Numberphile v. Math: the truth about 1+2+3+...=-1/12

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Confused 1+2+3+…=-1/12 comments originating from that infamous Numberphile video keep flooding the comment sections of my and other math RU-vidrs videos. And so I think it’s time to have another serious go at setting the record straight by having a really close look at the bizarre calculation at the center of the Numberphile video, to state clearly what is wrong with it, how to fix it, and how to reconnect it to the genuine math that the Numberphile professors had in mind originally.
This is my second attempt at doing this topic justice. This video is partly in response to feedback that I got on my first video. What a lot of you were interested in were more details about the analytic continuation business and the strange Numberphile/Ramanujan calculations. Responding to these requests, in this video I am taking a very different approach from the first video and really go all out and don't hold back in any respect. The result is a video that is a crazy 41.44 (almost 42 :) minutes long.
Lots of amazing maths to look forward to: non-standard summation methods for divergent series, the eta function a very well-behaved sister of the zeta function, the gist of analytic continuation in simple words, etc.
00:00 Intro
23:42 Riemann zeta function: The connection between 1+2+3+... and -1/12.
38:00 Ramanujan
40:36 Teaser
The original Numberphile video is here
• ASTOUNDING: 1 + 2 + 3 ... . Also check out the links to further related Numberphile videos and write-ups in the description of that video.
Here is a link to Ramanujan’s notebook that contains his Numberphile-like 1+2+3+… = -1/12 calculation. www.imsc.res.in/~rao/ramanujan...
This notebook entry was also one of the starting points of my last video on this topic: • Ramanujan: Making sens...
Other good videos that deal with this strange “identity” include the following:
• Why -1/12 is a gold nu... (a Numberphile video featuring the mathematician Edward Frenkel who is also talking about the connection between the Riemann Zeta function and Ramanujan's crazy identity.)
• But what is the Rieman... (a nice 3Blue1Brown video about visualizing the analytic continuation of the Riemann Zeta function).
If you know some calculus and want to read up on all this, beyond what is readily available via the relevant Wiki pages and other internet resources, I recommend you read the last chapter of the book by Konrad Knopp, Theory and applications of infinite series, Dover books, 1990 (actually if you know German, read the extended version of this chapter in the 1924 (2nd) edition of the book "Theorie und Anwendung der unendlichen Reihen". The Dover book is a translation of the 4th German edition. The 5th German edition from 1964 can be found here: gdz.sub.uni-goettingen.de/id/....
People usually recommend Hardy's book, Divergent series, but I'd say only look at this after you've looked at Knopp's book which I find a lot more accessible. Having said that, Hardy's book does have quite a bit of detail on how Ramanujan summation applies to the Zeta function; see chapters 13.10. and 13.17.
The article by Terry Tao that I mentioned at the end of the video lives here: terrytao.wordpress.com/2010/0...
Thank you very much to my mathematician friend Marty Ross for all his feedback on the script of this video and for being the grumpy voice in the background and Danil Dmitriev the official Mathologer translator for Russian for his subtitles.
Enjoy :)
P.S.: Here is a scan of the page from that String theory book that is shown in the Numberphile video. Note, in particular, the use of equal signs and arrows on this page. www.qedcat.com/misc/String_the...
For today's maths t-shirts google: "zombie addition math t-shirt", "label your axes math t-shirt".

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1 июн 2024

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Комментарии : 10 тыс.

@nivednewalit8117 5 лет назад

This is the math equivalent of a diss track.

@goyonman9655 5 лет назад

Math Battle 😂😂

@bilalkhares9337 5 лет назад

loooooooooool

@jaytan531 5 лет назад

Universal Kombat dont you mean -1/12 more important things

@nowonmetube 5 лет назад

Yeah but the only misconception he got is that value = sum Which is not the case. Edit: To be fair, the numberphile video explained it horribly wrong if I remember correctly. They made an updated video called "why - 1/12 is a gold nugged" that one's much better in explaining.

@nowonmetube 5 лет назад

@Multorum Unum 😐

@smith22969 5 лет назад

Your German accent automatically raises your math credibility by 3 points.

@Mathologer 5 лет назад

@AbhijitZimare1 5 лет назад

If it was Asian, it would be +100

@schrodinger6991 5 лет назад

@@AbhijitZimare1 i don' belive you

@user-kx7do4fh2j 5 лет назад

One of my favorite mathemathians is Cantor. He was German. Too bad he died a broken man because he was bullied because of his theory about cardinality.

@paulcasino9511 5 лет назад

I thought it was Indian

@DemitriMorgan 2 года назад

I could swear, when I took number theory, one of the first homework problems was proving that the sum of two natural numbers is another natural number.

@spiderjerusalem4009 2 года назад

how did that go?

@praharmitra 2 года назад

Two, yes. Finite, yes. Infinite? No.

@scinary7052 2 года назад

@@praharmitra if 1+2 is natural, then the result, 3+4 must also be natural. It'll always be natural even when you do it infinite times.

@l.w.paradis2108 2 года назад

@@praharmitra 1. Every partial sum is, by recursion, the sum of two natural numbers, and hence must be a natural number. 2. The set of all partial sums is countably infinite.

@praharmitra 2 года назад

@@l.w.paradis2108 I don't understand what your point is. Rational numbers are countably infinite. The infinite sequence 3, 3.1, 3.14, 3.141, 3.1415, 3.14159, ... is a sequence of rational numbers and each element of this sequence is a rational number. Yet, the limit of this sequence is pi which is not a rational number. Same goes for the sequence 1, 1+1/2^2, 1+1/2^2+1/3^2, 1+1/2^2+1/3^2+1/4^2,... where every element is a rational number but the limit is not.

@charlesje1966 Год назад

Thanks. I never understood Numberphile's assumption that an infinite series can have a fixed value like 1/2. It seemed arbitrary to assign a value but the presenter acted like it was self evident.

@raimundomuthemba766 Год назад

Bro it was so poorly explained it seemed like they were just randomly throwing in series that would conveniently result in the desired -1/2. Laziness and math do not go hand in hand. Ever. Even on RU-vid... I was fortunate to immediately go into the numberphile comment section and see someone recommend this video.

@osmarfreitas8646 Год назад

The sum of an infinite series of numbers can be a fixed value if it is convergent (e.g. 1/2 + 1/4 + 1/8 + 1/16 + ... = 1) as the video explains

@osmarfreitas8646 Год назад

@@candylover6419 search for "sum of convergent series"

@anomaliecosmos 7 месяцев назад

Arguably it is assumable for some cases, because it is *true* for some cases - convergent series, as another reply states. But something does have to be a convergent series for things only true about convergent series to be true about it, so you have to at least have an intuition for whether a series will converge if you don't know for sure - and while my own test isn't 100% accurate, it DEFINITELY rules out series whose terms *increase rather than decrease*. My point being I agree that here was not the place to act like that was a given.

@l.w.paradis2108 6 месяцев назад

You did this in grammar school when you divided 1 by 3 and got 0.3333 . . . and so on to infinity. This means 3/10 + 3/100 + 3/1000 + 3/10,000 + . . . + 3/10^n + 3/10^(n +1) . . . for all *_N_*

@Josh-zu8cr 4 года назад

Never I thought I would see the day that a maths channel gets exposed by another maths channel

@chrisven899 4 года назад

@Mika Hamari Could you somehow explain it to me? I am a high school student and my basic logic skills say that it is impossible to reach a negative result with positive additions. (Also english isn't my native language, so excuse some grammar or vocabulary mistakes).

@chrisven899 4 года назад

@Mika Hamari So, is there a fault on the calculations?

@ElectroMathExp 4 года назад

yes they had a contradiction . the series doesn't converges .but they assumed it does converges and they used the properties of convergent series to find -1/12 .which is impossible since we are summing a positive integers . and the correct answer is that the sum approches infinity when n goes larger and larger .but what is more interesting is some how -1/12 is related to the series and it has applications in string theory and quantum mechanics even though it came from wrong assumption

@lupsik1 4 года назад

Mika Hamari You can disprove convergence of all of those with all basic tests like D’alambert, Cauchy, Integral test and Leibniz for the +/- series, which are tools people learn on the 1st year of technical college. Really scary how few people talked about how flawed the numberphile video was

@supersonicgamerguru 4 года назад

@@lupsik1 I think the big thing is that the majority of people are divided into two categories: People that have seen this all before in math classes but forgot some of the specifics and caveats, and people who haven't and trust professional mathematicians more than their own intuition. The latter group are the ones that would have been confused and bugging all the other math channels to explain it or something, which is what caused any of this. In reality, the numberphile video isn't "debunked", just properly contextualized and constrained. The issue with people bothering other math channels about the confusion is really the full extent of any damage that could have been done, at least that anybody should care about. If you're taking stuff from a youtube video and using it as the sole justification for anything you do on any math exam or really anything ever, then you have a bigger problem.

@CoryMck 6 лет назад

Things are heating up in the Math community of RU-vid.

@pentacles__ 6 лет назад

Things about to get lukewarm up in this piece

@proghostbusters1627 6 лет назад

Waiting for Numberphile's response.

@turtle7562 6 лет назад

keemstar and scarce will be all over this in no time.

@CoryMck 6 лет назад

I'm waiting for the disstrack

@doubtfulguest5450 6 лет назад

The maths drama is the best drama. These guys don't mess around. Watch out for the diss equations - they can be savage.

@joshuastucky 6 месяцев назад

As someone who holds a PhD in analytic number theory, I appreciate the exposition here. The ideas are clearly presented and give a relatively complete explanation of the phenomenon occurring with -1/12. The explanation of analytic continuation was particularly nice, as this is a concept that's definitely tricky to pin down if you want to get into the technicalities around it. Glad to see some quality mathematics communication concerning the infamous Numberphile video.

@user-yi5cc9wn5c 6 месяцев назад

Can I ask you something?

@joshuastucky 6 месяцев назад

@@user-yi5cc9wn5c sure

@PC_Simo Год назад

39:20 Also; even Ramanujan, for all the formal education he lacked, didn’t call the identity: ”Sum”, in his personal notes. He used the notation: ”c”, for: ”Constant”.

@samueldeandrade8535 8 месяцев назад

Kinda po-tei-to, po-tah-to. But, yeah, was a careful move.

@PC_Simo 8 месяцев назад

@@samueldeandrade8535 I agree. It *_IS_* a kind of a small thing. But a lot of people just want to misunderstand others, and will take any excuse to do so, however minor. That was a careful and smart move, to disarm such people.

@dustein4221 3 года назад

Another way to put this is this: the sum of all positive integers equals -1/12, for very specific definitions of the words "sum", "positive", "integers", and "equals".

@chetricker 3 года назад

Mainly sum and equals but yeah

@KRYMauL 3 года назад

Or just use lim x-> 0 x+1 bc 0+1 = 1 the series is divergent.

@baruchben-david4196 3 года назад

Also, 1/12

@jensrenders4994 3 года назад

No, only sum.

