that's why 0 isn't a natural number :)

[@bradenweber3654]

That's only using the Cesaro convergent, which deals with averages rather than sums. Such could make sense in the regular mathematical system of an alien planet, but we decided our system would be different. That doesn't stop mathematicians from find answers outside of our mathematical system though.

[@fdagpigj]

Unless I misunderstood, the Cesaro convergent only deals with the averages which are still positive, it's the analytic continuation of the Riemann function that gives the sum a negative value.

[@chizhang2765]

I find the use of word "alien" quite confusing. Isn't it just some sort of fun experiments with maths? Or do they describe actual physical realities in some way, that addition→finding converging point of average of partial sums would actually be useful in simplifying physical equations in certain cases? Please do tell me more. I would be really grateful.

[@jwinburn]

Ramanujen once wrote just for fun that the sum of the numbers from one was not aleph naught as we might have all thought but one twelfth times negative one

[@calebberhow2203]

This is the best math poem I have ever read. (Haven't read many, but I had a laugh.)

[@gaymerjerry]

aleph null isnt the sum of all the natural number aleph null is the total number of natural number

[@petrie911]

@@gaymerjerry In cardinal arithmetic, the sum of all natural numbers is indeed aleph null. In ordinal arithmetic, it's omega, which is equal to aleph null in the most common formulation of set theory.

[@gaymerjerry]

@@petrie911 i dont think you know what cardinal and ordinal means if you said that cardinality literally mean sizeof or absolute value while ordinal means ordered lists you can have -omega but -aleph null is nonsense aleph null is the amount of counting numbers there are while omega is ordering them then adding another one at the end of the list (as adding another one at the beginning of the list still cant really make omega as ordinal infinity is communitive)

[@gaymerjerry]

@@petrie911 and also its not called cardinal arithmetic "Cardinal numbers" are a generalization of the natural numbers to describe sets of infinite size in set theory thats it if you ever put on any math test that all natural numbers add to a cardinal number you will get 0 points because the act of adding number is ordinal numbers if you were saying you can combine 2 sets to get a larger set you can do that to prove 2 * aleph null = aleph null you have the set of {0,2,4,6,8,10,12,14,16,....} and {1,3,5,7,9,11,13,15,...} now according to standard axioms if a set is infinite but can still be made into an ordered list its size is aleph null set one is clearly listed where each term matched to 2x and set to each term matches to 2x+1 and the fact i could put a ... to imply more number further proves its an ordered list so combine these 2 sets of size aleph null creates {0,1,2,3,4,5,6.....} which is the natural numbers which is the set that aleph null is based on but can also map each term to x so combining these 2 sets we show 2 * aleph null = 0

[@Hecatonicosachoron]

Those interjections by Ganesha are terrifying

[@xendu-d9v]

Jay shri Ganesh

[@HinduWarriorForever]

"You are encapsulated by evil energies. Pray to Bhagwan Narsimha. Stay calm. Respect fellow people. Don't get into alcoholism.*

[@crewrangergaming9582]

Ramanujan was no doubt a god-send, whose mind formed in such a beautiful way that these came to him out of nowhere. It is spiritual, it came to him by itself, as if Universe conspired to make him aware of these nuances of maths in the Universe.

[@Eric06410]

He does this to show off his t-shirt collection

[@Mathologer]

There are easier ways to show off ones t-shirt collection :)

[@gourabdas8560]

😂😂😂

[@wolframalpha8634]

😂😂😂

[@bharth_yadav]

Lol

[@sillybears4673]

He is laughed at this comment because all his shirts are awesome and related to the videos usually lol

[@DDF546D5F4]

I think it is similar with defining sin(-30°). We cant see an angle like -30° in triangle, but we use it because it is useful and continuous. I think 1+2+3... is similar with my example. We think it diverges, but if we define sum of series in other way, it is -1/12 and it is continous and useful.

[@MahipalSingh-nv1bq]

whenever such a sum appears anywhere in physics - I explicitly emphasize that not just in string theory, also in experimentally doable measurements of the [Casimir force] (between parallel metals resulting from quantized standing electromagnetic waves in between) - and one knows that the result is finite, the only possible finite part of the result that may be consistent with other symmetries of the problem (and that is actually confirmed experimentally whenever it is possible) is equal to −1/12. It's another widespread misconception (see all the incorrect comments right below your question) that the zeta-function regularization is the only way how to calculate the proper value.