@90800905675 3 года назад

Very much agree with this one, context is everything

@dk6024 4 года назад

"For every difficult problem there is a solution that is simple, easily understood, and wrong." H L Mencken

@otoyana 4 года назад

This sounds relevant only when you don't know who the author of the quote is.

@poogmaster1 4 года назад

Minakami Yuki What’s wrong with Mencken?

@sottallu 4 года назад

The original solution is also simple and easily understood by mathematicians of this era. Does that mean that even the original solution is wrong?

@dk6024 4 года назад

@@sottallu It asserts such "solutions" exist but makes to claim as to which "solutions" those are. It's merely a warning not to be fooled by simplicity.

@patjvr 4 года назад

Kinda like the opposite of Occam's razor

@tomaszberent801 Год назад

The best complex logics/math film I have ever seen. By “complex” I mean “consisting of many, sometimes, non-trivial elements”. If I confess I am awarded Best University Lecturer for many years, it is only to pay tribute to the quality of this film - to keep things so ordered and clear is SIMPLY AMAZING! I do appreciate the apologies for not explaining why complex numbers needed to be introduced (but no fully explained) when analytical functions were being talked about. It gives a lot of security to a lay listener that all vital things were introduced even if no all were fully developed. Yes, the content still can be completely wrong (I am not an expert to judge) but it is certainly “CONSISTENT and COMPLETE” - in contrast to the film it was commenting. The detailed and well paced debate with the statements of Numberphile content were excellent. Well, it was really impressive. I do not subscribe to any channels and social media but believe me, I will be watching you regularly!!! Well done (you know it 😊).

@jceepf Год назад

Absolutely agree with you, I am a professional physicist so I can judge this video with some degree of expertise. It is absolutely brilliant. I was wondering how he would justify analytic continuation.... he succeeds even for a high school level educated person in my view. I am still dazed by the level of pedagogical expertise.

@margodphd 4 месяца назад

I have a slight suspicion who You are, and If I am correct - we might have passed eachother a few times on Madalinskiego. My late father spoke very highly of You. Odd, getting teary eyed under math video, of all things.. With the current level of growing mistrust of science, I am eternally grateful for those smarter than me being on guard for falsehoods. I understand the desire to simplify complex subjects but this is unacceptable, not because it's a mistake -as these happen to best of us, but because it seems to be almost consciously feeding into the "stupid scientists, power to the simple minds, they are hiding truths from you" type of the political climate and I viscerally hate anything that creates artificial divides between people, some of whom perhaps could be lured into the dark side of learning and reason still. Thank You, Mathologer.

@jeffbguarino 3 месяца назад

Yes but he still assumes induction is valid forever and it isn't . The universe will stop you at a large number. You can't count forever. It is impossible. Physics will stop you from adding "one" to some large number and that will be the biggest number possible. You can't escape the universe.

@monkerud2108 2 года назад

Having rewatched this for nostalgia:) it really reminds me of early math education in primary school, where you just get told stuff with no justification and even though most of the methods you learn there are common sensical, the point of math is to connect common sense with rigorous logic. And pretending something makes sense out of the blue is a really hard thing to unlearn and i think that sets a bunch of kids up to hate maths. Which is really a sad thing.

@misanthrophex 11 месяцев назад

Not much philosophizing in primary school math though... Some people just don't like math, some people just don't like poetry. Some like both.

@pugsnhogz 10 месяцев назад

@@misanthrophexI have a BA in creative writing/English and now as a tutor, I also teach marh I can say with confidence that if primary school math involved more "philosophizing," the number of kids who "just don't like" it would drop significantly

@Acetyl53 10 месяцев назад

@@misanthrophex Arguing for uncaused causes.

@scott1564 9 месяцев назад

@@pugsnhogz I would strongly argue it would be the opposite. The mere seconds (if that) of attention span these kids have precludes virtually any form of philosophizing as it relates to much of anything, especially math. Putting that aside, they probably wouldn't get it anyway. These are, for the most part, people who, when presented with math word problems, freak out. I've never understood why anyone would have an issue with word problems, but then again, I've never had an issue with math. I had to study for Calculus, etc. but very little in math classes prior to that.

@TomCruz54321 9 месяцев назад

The reason many teachers don't explain the equation is because they themselves do not know the explanation of the equation. They just pull out the book and tell the kids to memorize the equations and methods, and this is a very boring way to learn math.

@DavidSmyth666 6 лет назад

Forget Logan Paul and Shane Dawson, numberphile vs mathologer is the real youtube drama of 2018

@steliostoulis1875 6 лет назад

There is no drama just mistakes

@alephbunchofnumbers 6 лет назад

Don't forget #shitholegate lmao Or rather, don't forget to forget it

@carbrickscity 6 лет назад

Numberphile just made the mistakes of picking Physics professors instead of real mathematicians to present some of their videos.

@frankschneider6156 6 лет назад

The interesting thing about it is that physicists often really don't understand the deep subtleties of the maths they apply, abuse the maths in a way that makes every mathematician cringe, and get out a result, which is exactly in-line with how nature behaves (just think of normalization in QED).

@cunningwolf4516 6 лет назад

DavidSmyth666 so this is what future arguments look like

@vaneck4438 4 года назад

*start of video* "This is a serious video so I'm wearing black" *later* Zombie + Human = 2 Zombies

@lokithecat7225 4 года назад

You forgot; "Und now we discuss Supersum" and switches into Black Superman shirt.

@RalfsBalodis 4 года назад

One does not simply change t-shirt 4 times in a video and gets away with it... oh wait. He did.

@alexandren.9346 4 года назад

@- RedBlazerFlame - The Zombie is like an Extension of the normal world: Your mathematical rules don't work here, human! 😈 Or you could say: This is the value you expect. The human is "converted" into a zombie, which actually makes sense

@MsJavaWolf 4 года назад

@- RedBlazerFlame - Other types don't have the exact same properties as numbers.

@mahmoodemami7466 4 года назад

Obviously the. Total of positive numbers is not equal to a negative number. There is at least one step wrong . It should be found.

@MrPLC999 3 года назад

I have a lot of respect for Eddie Woo who also did the -1/12 proof. I knew there was something wrong with his strategy, and now I know exactly what it is. Thank you.

@Entropy3ko 2 года назад

I just find it a bit dishonest (or very sloppy) they do not specify when the "super sum" (which is called I think Cesaro Summation), which assigns values to some infinite sums that are not necessarily convergent in the usual sense. The term "summation" needs also a big asterisk, since it's not the conventional sum you learn in primary school. In fact it's a swindle... the "Eilenberg-Mazur swindle", hehe

@yasyasmarangoz3577 2 года назад

I don't think you did.

@andreicecold4379 2 года назад

@@utkarshsaini5650, not even Ramanujan, it was Euler who first proved it, in the 1700s. This math has been around for years and there are multiple branches of physics-based around it, so if this video was accurate, which it's not, it would be one of the largest revelations for complex physics in the past 100 years

@jacobpeters5458 2 года назад

mathologer is great. as he points out, the shift in S2 is the culprit. if you did 3S2 where the last line got shifted back to the left, you get S2=-1/4, an S=1/12; also if you shift the 2nd line in 2S2 to the right twice instead of once, you get 2S2=-2S2-1, which also makes S2=-1/4

@hutsku1860 Год назад

To be fair, he never said that this result was true, at last with the standard definition of a sum. He just redemonstrate the result to make people think about the mathematical logic, never saying if it's true or not

@anhhoanginh4763 3 месяца назад

man, we really need new video for this "Does -1/12 Protect Us From Infinity? - Numberphile"

@Daspied 4 года назад

Numberphile is like the fun uncle. Whereas Mathologer is the Dad who smacks you on the head and says "get real son"

@MrOllitheOne 4 года назад

i^2

@aaronleperspicace1704 4 года назад

@@MrOllitheOne = -1

@MrOllitheOne 4 года назад

shit just became real

@AlgyCuber 4 года назад

hey i, get real! i : (grabs friend)

@balsoft01 4 года назад

In a matter of fact, Mathologer told us to quit being real and start seeing imaginary! It's Numberphile who tried to project the power of complex and imaginary to the simplicity of real, hereby resulting in nonsense.

@kristoferkoessel4354 4 года назад

Numberphile (Brits): It’s -1/12th Mathologer (Germans): Halt mein Bier

@leonhardeuler6811 4 года назад

*-1/12th

@MattixHQ 4 года назад

It's '' halt mein Bier''*

@kristoferkoessel4354 4 года назад

MattixHQ Sorry guys 😂 you get the point...

@kristoferkoessel4354 4 года назад

MattixHQ wait but halt=stop right? Halte=hold? Or am I just retarded please tell me...

@M3tag 4 года назад

@@kristoferkoessel4354 Halte would be correct too, but it is more formal, which doesn't make much sense in this context. And Halt also means stop. In English there is a similar relationship of words. If somebody tells you to put something on hold you will probably stop doing something. Or if you are supposed to hold a door open for someone you also stop the door from moving. So Halte makes sense and the person you are talking to will understand you, so it is not a real issue. That rule also does not only apply to Halte. The e is often dropped from the verb, if you are telling somebody to do something, I can't even think of a word right now where it usually isn't dropped

@DanielKRui 3 года назад

I keep coming back to this video every so often, and each time I am utterly amazed at how intuitive Burkard makes these complex topics. I appreciate that he is so careful with his terminology, and of course his graphics are awesome. It was so cool to have Burkard run down exactly the problems in the Numberphile calculation and how to "fix" them...when he did the transition from the Numberphile S-S_2 to zeta-eta I was blown away; in an instant, he transformed a simple, familiar, but false expression into a deep, rigorous, and true statement, highlighting the "simplicity" and "familiarity" behind things as complicated as power series in the complex plane. Literally one of the best math videos ever made.

@juancarlosortiz6756 9 месяцев назад

THANK YOU! The -1/12 meme has gone way too far.

@madlad4206 3 месяца назад

It's not a meme, it's used widely in physics and maths

@Doeff8 3 месяца назад

Nonsense comment. It's a perfectly valid evaluation of this series. Mathologer is an annoying pedantist.

@yiutungwong315 Месяц назад

41:20

@martint1775 5 лет назад

Numberphile on Schrödingers cat: The cat is half dead, meaning it's probably in a coma.

@blizzbee 5 лет назад

poor cat

@Dondala 5 лет назад

thats right what it is, he calculated an expected value, not a sum :-)

@nichitacruceanu9540 4 года назад

Lmao

@Alex-hj2jd 4 года назад

No they meant the cat is alive and dead. It was in a state of quantum uncertainty. Unless observed the cat is alive and dead not half dead.

@potman4581 4 года назад

@@Alex-hj2jd Yes, we know. It's a joke.

@Dreams_Of_Lavender 3 года назад

"And this is where Numberphile takes a bow... BUT" - 35 minutes left.

@amogorkon 3 года назад

...and then the real fun stuff starts!

@user-dg9eb4mc9t 2 года назад

@@amogorkon ...and then the imaginary fun stuff starts!

@anshumanagrawal346 2 года назад

@@user-dg9eb4mc9t lol

@RichConnerGMN 2 года назад

nice pfp

@jakeenvelopes9561 3 месяца назад

Yeah, I actually couldn't watch it. I'm ten minutes in and all he's done is slag off the numberphile video and it's been boring for a solid five minutes. I'm out.

@Jonathan-xb8yf 3 года назад

Wow, did not know about the sequence 1-1+1-1… not having a sum. Though it makes sense when u consider that one cannot evaluate oscillating functions, e.g. sinx or cosx, as they go to infinity.

@ScratRedemption 2 года назад

Indeed. The first thing i thought of when i saw that sequence was sin(x) which has no limit according to calculus.

@mcjon5477 Год назад

I thought it would be s={0,1}

@vgautamkrishna5197 11 месяцев назад

@@mcjon5477well sum should be a single value so you can't say it has a sum if it gives 2 different values

@viktorsmets29 2 месяца назад

That's what we call adherence points. These are points for which there exists an infinite subsequence with that point as its limit.

@foreverkurome 9 месяцев назад

This was like one of the first things they covered in undergrad, the series that alternates positive and negative 1 they told us to think about as a digital switch, it's either on (1) or it's off (0) and it can always be made to be in one of those states by adding an extra term but it can never behave like an analogue switch and be in a state that is some measure of two values it takes. Really helped me to understand why its sum cannot be assigned a value. This video made more clear outside of thay intuition.

@mayaq8324 5 лет назад

You killed my party trick

@christianrasmussen1 4 года назад

It'll be fine. You can still be an illusionist.

@bhavikshankar3235 4 года назад

Your part trick is still alive see from 41:15

@RedRad1990 4 года назад

Matt Parker's card trick, my friend :)

@cdavis7693 4 года назад

What kind of parties have you been going to?

@kristoferkoessel4354 3 года назад

Do 1=2 proof

@leonlu3147 6 лет назад

Numberphile: 1+2+3...=-1/12 Mathologer: Impressive, every word in that sentence was wrong.

@danildmitriev5884 6 лет назад

Ohhhhhh yesssss, Star Wars references ^_^

@deadaccount4221 6 лет назад

Mr Banana808 What is wrong with you

@NinjaoftheEnd 6 лет назад

Mr Banana808 Are you an actual banana?

@jesuslovespee 6 лет назад

Francesco Santi his pharmaceutical clock has dilated.

@Jotakumon 6 лет назад

So clearly what you wrote is all non-sense, but damn was it funny to read anyway. My favourite ones: "All scientists think light speed is c in the vacuum, they all wrong." Gee, I wonder what the light speed in vacuum is then... and what letter should we use to represent that value? "Iss is fake, AC systems cannot work in vacuum space" No, Iss is fake because there is no sound in space, so their alarm clocks wouldn't function properly. Get your facts straight. "If heat can radiate into space, [...], the whole universe will be at the same temperature, thermal equilibrium." *long stare* ... sure ... it's called heat death...

@Tekay37 3 месяца назад

With the new numberphile videos, I think this topic needs an update. :D

@ArnavTHR 3 месяца назад

which new vid

@Tekay37 3 месяца назад

@@ArnavTHR the one about -1/12 protecting us from infinity.

@v2ike6udik 3 месяца назад

2i/24, open your mind, open your mind. You live in a hologram. All who believe in infinite series are duped by reps. You know... Tiles. Reps-tiles.

@v2ike6udik 3 месяца назад

More data after contact. Cant share. ReptileAI deletes.

@v2ike6udik 3 месяца назад

Dang, already removed even the thing before that. Lets try it bitbybit.

@AmorLucisPhotography 2 года назад

Wonderful stuff! The second half was way above my mathematical pay-grade, but I still understand much more than I did before. Great work! I had been duped by the -1/12 stuff.

@wideeyedraven15 Год назад

Dupe isn’t the right word; this isn’t even necessarily a real rebuttal of the -1/12 sum. The result is controversial and this is a good argument against the result (which is counterintuitive which in itself isn’t meaningful). The whole thing, the controversy and the result, are more indicative of the clumsiness, errors and even perhaps uknowability of logic, math and the implicative language of trying to state it. The terms are very slippery and we get strange results in our minds when we try to manage it all. The argument made here is one, a robust and hardy one but it is no more ‘correct’ than other views.

@LeNoLi. 4 месяца назад

you haven't been duped. -1/12 is a meaningful value assigned to an infinite series. this "sum" is not an actual sum in the traditional sense, but it was derived using real methods. in the context of a youtube video teaching about infinite series, numberphile was correct. in the context of a mathematics course that requires rigor and proper definitions, it was incomplete. we know that -1/12 works because it can be used in real world applications of physics.

@AmorLucisPhotography 4 месяца назад

@@LeNoLi. This last comment is what really interests me. What does "-1/12 works" or its utility in real world physics tell us about mathematical truth? I have in mind the use of infinitesimals, in Newtonian calculus - i.e., before the introduction of a "limit". These "ghosts of departed quantities" (as George Berkeley memorably called them) "worked" in physics, despite being, at core, inconsistent. This suggests to me that having real world applications in physics really doesn't necessarily tell us much.

@sloaiza81 3 месяца назад

The irony. You are being duped by thinking that we were duped. Terrence Tao just should that the -1/12 is valid and their is another numberfile vid on it.

@AmorLucisPhotography 3 месяца назад

I think you misunderstand. By "duped" I mean that I misunderstood something about the proof. I in no way intended to suggest that it is not "valid", in its own terms, but simply that I misunderstood the terms of the proof.@@sloaiza81

@markgearhart1606 5 лет назад

Y'all so focused on James vs Tati vs Jeffrey while this right here is some high quality tea

@matthewboyea3860 5 лет назад

Thats a quality evaluation, Fonn the Human

@alexwang982 5 лет назад

Quali-tea

@user9287p 4 года назад

@@alexwang982 Shh.... you are not welcome here. You are not # e^(pi•i) after all.

@torontobud8902 4 года назад

Omg sisterrrrrr

@ashierapreston 4 года назад

Jason -e^(pi•i)

@jacfac9969 4 года назад

Everybody gangsta till there’s math RU-vidr drama.

@signorellil 3 года назад

I think this brilliant video shows how "math popularization" and "intuition" both have enormous limits. If you get below a certain rigour level, you're bound to make mistakes or say confusing or even totally false thing. Numberphile is a charming and even informative channel, but their format has some downside. When you get into stuff like power series and the zeta function you HAVE to dive into more "formal" math (that is the only math around!).

@marshallsweatherhiking1820 2 года назад

I think the original video was click-bait. It worked pretty well for that. It never made any sense to write down a bunch of infinite series without giving a solid definition of what you mean by the “sum”. Also, in introductory real analysis you at at least prove as a theorem something that states the conditions under which series can be added term by term. Non-convergent series are not included. The business of assigning numbers to non-convergent series is theoretically interesting, especially when you move out to the complex plane, but its not standard summation anymore.

@l.w.paradis2108 2 года назад

@@marshallsweatherhiking1820 thank uou

@alvarogoenaga3965 2 года назад

@@marshallsweatherhiking1820 . This -1/12 business is a more sophisticated trick than the 1=2 " proof"we know from our high school days.

@samueldeandrade8535 8 месяцев назад

... not really.

@dave6012 2 года назад

I’m learning data structures and algorithms and came to this video after a teacher told me to check out numberphile’s -1/12 video. So glad I pulled that thread and landed here where you made it all make “sense”. I would have laid awake in bed for far too long trying to wrap my head the bogus numberphile solution.

@benmcdaniel 6 лет назад

1+2+3+...=-1/12 is a Parker sum.

@C1Ansy 6 лет назад

Ben McDaniel And that is?

@minerscale 6 лет назад

A funny joke: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-aOT_bG-vWyg.html

@benmcdaniel 6 лет назад

When something in math isn't quite right, you name it after Matt Parker: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-aOT_bG-vWyg.html

@C1Ansy 6 лет назад

Ben McDaniel Ah, that guy. I recognize him. Thanks a lot.

@Tymon0000 6 лет назад

I LOLed :D

@trevorperkins4585 4 года назад

26:14 - "now let's play a game." Me: sweet I love games *Shows a graph* Me: is this some kind of German game that I'm not structured/organized enough to understand?

@irongolem5539 3 года назад

To some people (like me) gragh (maths) is a game

@nolann2382 3 года назад

@@irongolem5539 and you're losing

@markopolic9964 2 года назад

@@nolann2382 You are always losing a game of graphs

@WMHinsch 2 года назад

The Numberphile video in question seems to violate the principle, "Make it as simple as you can, but no simpler." Simplicity is a noble goal, and I laud those who try to make complex ideas understandable to a wider audience, but simplicity has boundaries beyond which it becomes simplistic or simply wrong.

@drsolo7 8 месяцев назад

The thing about maths is that mathematians always care about and give the general case whereas physicists in physics always cares about and give the special case And yes Richard Feynman said something like this

@jessers1712 4 года назад

"Kids in primary school should be able to follow it!" He should meet my coworkers...

@A_Box 3 года назад

what is your line of work tho?

@jessers1712 3 года назад

@@A_Box Physicist, sadly ;'(

@kotarojujo6365 3 года назад

Jesse Kucharek he should meet me.

@DrCorndog1 3 года назад

Emphasis on "should."

@segmentsAndCurves 2 года назад

@@jessers1712 Remember to blink twice.

@JayWez 4 года назад

I can't believe I am just now finding this video. The -1/12 thing has been confounding me for years. Well explained, thank you.

@rygerety8384 2 года назад

Same here, never made sense to me why all of the POSITIVE, INTEGERS sum to a NEGATIVE, FRACTION. Always seemed completely backwards, and +infinity makes far more sense

@veronicaacevedo4314 Год назад

Same here!

@lanchanoinguyen2914 Год назад

@@rygerety8384 (1-1+1-1...)=1 or 0 now 2(1-1+1-1...)=2 or 0 so it is undefined.It could be 0 or another number because it is an infinite structure of conditions.You can say an infinite number is not a number.We calculate base on renormalized numbers. Infinity is not real in real life maybe,because if the world is real so it must be a limited structure of numbers,an well defined number that represents for physics laws. Zeno had said,time or motion is not real and you can't prove he wrong,no mathematics or physics solution can prove the cause and effect work in such a infinite manner.

@ittipongchaisayun878 Год назад

same here

@l.w.paradis2108 Год назад

That Numberphile video was nothing short of vicious. I literally hate them for doing that.

@JusticeBackstrom 14 дней назад

The -1/12 thing always seemed more like a party trick than a genuine maths solution.