[@maxpower8460]

Hey I'm actually super curious to know more of what you mean! Any source I could dig in more? Can be an academic paper too.

[@igoranisimov6549]

Last time I heard string theory did not get experimental validation.

[@irokosalei5133]

@@igoranisimov6549 R e a d b e t t e r b e f o r e c o m m e n t i n g

[@igoranisimov6549]

@@irokosalei5133 I am professional physicist for the last 40 years. Am I talking to some kid who likes "anime"? Try doing something else in your life (sorry if I did not space every letter, don't have much time for cool things)

[@ArigatoPlays]

Man, I was really following the video understanding everything, being convinced at the end of this I'd understand the Riemann Zeta function only for you to hit me with that 32:27 lmao

[@HariPrasadindependantphysicist]

For anyone who wants to know what is the number shown in his shirt (from 16:53 to 26:09), its the expansion of *pi* in base 3 = 10.01021101222201.........

[@filipedelduque9407]

+Hari Prasad thanks XD

[@klobiforpresident2254]

He showed that one in his video about hating pi (not pie) already.

[@stefstefstef6789]

That's base 4

[@klobiforpresident2254]

stefstefstef6789 it should be base three, ...

[@jeymsie2474]

Thank you. Got confused by that number

[@Bleubear3]

How did I even get here. Autoplay set me up. I gave it a go, I'm at 16:31 and I give up LOL "This makes zero sense to me." That's all I've been saying the past couple of minutes. I'll let you mathematicians be. You're doing us a great service.

[@Mathologer]

+BleuBear Well, since you made it to 16:31 you get my seal of approval anyway :)

[@smb123211]

There are several articles on the web explaining this trick. Scientific American (for example) noted the intellectual dishonesty of redefining terms without telling the reader. But the best (and funniest!) explanation is at Physics Buzz blog. it clearly shows that "-1/12" is an area obtained by graphing the integration of x(1+x) / 2. (This is the formula alluded to by SR) Raising x to a power (in the formula) gives various answers but what's interesting is that x to the 13th gives...-1/12 - the same as x to the 1. LOL

[@roughysk9851]

Ramanujan was genuinely a masterpiece

[@AnneoSeneca1993]

I'm a PhD student in Theoretical Physics and I found this video really helpful and entertaining, thank you so much! :)

[@philippedavid0]

Great video, but man you made me jumpscare with those " *BOOM* Elephant speech bubbles" !

[@mtomazza]

+Philippe David me too

[@Jay-cn3js]

4:23 blew my mind, the way that man' mind must of worked. How he approched math, crazy. Talk about thinking outside the box.

[@AdrienLegendre]

Thanks for an excellent presentation. A related area that I am reading about now is Borel-Laplace summation and transseries. It is amazing that Ramanujan uses grade school arithmetic to perform the difficult task to extend the Riemman zeta function in the complex plane to z= -1. Another approach for the early part of the video: f(x)=1/(1+x) = 1-x+x^2-x^3+..., take the derivative df(x)/dx= -1/(1+x)^2 = -1+2x-3x+4x..., and set x=1 so result is 1/4=1-2+3-4+...

[@mylord6868]

i am hindu and i am totally ok with Ganesh thing, and i am not going to Charlie hebdo office😀. btw nice explanation professor👍👍

[@hank1519]

A little too soon for the joke, I fear.

[@DharamG]

My lord, Who is asking you what you are? U can't give any certificate pretending to belong to a community

[@BBCBullshit]

@@deka0014 😂😂😂😂😂😂

@@deka0014 you are as fake as your name is. Plus you're a coward (and obviously not a hindu) who had to hide while doing it.

[@alvarogoenaga3965]

Cool. After all, all deities are probably and possibly products of some people's imaginations.