@Owlrrex Год назад

The way I always explained the "nonsensical" result of -1/12 coming from the Zeta function was this: The original zeta function is defined as the given sum, for only Re(z)>1. The analytically continued Zeta Function takes those same values for Re(z)>1, but is _not_ defined by the sum over its whole domain. I don't know if we know the closed form of the extended Zeta, but that form would relate -1 to -1/12 - and have nothing to do with the 1+2+3... Sum.

@fblio7146 6 лет назад

I remember explaining how 1+2+3+... diverges in the comment section and people responded that I'm wrong since I'm not a university professor. So thank you very much for this video! Math is about truth, not educational authority.

@Noah-fn5jq 6 лет назад

But... they are! en.wikipedia.org/wiki/Indiana_Pi_Bill (end sarcasm) That was a sad day

@vacuumdiagrams652 6 лет назад

"I remember explaining how 1+2+3+... diverges in the comment section " It does diverge. Everybody agrees that it diverges. The question of what it "equals" is conceptually separate and requires agreeing beforehand on what the word "equal" means. It's not at all true that the only possible meaning of "equal" for an infinite series is that of the limit of the partial sums. That is a choice, one which makes sense in many circumstances, but sometimes you may want a different one.

@fblio7146 6 лет назад

Vacuum Diagrams yes but then one has to make it very clear what equal means in a certain context, especially when the large amount of viewers might not be math students

@ShinAk1raSama 6 лет назад

I'm pretty sure Appealing to Authority is a logical fallacy. So, I wonder why people use it...

@vacuumdiagrams652 6 лет назад

"yes but then one has to make it very clear what equal means in a certain context" Indeed, but this applies to _convergent_ sums just in the same way. When I say that 2 + 2 = 4, I mean something quite different than when I say that 1 + 1/2 + 1/4 + 1/8 + ... = 2. The former is the result of a single addition, while the latter is a statement about convergence and limits. It's a nonstandard use of the equal sign, just like the use in 1 + 2 + 3 + 4 + ... = -1/12 is nonstandard.

@Purin1023 6 лет назад

Oh god, mathematical hell is gotta be like 10 times worse than regular hell.

@Mathologer 6 лет назад

-1/12 time worse :)

@skhumbuzocele1330 6 лет назад

😂😂😂😂😂😂😂

@metacylinder 6 лет назад

All you do is math problems there...chilling

@TheLK641 6 лет назад

I would have said pi time worse.

@ilpinto4925 6 лет назад

it is the analytical extension of regular hell

@mattsgamingstuff5867 2 года назад

Nice to see someone do this. I randomly stumbled across someone still in university (I think an engineering program) bringing up these sums, I think as fun puzzles. I quickly put up proofs of their divergence, I might be a chemist but I was taught well enough to test a series for convergence before running off with it in my math classes (that and the sum of all natural numbers is obviously divergent). I was vaguely aware of non-standard summations such as cesaro sums and brought up that those series can be assigned summation values, but struggled to explain the nuance of the difference between being able to assign a value and the sum being that value. If only I could go back in time and have actually studied mathematics instead of science.

@69k_gold 2 года назад

So much attention to detail in a long video. Great work

@j03man44 3 года назад

Reminds me of the first time i learned about the dirac delta function in physics. I was basically told "there's some complicated math that proves this is correct but it works and that's all we really care about."

@keineangabe8993 2 года назад

Well in the case of the Dirac delta, they are at least not giving wrong arguments why it works, do they? Btw: the foundations of distribution theory are really nice imo, worth checking out.

@schizoframia4874 2 года назад

Not satisfying at all

@davidr1138 Год назад

I remember loving Laplace Transformation until I found the Dirac Delta function felt like a brick wall.

@thewatchman_returns Год назад

Physicists being physicists

@PC_Simo Год назад

@@keineangabe8993 And at least they don’t try to change the definitions; e.g., try to pass off Ramanujan-summation as standard summation 😅.

@swerasnym 6 лет назад

Z -> Q loses single representation, Q -> R loses countability of the set, R -> C loses the order of numbers, C -> H loses commutativity of multiplication, H -> O loses associativity of multiplication. EDIT: s/looses/loses/g

@Mathologer 6 лет назад

Cool :)

@swerasnym 6 лет назад

Must admit i had to look up octonions, but had enough knowledge to do the rest!

@GSandSDS 6 лет назад

Why stopping there? We also have the Sedenions. ;) O -> S looses alternativety of multiplication.

@Stefan1of3 6 лет назад

What do we loose going from Reals to Surreals? (Honest question. Those exist.)

@DanielBeecham 6 лет назад

Heyo, cool!

@stevekeiretsu 3 года назад

When the numberphile guys said "so this series alternates between 1 and 0, so the sum must be 0.5" I was like, "what, no, it doesn't work like that", but since I only have 'high school' maths and they're professors, I went along with it. I am feeling relieved and validated now that youtube has recommend me this. I'll be honest, started to struggle to follow around the zeta/eta part, but at least thanks to the first half of this vid I can rest assured the 0.5 thing was indeed nonsense

@tomsvoboda2309 2 года назад

One can do all kinds of stuff with the Grandi's series, for example I can make it equal to 1 by writing 1-1+1-1+... = 1 - (1-1) - (1-1) - .. = 1 + 0 + 0 + .. = 1 and I can take it even further and make it equal to any number X by writing 1-1+1-1+... = (1-1) + (1-1) + .. = 0 + 0 + ... = (X-X) + (X-X) + ... = X - (X-X) - (X-X) = X - 0 - 0 -.. = X This series is actually the most profound counter example for unjustified arithmetic operations with infinite series. It's one of the first things a math major learns in the theory of infinite series. It's incredible how dishonest that Numberphile video was in that regard.

@LifeInZadar Год назад

This reminds me of the story the little engine that could. Gotta have some faith in yourself. Be a fucking Zaibatsu.

@jacobbabcock8943 2 года назад

God bless finally, I was sick and tired of hearing people try to tell me that adding infinitely positive numbers equals a negative number.

@elasiduo108 4 года назад

I think Mathologer deserves no criticism for this video. I like the Numberphile guys, but in that video, they presented a very misleading argument for the "sum" of these divergent series. The first rule in any, ANY argument regarding series is: "you can make some algebraic manipulations with series ONLY IF they converge". Notice the "IF". This is very important, because, with divergent series, you'll end up with nonsensical results applying algebraic manipulation. Let us check a stupid example. Let us suppose that I don't know if the following two series are convergent or divergent. S1 = 1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6... S2 = 1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6... Now, let us take, S1-S2, which, computating term by term, we get: S1 - S2 = (1/2 + 1/2) + (1/4+1/4) + (1/6+1/6) + ... = 1 + 1/2 + 1/3 + ... = S1 So, S1-S2 = S1, and thus, clearly, S2 = 0. Right?. WRONG. S2, as Leibniz discovered, converges to ln(2). The argument is invalid because S1 is a divergent series. So, my algebraic manipulation is invalid. The Numberphile guys should have made that very clear in the video, saying "these algebraic rules are only valid if the series are convergent. But, we'll be playful, and let's see what strange shennanigans happen if we ignore the convergence criteria". With that disclaimer, everything would be completely fine, but they failed to do so, so they deserve criticism in that regard.

@SparelWood 4 года назад

And they further state their math is valid because it "shows up in physics." Thats the part that irritated me.

@elasiduo108 4 года назад

@@SparelWood I think the Numberphile guys were trying to be informative regarding these "strange sums" which appear in advanced mathematics. But, of course, without any disclaimer, these identities are just nonsense. For example, we all know that "S1 = 1+1+1+1... = infinity". In fact, that is the main definition we use to explain people what infinity is!. But, let us again ignore any rules regarding convergence. S1 = 1+1+1+1+1+1+... S2 = 1-1+1-1+1-1+1-... S1 + S2 = 2+2+2+2+2+... = 2*S1 S1 = S2 So, given that we "know" that S2 = (1/2), then, S1 = (1/2). And thus, "infinity = (1/2)". So, even it is true that some process in physics in which the partial sum of a value can be considered "averaged" occurs in reality, but that is NOT an argument for justifying this kind of nonsense.

@MrTiti 3 года назад

@@elasiduo108 ....... " because it shows up in physics" ...... LMAO.

@Wyverald 3 года назад

what a beautiful comment, and great counterexample. well said!

@jstodd4398 2 года назад

This is the best counterexample ive seen

@MathManMcGreal 6 лет назад

Yooooo Mathologer throwing the shade at Numberphile... This calls for a math off!!!

@mheermance 6 лет назад

I think they would prefer a maths off.

@playscirox2129 6 лет назад

Geez that would be a close call, depending who from Numberphile would fight Mathologer.

@awsomebot1 6 лет назад

I've heard "math duels" were the main income source of mathematicians from few centuries ago.

@alexanderf8451 6 лет назад

*sharpens division symbols*

@IllumTheMessage 6 лет назад

Now if we can get the Vatican in on this fight we'll have the scene set for some epic Math Drama!

@re5o28 Год назад

I've always had a fascination w/ Euler's infinite summing. I've never been able to reconcile the shifting of the equations to apply basic algebra to rewrite the initial equation (as done in your video) to something more useful w/ other equations when it comes to right before infinity and infinity (The former behaving finitely and the latter not). A sum of integers that approach infinity would seemingly approach infinity faster if each is e.g. squared than if not. So, my basic understanding of HOW every equation that yields infinity just doesn't seem like it's equal to another equation that equals infinity, yet gets there faster. Will you kindly provide some resources that will help understand how this works?

@beelzzebub 2 года назад

Does he respond to the "little puzzle" at 22:08? He says if we add infinitely many zeroes (and shows the new sum) the super sum is no longer 1/2 - but I worked it out, it IS still 1/2. Did they use an incorrect question to demonstrate their point? Perhaps if they added a 0 after every +1 but not any of the -1 terms, then he would be correct (and it would still be infinitely many zeroes).

@JohnDoe-ti2np 2 года назад

Good catch! You're quite right. He probably meant to do what you suggested; that would lead to a supersum of 2/3.

@telaferrum Год назад

I got the same result. I'm glad I came across your comment. I trying to figure out whether I was missing something but this is the first comment I found actually trying the puzzle.

@jorgenharmse4752 9 месяцев назад

I forget which sum he wrote, but you can make it come to anything between 0 and 1 if you put the zeros in the right places. (Each zero causes a repetition of the previous partial sum, and that changes the average.) I think you can even make it not super summable.

@BenDRobinson 5 месяцев назад

Yay! I had to scroll a long way to find someone who answered this. I quickly concluded exactly the same thing, so I think that is a genuine mistake in the video.

@BenDRobinson 5 месяцев назад

@@JohnDoe-ti2npindeed - perhaps her just mucked up when doing the graphic

@macronencer 6 лет назад

Excellent video. Unlike some, I don't think you were being harsh. When millions have viewed flawed information, a clear refutation can be seen as a public service.

@Mathologer 6 лет назад

That's the way I look at it :)

@CGoody564 6 лет назад

Agreed. Can't fix a problem if you won't admit there is one.

@screwhalunderhill885 6 лет назад

Thanks a lot for your effort. I saw that numberphile video years ago when I began my studies and it confused me a lot because we've all been told you cannot do anything with divergent series. This video finally cleared things up for me.

@johnblah1234 6 лет назад

ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-0Oazb7IWzbA.html

@macronencer 6 лет назад

John Deacon - that is a nicely-worded response, but it is, after all, written from the point of view of a physicist. I understand the points he makes, and he's quite right about the usefulness of analytic continuation - but that isn't the point. The point is that the audience of the video may have been given the impression that such things can be stated without context, as being strictly true. To me, it is clear that summing the natural numbers cannot possibly result in -1/12, UNLESS you state clearly that your context is one of analytic continuation. This is a subtlety unlikely to be understood by a general audience, and the complaint was that this was not made clear. I think this was a fair complaint. I differ from you about the style of Mathologer's video too - I don't think it was unpleasant. But of course, that is subjective and therefore not open to debate.

@rcb3921 6 лет назад

In (slight) defense of Numberphile, they did follow up with a much more informative discussion with Prof Edward Frenkel. Some aknowledgement of the flaws in that video that Mathologer is complaining about; the first thing we hear is Frenkel saying with some dismay "Oh... it's /you/ who made that video." He chuckles and shakes his head. Then what follows is some explanation of assignment rather than summing. They are very explicit: "[-1/12] is certainly not the result of summation of these numbers [1+2+3....]. It is something else, but what is it?" ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-0Oazb7IWzbA.html

@Mathologer 6 лет назад

Yes, I actually like that video with Edward Frenkel, he is a very good mathematician and really knows what he is talking about :)

@ragnkja 6 лет назад

Lesson learned: Don't ask a physicist to explain number theory.

@TomJacobW 6 лет назад

Nillie I still think they were meming hard and were just joking in that video. ^^

@ExpIohd 6 лет назад

There is also the 'extra footage' video on Numberphile 2 which goes into greater depth of the math on the original- ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-E-d9mgo8FGk.html

@AzCcc 6 лет назад

In this video (Frenkel's @ 10:19), Brady asks "My understanding of Math is it's very rigid and rigorous and it's never arbitrary, how can you throw away the dirt and keep the gold?". This question is the reason why I hated the 1+2+3...= -1/12 from the very first moment. Because that kind of destroys my view of Math (as the only concrete, unambiguous and objectively true tool we have). Mathologer if you're going to make a discussion video about this subject, PLEASE address this question.

@king_noah_2692 2 года назад

Bookmarks: Starts at 2:50, gives explanation of Numberphile’s logic. 5:30 “These three identities are false.” 10:28 Properties of convergent infinite series. 13:22 “Does this prove that M is 1? No.” The series must be convergent (not just assumed to be) in the first place to do this kind of calculation. 16:10 Super Sum properties 19:03 if ANY of these new series converge, the super sum of the original series converges to that. 20:54 RECAP 24:08 Super Sum is more like a super average than a summy sum. 24:45 RIEMANN-ZETA FUNCTION 26:10 “Rough and ready intro” to Analytic Continuation. 30:22 Combining two extension ideas. 33:55 How Numberphile used Riemann Zeta trick. 36:28 the punchline 38:45 wrapping up 40:53 -1/12

@PC_Simo 11 месяцев назад

Thank you for devoting the effort to put up all these bookmarks, it must have been quite a bit of work 🙏🏻🙇🏼‍♂️.

@king_noah_2692 11 месяцев назад

@@PC_Simo I did it just for you

@PC_Simo 11 месяцев назад

@@king_noah_2692 Thank you 😌👍🏻.

@sahiltrivedi69 2 года назад

This video also explains why certain applications in theoretical physics might assume the sum of the positive integers converges. I suspect it might be a consequence of following the statistical approach to calculate the average of values over a set of objects. We do this is in thermodynamics all the time. Great video 👍

@ugopinho2121 6 лет назад

TOP 10 ANIME FIGHTS OF ALL TIME

@1stPCFerret 6 лет назад

Anime?

@MrPointness 6 лет назад

The strongest attack in his arsenal: Serious Series: Infinite Sum!!

@glowingdawn9179 6 лет назад

respect

@hernandojosedeavilapereira511 6 лет назад

jajajajajaaj

@solarisone1082 5 лет назад

Vegetto vs Buuhan: Mathematics Edition.

@markstgeorge405 4 года назад

The fallacy of the first series reminds me of the analysis of the human race that concludes the average human has one boob and one ball.

@jedinxf7 3 года назад

lol

@thelickpolice1210 3 года назад

Underrated comment, that's actually funny as hell, I was thinking of an analogy and this is a perfect one!

@jedinxf7 3 года назад

that's really just a bimodal distribution situation, not sure if it's quite applicable to the fallacy at work here. but it's funny as hell

@karlkiili1572 3 года назад

PFFFFFTTTT dang!

@russell2952 3 года назад

The average human has 9.x fingers and 9.y toes. Averages never claim to represent a single one of the values that went into calculating them. Another good example are population BMIs (body mass indexes) being applied to individuals. It's almost always wrong.

@justinthejerkoff Год назад

I just found the number numberphile video yesterday and now RU-vid recommended this video to me. Thanks for doing this!

@SunnyKimDev Год назад

22:56 examples of properties lost when expanding the number system: N->Z (positive -> integer) Prime Property "All numbers are a prime, composite or 1." "There is no two numbers with equal distances from zero." Z -> Q (integer -> rational) Odd/Even Property "All numbers are odd or even." Q -> R (rational -> real) Sane Representation "All numbers can be represented by a combination of digits." R -> C (real -> complex) Positivity/Negativity, Size(> H (complex -> quaternion) Commutativity "A * B = B * A."

@shantanubadve4668 5 лет назад

I was watching 8 mile ending rap battles and this came up Not disappointed this is a very mathematical diss track

@XavierDesroches 5 лет назад

Did you end up finishing 8 miles, or was that too much of a diss-track-tion? Alright, I'll go hide...

@Caribbeanmax 5 лет назад

@@XavierDesroches

@realdragon 5 лет назад

This is math war, very brutal war

@crabsynth3480 5 лет назад

Screw nitwit 8 mile crap... this is real rhyme and reason not just random rhyming words by a dumb rapper looking for a pissing contest.

@natevanderw 4 года назад

Crab Synth whoosh

@eyepatch2696 6 лет назад

Mathematics equivalent of a diss video

@jasonbucy 6 лет назад

haha yes! Mathologer is basically Eminem

@88michaelandersen 6 лет назад

Mathematicians reuse the same symbols with different meanings all of the time. It is much easier to say, here is this idea I am working with, and here is a nice symbol for it, than to come up with a brand new symbol for everything. Numberphile's problem was not putting a disclaimer up saying "Here is the standard meaning for this notation, and here is another idea that uses the same notation, but isn't the same thing." They should have made the distinction clear, instead of not mentioning it.

@___xyz___ 6 лет назад

Obviously it's not always a great honour to be corrected in science. Some of the most renowned scientists of all time, including Newton, Kelvin, Edison were all challenged after having reached fame; their ideas about the universe and the contents of papers they had published were corrected, but they refused to accept and acknowledge these discoveries, many of which were ignored for a century before finally resurfacing providing solutions in other sciences. A great deal of this was the fact that basically all people are stubborn and will give in to power and fortune. You can think of it as great scientists being corrupted, or there being little to no difference in science emotionally from other endeavours. If you can acknowledge that you were indeed mistaken in your assumptions, then standing corrected may be a personal honour. But that actually has very little to do with being wrong. Most researchers for instance do not care about being right or wrong at all: providing an argument in the publishing of a discovery is just a formality. Being recognised for posing the right question and having the idea that sparked the study is a much greater honour. And when then someone comes afterwards and points out a mistake in a study you were the mind behind, you are quite simply flattered. Feeling honoured for being dissed in science is the worst pseudo spiritual zen bullshit myth I have to live with. It's just a mindset overrepresented by Hollywood movies.

@hellfrost333 6 лет назад

Math isn't a rational subject: It's a system "we" created based off axioms which are accepted as true. (When a Contradiction occurs in Math- we either correct for the contradiction or avoid doing what caused error) Eugene Wigner wrote a really famous paper called: "The unreasonable effectiveness of mathematics in the natural sciences." *If there is an infinite amount of numbers between 1 & 2 (How do you get to Two?) *If it's Zero degrees outside and the weather man says it's going be twice as cold tomorrow as it is today. (What's the temperature going to be tomorrow? [ 2 x 0 = ? ] ~Not Zero you need to switch the formula. 1+1=3 When a Man and a Women enter a Dark-room- Nine-months later you have Three people... 'Math is litterally the Definition of *close enough;* The Great Pyramid of Giza is the most accurately aligned structure on earth- and it's still off 3/6 a degree True-North. (Rolls eyes) Don't get me wrong- Math is extremely important: Without Math we'd suck at 4th dimensional physics. But there's really only one number and that number is: *EVERYTHING*

@TrickyTrickyFox 5 лет назад

Math is an observational tool, and while yes, we agreed to 1 = one object, 2 = two objects and so on to be the case, it doesn't change the fact that there was two objects in the first place. For your points: 1. Eugene Wigner, while being a wonderful physicist bringing light and joy to people arround the globe by some of his greater projects (sarcasm, obvs), absolutely did that. And he also has several others - "Maths being shit in economics", "Maths being shit in everything" and so on (obvious hyperboly is obvious). Reading through those articles (thank you for bringing it up in the first place, was an interesting read) - I came to a conclusion, that either: A - he is not aware, why does physics need some of the cooler stuff and how mathematics and physics are connected or B - he was just a hater for the sakes of it (especially when it comes to economics one, since Eugene seems to be fairly low knowledgable in the field). 2. By defining the step of your infinity in the first place. The one you mentioned is an uncountable (1;2) infinity 3. Extendanding an example to the concept - is a logical failure on your behalf (or wherever you took the quote from). One guy saying, that it will be twice as cold tommorow, when it is 0 today - isn't really the best example of human brain functioning in the first place 4. That is not really how babies work. If you want to be tehnical - throw in all of the variables (the baby doesn't appear out of nowhere, it has energy consumption throughout the whole process). Otherwise, I will extend your example on two rocks being left alone in the dark room for 9 months - and after that a third rock would magically appear 5. Great Pyramid of Giza - is "close enough" in your statement, not the other way around 6. You wouldn't be able to write your comment in the first place without math. Or watch the video for that matter. Or use RU-vid. Assuming you'd have Internet to open RU-vid. And an internet connection in the first place - to your PC, of course, if it'd exist 7. Hey look, I used numbers to make my comment easy to read. When were you born tho? Answer me in everythings please ^^ And also, if 0 degrees outside - you are a flat earther!