[@vaibhavtiwari8865]

Remember Ramanujan don't have any formal education even after that He was a Genius ❤️

[@AaronBruffett]

Mathematician Love poetry: How do I love Thee let me count the ways -1/12 ways

[@teressacooks7928]

2:43 the most telling and interesting of this entire segment👀! Insight into Ramanujan himself (see graphology) ✍

[@arnabpaulchoudhury8455]

Lord Ganesha surely knows his stuff... :D

[@aaronleperspicace1704]

Shiva taught him

[@enejidjsi5939]

@@aaronleperspicace1704 ohk

[@BillyBuerger]

The part that bugs me about this and other things I've seen about infinite series is that people just re-order the sequence and say things like "oh, we'll just put these here." as if that is fine and doesn't have any affect on the output. So you do something that doesn't make sense and then use that to prove something that doesn't make sense. Sure. For instance, if... C = 1, 2, 3, 4, 5... Then 4C would be ... 4C = 4, 8,12, 16, 20... But instead, you decide you will change 4C to be... 4C = 0, 4, 0, 8, 0, 12, 0, 16, 0, 20... By lining up the number however you please, you are ignoring the sequence and changing it. So of course when you then subtract your adjusted sequence from your initial sequence you don't get what would make sense which is... -3C = -3,-6,-9,-12,-15... Maybe there is a reason to change the sequence. But if so, no proof was supplied for doing so. Is there a proof for infinite series that explains why you can change the sequence?

[@Mathologer]

If you listen carefully it should become clear that the "just do it" approach that you are worried about never counts as a proof to start with and is only used as a quick tool to build some intuition for what to aim for. When it is and when it is not okay to reorder and rearrange series is something that is sorted out whenever one introduces one of these different ways to sum series :)

[@BillyBuerger]

Thanks for the response. I appreciate you taking the time to answer. Although it leaves me wondering what the point of this is?

[@Mathologer]

Well, one of the points is to find this analytic continuation of the Riemann zeta function which is a very big deal in math :)

[@BillyBuerger]

Okay, I did some more reading and some articles that are critical of the Numberphile video on this same topic. Seems the confusion being the = part. As you state near the end, the answer on a test of 1+2+3+4... would not be -1/12. But you can use the methods you discuss to derive a value from divergent series that otherwise doesn't have a values.

[@Mathologer]

Infinity has many VERY PRECISE meanings within mathematics depending on the context it is used in. One thing it definitely does not mean is "undefined", at least not in mathematics. Also, clearly, none of this can possibly originate in the work of string theorist since there was no string theory in Ramanujan's time. In fact, most of what I talk about in this video dates back more than a hundred years (Cesaro sums, Zeta function). If you are interested in historical details maybe just google some of the terms that pop up in this video :)

[@prateekkumartiwari7294]

Ramanujan as one of the greatest mathematicians of all time. The sheer genius of Ramanujan in number theory always fascinated. The great man often ignored proofs for many of the derivations. It’s like this; a genius looks at a problem, the path looks obvious, so he skips the steps, reaches the solution, but the mathematicians are still pondering on the left conjectures.(like the professor above) So yes Ramanujan also did interesting mathematics in the field of infinite summation Amazing!!! Our sum is negative! It looks like god plays with numbers in a bizarre way. This shocking result is not known to many non-mathematicians. Number-theorists call it “One of the most remarkable formulae in science”. This summation is a secret of mathematics kept away from layman. Further, it is interesting to know ‘S’=-1/12 has been used to derive the equations in “string theory”, quantum field theory and in some complex analytics. So now you know sum of positives can be negative. We knew much less during schooling about summation of numbers. The result teaches how universe can be more complicated than we think and how we need to keep an open mind to learn more. And everything is correct in the real world. But some people will never get it like the professor here

[@prateekkumartiwari7294]

Must read-- “The Ramanujan Summation: 1 + 2 + 3 + ⋯ + ∞ = -1/12?” by Mark Dodds link.medium.com/bK8txtm760

[@prateekkumartiwari7294]

Professional mathematicians certainly make mistakes and fool themselves, but this was never the case here. Neither Hardy, nor Littlewood, nor Ramanjuan ever thought that they “proved” that the natural numbers add up to a negative magnitude.