@NolimitsNinja 2 года назад

One thing I'm struggling with, which I'm sure I could fix it in my head if given enough time, but I don't! so asking away here. When we take eta from zeta, we seem to just dispose of the zeros. BUT, as this is a divergent series where we are using supersumming as part of these identities, why do we not have to keep the zeros? Thanks in advance to anyone who can offer a lovely answer to this question!

@DarioVolaric Год назад

Never doubt someone who explains math in a german accent.

@grantorino2325 7 месяцев назад

Indeed! Just keep him safely away from his stupid sister, DeeDee. 👱🏻‍♀️

@SmileyMPV 6 лет назад

Oh my god this video is amazing thank you very much for making this. Here are my answers to your challenges and some question I have at the end of this comment. On 22:22: Series: 1+0-1+0+1+0-1+0+1+... Partial sums: 1, 1, 0, 0, 1, 1, 0, 0, 1, ... Partial averages of partial sums: 1, 1, 2/3, 1/2, 3/5, 2/3, 4/7, 1/2, 5/9, ... -> 1/2 Therefore the supersum of the series is 1/2. So I think you made a minor mistake taking the wrong example as this does not prove your point. Here is an example which does prove your point: Series: 1-1+0+1-1+0+1-1+0+1-1+0+... Partial sums: 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, ... Partial averages of partial sums: 1, 1/2, 1/3, 1/2, 2/5, 1/3, 3/7, 3/8, 1/3, 2/5, 4/11, 1/3, ... -> 1/3 Therefore the supersum of the series is 1/3. Therefore supersumming is not invariant under adding infinitely many zeroes. On 23:10: Funnily enough, every extension from N to Z to Q to R to C is mostly invented in order to add structure. The structures added are additive inverse, multiplicative inverse, completion and roots respectively. Some things you might consider a loss could be the following: You lose well-orderedness, completion, countability (but regain completion) and uniqueness of roots and logarithms respectively. On 23:25: If 1+2+3+4+... supersums to some S, then: 0=S-2S+S= 1+2+3+4+... ...-2-4-6-... ......+1+2+... =1+0+0+...=1. This is obviously a contradiction. From this we can conclude that it is impossible to define some ubersum with the three desired properties such that the series 1+2+3+4+... falls in the domain of the ubersum. From this we can conclude that the series has no supersum, because supersums have the three desired properties. On 38:40: Do I understand correctly that this means that if Re(z)>0 then zeta(z)=0 if, and only if, eta(z)=0? And because Re(z)>0 implies eta(z)=\sum_{n=1}^\infty((-1)^(n+1)/n^z), finding zeroes for the Riemann-zeta function just corresponds to finding z with Re(z)=1/2 such that this series is 0? (Assuming the Riemann hypothesis.) Because that is simply amazing! Edit: I really want to thank you for this video, because I was always very curious how it is possible that the argument given in the numberphile video just happens to give the same result as analytic continuation. I always refused to believe this is a coincidence. So thanks so much for showing why this is actually not a coincidence!

@qwertz12345654321 6 лет назад

Very nice summary of most important points. Should be stickied

@ikaro342 6 лет назад

The partial averages are wrong. The second aveeage isn't 1, but 1/2

@SmileyMPV 6 лет назад

Manuel Ochoa (1+1)/2=1/2?

@Blananas2 5 лет назад

"This is not mathematics. Don't use it. Otherwise, you will burn in mathematical hell." xD

@srimaryati337 4 года назад

Blananas2 wow a new religion have been born is Math Religion.

@srimaryati337 4 года назад

Blananas2 wow a new religion have been born is Math Religion.

@hypehuman 4 года назад

Mathematical Hell = Being doomed to make wrong predictions about the world

@jkellyk7920 4 года назад

You are tortured with people using 3 for pi and x for sin(x)

@pavanato 4 года назад

OMG 314 LIKES

@kevinaustin6971 2 года назад

Explained really really clearly to someone with a limited math background, nice job

@jgallantyt 2 года назад

I have an issue with the zombie shirt. That's a reaction, not an equation. The equals should be an arrow. Zombie plus human YIELDS two zombies.

@Mathologer 2 года назад

I think you have an issue ... :)

@drewkavi6327 6 лет назад

Mathematical équivalent of a diss track

@bluthemeth 3 года назад

Teacher: “What’s 1+2+3... forever?” Me: “Infinity” Teacher: “Wrong. It’s -1/12” Me: *_”DID I STUTTER.”_*

@grantorino2325 3 года назад

MY AUNT: But, the way that I calculated it, you owe me money for my purchasing all of this. *Everyone stares at us.* ME: Please excuse my dear Aunt Sally.

@rohangeorge712 2 года назад

you may me 10000000000000000000000000000000000000000000000000000000000000000 dolllars. i tell u to keep giving me money and i will pay u back. soon enough i keep getting money from u infinitely and i say it can be represented by 1 + 2 + 3..... and he is like yea whtver give me back my money. and i say nope, i owe u -1/12 of a dollar, which means u owe me 1/12 of a dollar GG (ps: ty for all the money hehe

@PlatonicPluto 2 года назад

@@grantorino2325 :O

@roseCatcher_ Год назад

This video proves you wrong too.

@NTNscrub Год назад

@@roseCatcher_How so?

@Fallkhar 2 года назад

This is such a brilliant video. I am so happy I watched it. Initially I wanted to watch it in two sittings but I could not take my eye off it.

@NerdWithLaptop 2 года назад

Don’t give them -1/12 marks. They’ll take it as you giving them 1 + 2 + 3… marks.

@FriedChckn13 5 лет назад

“On my home planet, this symbol stands for S U P E R S U M”

@DamianReloaded 6 лет назад

**stares at the length of the video** **stares at the fully loaded coffee machine** **unpants** **presses play**

@DanJan09 6 лет назад

unpants? ok, you do you ;P

@AndreiNeacsu 6 лет назад

Panting = breathing quickly. unpanting = not breathing quickly. So, "he unpants" could be interpreted as "he calms down and no longer pants". www.dictionary.com/browse/panting

@DamianReloaded 6 лет назад

Nah I just fap while I drink coffee and think about math. XD

@VeteranVandal 6 лет назад

This is hardcore math.

@JLConawayII 6 лет назад

Do you actually think anybody on the internet is wearing pants?

@brianfriis4784 Год назад

In the video "Why -1/12 is a gold nugget" on numberphile there is a pretty sound explanation (in my amateurish opinion) on this peculiar "illegal" operation on divergent series with references to e.g. Eulers work.

@gogobram Год назад

I saw the numberphile video hours ago, and immediately commented "is this an april fool joke", then noticed they were talking about those zeta functions, watched that video, and removed my earlier comments, but still kind of felt like it was syntactically wrong. And finally arrived here, where you fortunately support my claim, restoring my universe. :-)

@Hexanitrobenzene 5 лет назад

To all commenters. I'm sorry that this comment is so long and ask you to be patient. The debate in the comment section whether Mathologer is rude/too late/ignoring other Numberphile videos on the subject is making me smile, so I'll put my two cents, too :) Numberphile made a video about a subject which is completely counter-intuitive. So it went viral, to the point that my father, who is 50+ years old electrical engineer, completely unconcerned with mathematics other than that helps to do his job in reality and barely speaking English, and even some medical doctor I went to (knowing that I studied physics), both claimed to me that the sum of all positive numbers is -1/12 ... That doctor even stated that nowadays mathematics is incomprehensible :) That's exactly the point which drives people like Mathologer out of their minds - claiming such counter-intuitive statements without proper disclaimers (I'm not even saying proper context, like Zeta function and analytical continuation). One guy in comments says (I'm paraphrasing) "All natural numbers can be written as a sum of 1s. So, 1+2+3+4+...=1+(1+1)+(1+1+1)+...=1+1+1+1+1... You say that 1+2+3+4=-1/12 and 1+1+1+1=-1/2. So now -1/12=-1/2 ??? " I guess that some people, uninvolved in mathematics, thought to themselves after seeing that video "And these people get paid for that ?" Numberphile should have added only one minute, saying that: "equals sign in these equations should be understood as "is assigned to", not "is equal to" " and "these calculations are not intended as a proof, they merely show what answer is to be expected from more rigorous methods". That's it. Everyone (almost) would be happy. Instead, all we heard was "astounding", "amazing" and "correct". Someone says (I'm paraphrasing) "How dares Mathologer cite Numberphile out of context? Numberphile did two other videos on the subject, which (more or less) address the issues with the first video. Mathologer ignores that. " Mathologer is perfectly aware of this. He even links one of them ("Why -1/12 is a gold nugget") in his description. The reason is simple: view count. The first two Numberphile videos on that subject, which completely miss to point out the crucial distinction between "is equal to" and "is assigned to" have been viewed 7.7 M times combined as of 2018 July. The one which discuses the subject properly ("Why -1/12 is a gold nugget") has been viewed only 1.6 M times. The difference is those confused people inundating comment sections. Another person says (I'm para...) " The goal of Numberphile channel is to make mathematics interesting to wider audience. Don't expect rigour there. Anyone who is wiling to get deeper understanding should follow the links and research themselves." Well, this youtuber forgot that he is commenting in ... RU-vid :) Content providers in RU-vid, especially those who want to appeal to "wider audience", should keep in mind "least action principle" - most people these days will spend the least effort to get information. Those who will research seriously, I assume, are those who already find mathematics interesting + small minority newly engaged. Most people, I guess, come there just to see "what interesting video did Numberphile upload today ?" I even suspect that many people rejected the video as nonsense, not wanting to have anything to do with divergent sums anymore, barring further research. All in all, I don't think that Mathologer is rude or incorect, I think he is right on the money (except that cameraman. He should have kept his jokes off-record.)

@adamzeggai5506 5 лет назад

lol

@seacaptain72 5 лет назад

This is the most precise explanation I've read in this whole comment war. Well done.

@Hexanitrobenzene 5 лет назад

seacaptain72 Thank you.

@badlydrawnturtle8484 5 лет назад

1. You fail to actually address the rudeness. There is a clear tone of condescension throughout the video, not just from the cameraman. Who is factually correct is irrelevant to whether Mathologer was rude, which he was, by standard observation of tonality and wording. Your comment rather comes off as ‘I think he's right, therefore he wasn't rude’, which is a nonsense argument. 2. Your argument is essentially that this video is to address misconceptions of people who viewed the Numberphile video and misunderstood it. Meanwhile, this video actually directly tells Numberphile they are wrong, repeatedly. For what? Not being able to control what their viewers say and do? No. You don't get to blame Numberphile for that. Your suggestions for what they should have said may have affected things… but you fail to provide a reason why they would know those suggestions would be necessary BEFORE THE VIDEO WAS MADE AND PUBLISHED. Funny; those suggestions are followed in the other videos that both you and Mathologer handwave away… almost like it doesn't matter what Numberphile does or doesn't do, they're just wrong because of what people watching them do. Either your understanding of this video's purpose is incorrect, or both your and Mathologer's understanding of responsibility is crude.

@Hexanitrobenzene 5 лет назад

Badly Drawn Turtle Hm, on a second thought I guess I gave Mathologer a pass to being condescending, because he is right. Ok, I can somewhat concede this point. However, that first Numberphile video was just doomed to be interpreted incorectly. I believe this was because he was asking physicists to explain it. Physicists are less concerned with nuances in mathematics, and more concerned with applications, which in this case was knowing what number can be assigned to this sum. When Numberphile came to mathematician, namely Edward Frenkel, who has seen the video, Edward immediately understood that the solution was not explaining rigour, details, zeta function and all that, but an abstract meaning of that hapless equals sign. In fact, an advanced physics textbook is shown in an original video, and there is an arrow instead of equals sign. They did not explain that crucial detail which would have made a lot of people happier.

@Mathologer 6 лет назад

Confused 1+2+3+…=-1/12 comments originating from that infamous 2014 Numberphile video keep flooding the comment sections of my and other math RU-vidrs videos. And so I think it’s time to have another serious go at setting the record straight. In this video I’ll do just that by having a really close look at the bizarre calculation at the center of the Numberphile video and then stating clearly what is wrong with it, how to fix it, and how to reconnect it to the genuine math that the Numberphile professors had in mind originally. Lots of nice maths to look forward to: non-standard summation methods for divergent series, the eta function a very well-behaved sister of the zeta function, the gist of analytic continuation in simple words, some more of Euler’s mathemagical tricks, etc. This is my second attempt at doing this topic justice. This video is partly in response to feedback that I got on my first video. What a lot of you were interested in were more details about the analytic continuation business and the strange Numberphile/Ramanujan calculations. Responding to these requests, in this video I am taking a very different approach from the first video and really go all out and don't hold back in any respect. The result is a video that is a crazy 41.44 (almost 42 :) minutes long.

@volvoxfraktalion5225 6 лет назад

Thanks for that. I'm not realy mathematicly educated, but i enjoy watching your videos and thank you for clearing that myth out which i myself believed

@dantom5232 6 лет назад

Mathologer what happened to the plain black shirt at start 😁

@Nmmoinn 6 лет назад

Sorry to be a dick but 41.44 minutes /= 41 minutes 44 seconds

@frederickm9823 6 лет назад

Didn't you mean a "series go" :)

@alejandrolopeztobon1643 6 лет назад

Thanks for your video. I regularly watch both numberphile and your videos and love them both. Not being a mathematician but being in science I really appreciate them. Likewise I know that in science arrogance spurs easily and often egos simple don't match even where facts have the reason. I was a bit surprised by the aggressive nature of your video, I just hope you pointed out their mistake directly to numberphile guys before doing this video. I reckon that may have been the case and they didn't took it well and that led to the tone of this video.

@several9286 2 года назад

S(infinity) only exists when the modulus of the common ratio of elements in a set is between 0 and 1. The set of {1, -1, 1, -1,...} has a common ratio of (-1) between elements of the set and thus has no sum to infinity

@collegemathematics6698 3 года назад

Prof. Pulster you are amazing. I learned alot from you. Thanks. 🌹

@How-Do-I-Nezzy 5 лет назад

Video is pretty good, if long, but I was not a fan of Grumpy Background Voice, who didn't seem to be making any actual contribution to the content, just kind of dissing half-heartedly.

@innamordo 5 лет назад

couldn't agree more about the pot shots coming from the Henchman

@Dondala 5 лет назад

your right, thats not smart, but I understand his point. It is like when Sheldon tries to trap his rage about schrödingers cat.

@MrYourDry 5 лет назад

Couldn't agree more, he should've been dissing with all his heart.

@inyobill 4 года назад

This is the Mathologer's video, he doesn't have a problem with it, and the videographer actually does contribute.

@tommyvasec5216 4 года назад

He is contributing, representing you the ignorant public.

@Loonce 6 лет назад

There was a video made by Numberphile called, "Why -1/12 is a gold nugget", where the professor, Edward Frankel, made it clear on what the identity "1+2+3+...=-1/12" really meant.

@Mathologer 6 лет назад

Yes, a very nice video :)

@MDelorean 6 лет назад

Would be fair to mention that video as well. Otherwise the term 'misled' could be partially true for your video. It's clear math videos like to be 0 or 1 :) Great video, my issue is just a small footnote.

@Mathologer 6 лет назад

I link to it and lost of other relevant thing in the description. There is only so much you can say in a video :)

@MDelorean 6 лет назад

Yes, that's also the case with Numberphile of course, but their videos are shorter so they cut (too many) corners. I just like the 'gold nugget' metaphor and wanted your opinion. Maybe you have another (better) metaphor. But like I said before, it's only a footnote in an otherwise very well made video, the effort really shows!

@setha3287 4 года назад

Isn't that the video that compared the infinity-ness of the series as a bunch of dirt that can be swept away, leaving a gold nugget behind. I found that almost as troubling as the first. It was like an explanation why it's true without explaining how it's true.

@habarvaz3142 2 месяца назад

I love this channel and everyone makes mistakes You just gotta remember how limits and sums work, 1-1+1-1+1-1.... does not converge as it doesn't have a limit as it's value simply changes. Also if you were to assign it a value I would argue for 0 and not 1/2, but it's absolutely undefined.

@pablofernandezpena1045 13 дней назад

The last part where explains the zeta function in terms of the eta function which in turn is based on the average summation is pure gold!

@kueist8952 5 лет назад

"If you've made it this far you know..." I stopped knowing at the 10 minute mark

@constantly-confused5736 4 года назад

Well, I found it releatively easy to follow along.... then again... I have a math degree ;P

@jamest3828 4 года назад

@@constantly-confused5736 I'm 14 and I understood it

@alexandrubragari1537 4 года назад

Me too and i actually like the video and seen until the end and i just completed high school and some shit calculus and algebra from computer science.. Many time i wish i choosed math or phisics instead of cs

@hassanakhtar7874 4 года назад

@@alexandrubragari1537 rip bro

@1992WLK 4 года назад

I stopped at the 10 minute mark too. Cause it felt he was done explaining the wrongness. "What else is there? An extra 30 minutes! What the hell... I don't remember signing up for this."

@EvgeniiNeumerzhitckii 6 лет назад

Awesome video! My summary: 1+2+3+... = -1/12 is wrong when we use the standard summation for the infinite series from 1st year calculus, since 1+2+3+... series diverges. However, if we use a very different Ramanujan summation method, then 1+2+3+... = -1/12 is true. The problem with the Numberphile video is that they used the standard summation method incorrectly to prove the -1/12 result. This might give millions of people a false idea that 1+2+3+... = -1/12 is true for standard summation. I think making mistakes is ok. Only those who do nothing don't make them. So good job Numberphile, keep the ball rolling! :)

@douggwyn9656 6 лет назад

It should also be said that there are standard meanings for symbols like + and =, and if somebody intends to change their meanings he is socially obliged to explain them in advance. It is much better to retain the standard meanings and use different symbols for operations or things that have nonstandard properties. For example, a right-pointing arrow can represent a transformation that does not preserve equality, as in the string-theory text exhibited on the Numberphile "ASTOUNDING" video.

@EvgeniiNeumerzhitckii 6 лет назад

Well said! And good point about the arrow, it would be more suitable here, since 1+2+3+... is not equal to -1/12 in the usual sense. But we can say that we can use so-and-so method to associate the -1/12 number with the 1+2+3+... series.

@douggwyn9656 6 лет назад

Yeah, I wish I knew a rigorous definition for "associating a number with" that supports substituting the number for.

@MyRahulpathak 5 лет назад

its a interdimensional maths

@pentachronic 5 лет назад

I disagree about using another method to make a mathematical claim. That’s like saying, I scrambled the letters of the English language into my crypto code, made some sentences and then descrambled it and came up with the perfectly reasonable answer to your question (which is exactly wrong!!). Mathematics is a rigorous scientific method. You obey the rules. It’s that simple.

@sentzeu 2 года назад

I agree it was their worst video, but they never retracted it, nor did they ever explain it or give any indication that they understand the difference between a divergent sum and a Ramanujan summation.

@dominicmarazita6846 2 года назад

Actually, they explained all of that and more in separate videos that they tell you about in the video. I.e. it references the proof for S1 in the ghandi’s series video. They also talk about their extra cut where they talk about the rest of the sums and Riemann’s Zeta function. They explain everything within this video.

@l.w.paradis2108 2 года назад

@@dominicmarazita6846 True, they finally did, but the original video they posted was a disgrace. Their insinuation that mathematics is some esoteric knowledge rather than the universal deductive science that it is, ultimately accessible to all (see, e.g., Plato's Meno), was the most disgraceful part of an all-around disgraceful performance.

@SanePerson1 Год назад

One of my last graduate students (in chemistry) and I had a heated and, at times, acrimonious argument over the Numberphile video and the NY Times article about it that came out soon afterwards. I was of the opinion that the Numberphile video was trash, and - worse - it undermined one of it's main purposes, which is to spark interest in mathematics. The student was so impressed that the series summation appeared (stripped of context) in a book about string theory that he insisted that I, a mere chemistry professor, was full of s**t. Thanks for your much calmer, much more amusing, presentation on this whole episode and your very nice and accessible (intuitive) discussion of analytic continuation.

@PC_Simo 11 месяцев назад

*_WOW!_* He sure was arrogant; especially for someone, who’s royally wrong. 😮

@samueldeandrade8535 8 месяцев назад

It actually sounds like you are full of s**t.

@faith3174 6 лет назад

Thank you for explaining analytic continuation in an actually good way. I've seen so many math RU-vidrs talk about it and every time it boils down to "the most natural extension of a specific function," which, I imagine, would leave many questions in the audience's head. I can see myself understand this when I didn't already know what analytic continuation or any kind of analysis deals with. Really shows why derivatives shape a function which is not traditionally defined. Great job!

@General12th 6 лет назад

3blue1brown defines it pretty well. It's most natural because the derivative is constant and it preserves angles.

@jbiasutti 6 лет назад

The exact definition of the analytic continuation is that the value and derivative of the function is the same as the data given at all point.

@alvaroaguado3 6 лет назад

People taking this video as offensive have little respect for mathematics. In the mathematical community proofs must be truths not follower fights in terms of what channel i like better. The way is presented may get some angry but the proof seems to be correctly developed

@oenrn 6 лет назад

Welcome to the snowflake generation. Where the truth doesn't matter anymore, only if you "hurt people's feelings" (TM)

@TheVergile 6 лет назад

the problem is not his proof, but something no serious scientist would do: quoting parts of someone else work without considering the other half of their work. Numberphile themselves added two more videos to their introductory video which went viral. In these videos (esp. "why -1/12 is a gold nugget") they explain in more detail how -1/12 actually differs from a convergent sum and why it is still meaningful. What Mythologer does here is quoting and attacking (yes attacking. The headline of this video and the way it is presented is sensationalist and honestly a bit disappointing, since it is in general good content) part of someones work, ignoring other parts completely. Especially if the part of work you quote is a video made to introduce non math-PhD people on the internet to interesting and "mindblowing" concepts in mathmatics.