[@Christian_Martel]

Mathematics are similar to these granular tests we do at engineering university. Out of a boring standard soil, you’ll always all sorts of gems at every filtering layer… From the natural numbers to Ramnujan summations, through fractions, irrational numbers and the complex domain, a wonderful world of discovery!

[@carlosffm]

It is actually nice to see a video that so elegantly shows the beauty of math, and in such a simple and fun way too, thank you very much

[@UltraRik]

Thank you for teaching us the complex math of blue aliens

[@HDitzzDH]

7:28 Correct me if I'm wrong, but when it comes to the geometric series the absolute valur of r must be strictly

[@u.v.s.5583]

But now imagine you are doing a hard physics problem and get this sum as the answer, and you know the sum exists. Do you say, no, the sum does not exist, or do you try to generalize the very concept of the sum? Of course, you generalize the concept of the sum, and this is what one does here. College calculus is out at this point.

[@parshvpatel9644]

LoL I tried praying Lord Ganesha and I found myself more concentrated with some inner voice. Will love to do maths with Lord Ganesha in future :::))) My extraordinary wisdom source.

[@jimmyalderson1639]

It's hard enough learning this, just imagine how hard it was to solve in the first place

[@UdayDesai]

Perhaps this is the best presentation I ever have seen on Math or Science. Awesome!!

[@HeinrichTsanov]

The elephant god jumpscared me for many times...

[@KL-sq9hg]

his name is lord ganesha well dont say like he is bad i and many more than a billion people around the world worship him

[@ullqiorracifer9935]

+Firicious Gaming- Clash Royale and More i dont think everyone in india is a hindu

[@nalawade]

around 80 % are

[@KL-sq9hg]

***** well i said around the world just google it

[@autodidactusplaysjrpgs7614]

Stimulus and response

[@tensevo]

This is extremely fascinating, and puts me deeper down the Riemann hypothesis rabbit hole. Has anyone considered how Ramanujan-sum could be used to explain black hole singularities? If I understand correctly, there is a way in complex number theory that infinity could sum to a constant number. Would this allow physics to work, that would otherwise be broken by singularities?

[@RamirodeSouza]

*WARNING*: Ganesha will scare the shit out of you.

[@joshuajurgensmeier4534]

+Ramiro de Souza Every time.

[@MereddynYT]

+Ramiro de Souza yes he did -_-

[@gobstompper13]

Enjoy your channel. I'm sorry to point out that your elephant sound byte scares me.

[@duvanm2483]

thank you, this is the most complete video about this subject that has been killing my mind for 2 years...

[@handsomenubian]

Now I'm at home in the curfew I have never thought I would watch a math video and enjoy it like I have just done.

[@JasonFuller]

20:30 Again, just labeling something "Cesaro convergent to 1/2" causes a loss of information that could very well be of theoretical importance later. Means are statistical tools that aren't all that useful without knowing at least the standard deviation. Shouldn't the amplitude (and possibly periodicity) of the oscillation also be tracked when reducing a series to a Cesaro number?

[@furrball]

All I know as a programmer, is that if you keep adding up positives and get a negative result, you probably had an overflow. We can fix that by putting more bits into our sum accumulator. Unfortunately, if we're adding up an infinite amount of positives, there aren't enough resources in the universe to make the variable big enough to hold the result, hence it will always overflow. It's all clear now! yay me! ;-)

[@snbeast9545]

You could just define 1 + 2 + 3 + 4... as -1/12, just like how pi and e are usually defined in programs instead of calculated.

[@noobita4983]

Alessandro Ghignola Grats you broke the code!

[@dragonsdream4236]

But I don't think overflow applies here, ramanujam is just performing simple arithmetic operations on the whole infinite series, not adding it integer by integer as a program would.

[@anandsuralkar2947]

I m.studying computer science engineering but all i get to learn at my college is total shit help me..i want to learn cse

[@Slackow]

Alessandro Ghignola it’s not just negative, it’s also a fraction. So you’d never get a fraction from adding integers. Even with overflows. This has nothing to do with overflows.

[@ATLTraveler]

You should go over his theory on how to calculate the number of primes within a given integer, this is what really impressed Hardy.

[@pramodvora2067]

Congratulations!! Your attempt at this is indeed very successful. Many many thanks for this explanation both here and other planets!! We know our maths is just an approximation as we are not perfect. If we get enlightened, we will see things very clearly. This is the inspiration to lead the kindly light.

[@whatby101]

As a student only in pre-calc, i genuinely look forward to working with numbers such as these someday. On a side note i watched the whole video and everything made perfect sense, thanks for making it so clear.

[@kaihulud87]

+whatby101 Mathematics is a very interesting field and its amazing that people can be so intrigued by it no matter their education! I struggled to love maths for awhile now but I like to think I see the inherent beauty in it now. Before he announced it but showed us the complex realm, I was freaking out because it dawned on me that he was about to mention the Riemann-Zeta function which in itself I tried to understand awhile back and walked away agreeing to disagree with the maths..hah

[@whatby101]

+Password1andUsername That's not really an option for me. Even if I myself am capable of just skipping calculus, I am only in high school. Ap Calc is the highest level class they offer. I will take that junior year, then the following year I plan to take 3 different science classes and take a class on statistics. Sadly in any public school, there are not enough mathematically gifted students for there to be an even higher level course. At best I can take a summer class at a college either this or next year.

[@whynot1150]

This happened to me as well I took AP calc this past year as a junior, and I am not without a math class. To fix this problem I made a deal with a college nearby, and now I am able to take further math classes there. I don't know if that is a possibility for you, but it was the only way for me, for my school only offers two AP classes that I haven't yet taken. Hope you are able to take more maths instead of having a gap year.

[@lamnguyen-uh4tz]

I have an option for you. Hang out with us on math.stackexchange.com We have tons of smart people, and we're always willing to explain super weird topics (plus we can help with calc if you need.)

[@denjam2423]

I've just earned the mathologer seal of approval and I'm proud of it!

[@Mathologer]

+denjam Cool, I really wonder how many people watching this video can say this of themselves :)

[@V-for-Vendetta01]

brilliant. I love mathematical paradoxes and puzzles and this is just perfect for me.

[@trololollolololololl]

Rishabh Vailaya this aint a paradox

[@jestes7]

Filip what is it then

[@btdpro752]

@@trololollolololololl shhh let him stay in his feel good bubble

[@V-for-Vendetta01]

@@btdpro752 why so salty? Looking at others happy triggers your little brain?

[@btdpro752]

@@V-for-Vendetta01 The fuck do you mean. Reread what I wrote. I was saying let him be happy. My god what's wrong with the internet.

[@ralphallengamboa9394]

This is much better than an intense film. That Elephant jump scares.. 😂

[@MrDrgdf]

Hindu God Ganesh.

[@marksmod]

The elephant god of jump scares (aka. purple heart attack)

[@McRaylie]

+marksmod Ganesh?

[@SomeoneCommenting]

+McRaylie He means that every time that it appears the sound is VERY ANNOYING. I was using headphones when I was watching this video and it is a very unpleasant sound that makes you jump because it is unexpected and awkward.

[@QuantumConundrum]

+marksmod Ganesh is secretly trying to murder all the mathematicians.

[@Hect0rxP]

hahahahaha all this comments have made me laugh so hard!!! hahahaha and yeah! freaking Ganesh! my heart almost stops >_

[@shepaustad558]

Be kind

[@rajeshkoppa2367]

True knowledge comes from humblest of origin.....

[@rosaliacastro1551]

Not necessarily.

[@KodamatiUdayaPrabhuKiran]

@@rosaliacastro1551 necessarily

[@jovian304]

@@KodamatiUdayaPrabhuKiran not necessarily.

[@KodamatiUdayaPrabhuKiran]

@@jovian304 I guess your name says you are dumb and arrogant

[@jovian304]

@@KodamatiUdayaPrabhuKiran Well I know meaning of ur name is sunshine. There's nothing wrong in having name as "sunshine". Just don't judge others.

[@priyanshuindra4648]

You always make great content for viewers with your hard work. We appreciate that.

[@humanidrome]

This video took me on an interesting tour: up the insane-o-meter to the fourth degree, down to the first again, and at the same time up to the fourth level of maths enlightenment. Thank you Burkard & Ganesha

[@MrPoutsesMple]

It's amazing how complex analysis pops in...Really cool video. I enjoyed the visualizations, but the elephant God was a bit scary..

[@MyAce8]

+ΣΚΡΟΥΤΖ ΜΑΚ ΝΤΑΚ He came out of nowhere, I am going to have nightmares for weeks

[@rcksnxc361]

+Ace shinigami lolxd

[@MyAce8]

I guess it's more the sudden noise that accompanies him

[@TheJohnskinner]

Props to anybody who can understand this type of math , as for myself, I have no idea what’s going on but I watched the whole video, Absolutely mind boggling

[@PaulKruskamp]

I did the maths and came up with 42.

[@JakeIsEpicHavard]

😐

[@PaulKruskamp]

EpicGuyRandom you don't get it.

[@JakeIsEpicHavard]

+Dean Thorton no i do 😂 but that is the answer to everything ...

[@PaulKruskamp]

EpicGuyRandom lol, so silly

[@perryt2039]

I did it and came up with 69.6969696969696969696969...

[@1nd93dk3]

I reached the -1/12th level of enlightenment

[@Wow-vu8sf]

Thank you for making these really high quality mathematical videos

[@smritidipa]

I think you should give a image of NAMAGIRI the Hindu goddess on whom "Ramanujam" ( actual name not Ramanujan) had faith.

[@hello123s]

@Adarsh Jha it is.. She is a feminazzi.. Who just wants credits for no reason..

[@stuntxo3968]

Correct

[@anshul6168]

He had faith over all Hindu Gods In Hinduism you can worship any god but that doesnot mean people who worship Devi dont have faith in Ganesha Ganesha is known as obstacle remover

[@samchaco2]

Right? Its so poor research, he actually credits Goddress Namagiri for revelations. How easy would that have been to quote.

[@YEC999]

The Ganesha animation is like Pixar spent 2 years to make it perfect. It really gives me the chills when she comes.

[@bosongod2830]

*he

[@gigachad2419]

he*

[@adityanair7159]

Jai shree Ganesha Cutest God 🙏😇 The God of Wisdom .

[@xendu-d9v]

Jay shri Ganesh

[@catwithsunglasses3949]

@@KhvLeD yea what’s wrong with that? I’m an atheist but even I don’t think anything is wrong with believing in multiple gods lmao.

[@sttpt9217]

@Store Patternator 😂 lol we don't belive in man made religion like Islamism, Christianity ..

[@superman5742]

@Store Patternator nothing...just carry on with our respective lives 😂

[@Hwd405]

Thanks for this video - there aren't many other videos about explaining analytic continuation or what these divergent sums really mean, and you "summed" it all up really succinctly. Another very interesting place that these sums pop up are in real life - calculations in string theory and of the casamir effect seem to suggest that certain values are actually equal to divergent series like 1^3+2^3+3^3+..., and interestingly enough experiments to determine these values seem to agree with the value of the series when defined by the function's analytic continuation at that point. Spooky!

[@Mathologer]

+Hwd405 Actually this is something else I'd really like to do, explain how exactly the Casimir effect can be calculated using 1^3+2^3+3^3+... . The usual "just do it" physics explanations are nice but in the end just don't do it for me :)

[@nicoleayalon4640]

Very cool video! Just watched the entire thing 😁 I just recently learned a bit about complex functions in the context of a path-planning robotics problem where the robot's path had to be restricted to certain homotopy classes defined by obstacles in the world. All that to say, now I think I can see why analytic functions can be so versatile and useful 😄 Just when I thought I would start hating them... Anyway, thank you for making these videos they are so informative and easy to follow along with. This one was especially interesting! Also noticed the change of shirts, was it to go along with the topics lol is the last one for Super Sum? And was the first shirt meant to be read as 'Just overdo it'?

[@aee220phmunirabad]

-1/12 is closely related to string theory. Strings are basic elements of particles like electrons , protons and neutrons

[@SriNiVi]

Haha.

[@u.v.s.5583]

The first sentence is true, unlike the second.

[@TenzinLundrup]

I saw the Ramanujan sum mentioned in a talk on quantum field theory.