@WitchidWitchid 6 лет назад

But it's not the right answer. The correct answer is that the infinite series 1+2+3+4+... is divergent. It does not converge to -1/12. This is what Mathologer has pointed out. If an infinite series diverges it diverges. Stating "it diverges" is stating the correct answer.

@WitchidWitchid 6 лет назад

In a mathematical context it's not an attack nor is it sensationalist.It's only an attack if one is defending a channel or brand.

@WitchidWitchid 6 лет назад

I am not basing my conclusion on intiuition but rather on regular summation. If we derive an expression for the partial sums of 1+2+3+4+... (i.e. Sn=n(n+1)/2 ) we find that the partial sums get increasingly larger as n->infinity thus the series is divergent with respect to regular summation and is a valid and correct answer. If we use zeta function regularization (i.e Reimann Zeta function) / Reimann summation we can assign values to otherwise divergent summations. Applying such techniques we can indeed correctly answer 1+2+3+4+... + = -1/12. Such results have value and meaning in Physics and I stand corrected in my assertion that it is the wrong answer. n the contect of regular summation however we find ever increasing partial sums and we conclude the series s divergent which in this latter context is correct although not particularly useful if you're a Physicist. :) Nonetheless 1+2+3+4+5+6+... is divergent is correct with respect to it's regular sum which is proven when we look at the limit of the expression for partial sums S = n(n+1)/2 as n-> infinity which is clearly divergent therefor 1+2+3+4+5+6+... is divergent. Q.E.D.

@Lolwutdesu9000 3 месяца назад

We need a new video after the new Numberphile video!

@newstandardaccount 3 месяца назад

THANK YOU. It's been a while since I took calculus but I immediately recognized this as a series that would have to go to infinity. In fact, when I started off studying calculus, the first thing I learned was about limits. As a number increases without bound it must go to infinity, and therefore summing those same numbers must also go to infinity. However I wasn't sharp enough to know what was wrong with the calculations, only that they must be wrong (or, I must be crazy).

@jlhjlh 4 года назад

Thanks for this great video! I think there's also another way to reason about this: Given the infinite series S = 1 − 1 + 1 − 1 + 1… the conclusion was made (by summing it with a shifted copy of itself) that S + S = 1. However a silent assumption is made here that S is an actual number in the first place. It was assumed that S ∈ ℝ (or ℂ if you prefer) from which it follows that the expression S + S is a well-defined mathematical expression that has a meaning, from which one can conclude that S = ½ using the usual manipulations. However if S ∉ ℝ then what is S? Then the expression S + S lacks any definition of what it means and makes as much sense as the expressions "yesterday + the moon" or "the square root of yellow". Thus to complete the proof, one would have to show that symbolic manipulation on S have a meaningful definition and there exists a sequence of valid manipulations on it that lead to S = ½. That could for example be done by showing that S ∈ ℝ, but that is unfortunately not feasible. That is the missing part of the proof. And of course it's invalid to conclude that S ∈ ℝ because ½ ∈ ℝ ∧ S = ½, because that would be begging the question (a circular argument).

@ironmandedanadan9653 3 года назад

Yes you are right . There are many many problems in which we assume it to be a number by itself in the beginning and solve for that real value

@ironmandedanadan9653 3 года назад

But you should know that "while dealing with real numbers, addition and substraction on them results in real answer" but it is not always true for multiplication and division so as far as the series given in this video fall under this law we can consider them to be equal to s and (s€R)

@JohnRandomness105 3 года назад

Ever heard of the square root of a South American abacus?

@twobob 3 года назад

@@JohnRandomness105 The European Abacus flies faster though because the partial sums of it's constituent states are smaller, right?

@JohnRandomness105 3 года назад

@@twobob I never heard of that one before.

@PhilBagels 6 лет назад

a) If 1+2+3+...= -1/12, then b) 2+4+6+... = 2*(-1/12) = -1/6, and therefore c) 1+3+5+... = -1/12 - -1/6 = 1/12 d) Taking c-a gives us (1-1)+(3-2)+(5-3)+(7-4)+...= 1/12 -(-1/12) e) Simplifying, we get 0+1+2+3+...= 1/6 f) Therefore -1/12 = 1/6 And if you want to, you can keep going and generate all kinds of other values for 1+2+3+... I posted this (or something like it) on the Numberphile video.

@Mathologer 6 лет назад

Yes, that's the sort of contradictory calculation I was after :) Maybe try to come up with something that does not require zapping infinitely many interspersed 0s, so restrict yourself to the three basic properties that I highlighted in the video :)

@PhilBagels 6 лет назад

OK, how's this: a) 1+2+3+... = -1/12 b) 1-1+1-1+... = 1/2 c) a+b = -1/12 + 1/2 = 5/12 d) (1+1)+(2-1)+(3+1)+(4-1)+...= 5/12 e) Simplifying, we get, 2+1+4+3+6+5+...= 5/12, which is the same as 1+2+3+4+5+6+... f) Therefore -1/12 = 5/12. Or is re-ordering a problem also?

@PhilBagels 6 лет назад

Perhaps even better: a) 1+2+3+4+...= -1/12 b) Insert just one zero: 0+1+2+3+...= -1/12 c) Adding a+b gives 1+3+5+7+... = -1/6 d) Insert just one zero again: 0+1+3+5+... = -1/6 e) Add a+d giving 1+3+6+9+12+...= -1/4 f) Subtract that initial 1: 3+6+9+12+... = -5/4 g) Divide by 3: 1+2+3+4+...= -5/12 h) Therefore -1/12 = -5/12. Ha! Nothing infinite done. Added two zeros, one at a time, and subtracted one once.

@ferb1131 6 лет назад

Adding or subtracting sequences was permitted, but adding or subtracting an individual number like one wasn't a permitted operation was it? I'm more interested in the fact that a-b implies an infinite sequence of ones sums to zero. Shifting that new sequence again and subtracting from itself easily proves that 1=0, implying that the shift operation isn't allowed for these 'supersums'.

@ahmedouerfelli4709 6 лет назад

Do you think the sequences of ones supersums to 1? Maybe you mean infinity or do I miss something?

@ricardoguzman5014 Год назад

I just watched this video now, even though it's 4+ years old. Thank you for posting. My concern, like yours, is that false information is dispensed, potentially millions of people all around the world believe it, repeat it, and the misinformation accumulates. Personally, I have encountered 7 situations in the last 15 years or so where I have had to correct false mathematical conclusions. I have commented on 3 videos in recent weeks where well respected youtubers make incorrect statements. In the most recent, Stand-up Maths channel (Matt Parker) was talking about infinities and he used an example of numbering ping pong balls and adding and removing them from a box and stated a completely erroneous outcome. I commented on it using accurate mathematics. But the strange thing is, as I was scrolling through the comments, I didn't find any other comment exposing Parker's conclusion as false. I don't know how to contact Parker to point out his error in hopes he posts a correction video. He posted the video on October 31.

@MuffinsAPlenty Год назад

Matt Parker has a lot of comments on that video, and unfortunately, I can't find yours. Nonetheless, I consider Matt's answer in that video to be correct. He definitely didn't provide enough explanation to make everyone happy, but I could possibly fill in the gaps. Would you be able to give me the comment code for your comment so I could read it and reply? To get your comment code, if you find your reply and click on the "date" of it, like "1 month ago", you get a code to your comment. For the comment I'm _currently_ replying to, that code is watch?v=YuIIjLr6vUA&lc=UgwIsRIp9QSD1lGjphR4AaABAg

@ricardoguzman5014 Год назад

@@MuffinsAPlenty I didn't figure out how to get a code. In any case, Matt Parker is absolutely mistaken. I don't know why people think this. It is strange to me that people have believed this falsehood for almost 70 years. That's INSANE! What has happened to thinking mathematicians??? It literally boggles my mind, I can't say it enough. RU-vid videos, online posts, wikipedia article on the Ross-Littlewood paradox, they all got it WRONG. WHY???!!! OK, here's the bottom line. The so called Ross-Littlewood paradox is nothing more than a modified version of a conditionally convergent series. THAT'S ALL THAT IT IS. Don't people get that? It's just that whole numbers are used instead of fractions. I never heard of this paradox until I saw Parker's video on it. When he asked the question, how many balls are in the box after an infinite number of times, and then said the answer is zero, I thought about it for maybe 15 seconds before I thought, wait, that's not right, that can't be. Then I worked out the math for it and posted it as a comment on his video. Then over the next few days I figured out some more math related to it. Again, all it is a conditionally convergent series problem. We have known how to solve those for almost 200 years, and we know that if you change the order of the terms in the series, you get a different sum. And you are right, Matt doesn't even answer the question. He switches to, "oh, let me show you. What if we have a box with an infinite number of balls in it, and then dump it out" Um, That's not the same! Focus on the exact problem with the exact parameters that you described, not this switch up business. He says something like, "that's just how infinity works". WRONG. That is NOT how infinity works. Then he says, "at midnight, balls flying in, balls flying out". What kind of mathematics is that?? That's just fun theatrics and nothing else. He NEVER gets around to actually working out the mathematics of the exact problem he describes. I watched another video about it some days later by that female youtuber Up and Atom. She uses the actual scenario that Littlewood uses in his book (Parker uses a modified version). And she comes to the wrong conclusion because she doesn't understand the nature of the problem. Anyways, if you give me an email address, I can send you the comment that I posted on Parker's video. It uses very elementary math to show the correct answer. I also searched online for other work on the so-called paradox. And I actually found a guy that posted almost the same kind of correct logic that I found. If you google Ross-Littlewood paradox, one of the listings that shows up is titled "on resolving the littlewood-ross paradox - Project Euclid" and the website shown is projecteuclid.org. Thanks for your reply, chat again soon maybe. 12/17/2022--Forgot to mention that because whole numbers are used, mathematically it becomes a divergent series. Parker says the box would be empty, basically saying the number of balls in the box converges to 0. Wrong of course because divergent series do not converge.

@Rustam2704 Год назад

13:32 - "Does this prove that M is 1?" *video cut* *reassuring* "Nooo))00" *nervous laugh* (*yes*)

@papalyosha 4 года назад

22:09: The supersum 1+0-1+0+1+0-1+.... is still 1/2. However if you insert zeros like this: 1-1+0+1-1+0+1-1+0+... then the supersum indeed will change to 1/3

@Mathologer 4 года назад

Well spotted :)

@davidgould9431 3 года назад

I just worked through that, got ½, and naturally assumed I'd got it horribly wrong as usual. Thanks for the clarification.

@interestedparty00 3 года назад

Um, Alex was being sarcastic. He was asserting that adding zeroes could change their wrong answer to a different wrong answer.

@captainhd9741 3 года назад

Can’t you also get 1 if you say 1+(-1+1)+(-1+1)+(-1+1)...?

@captainhd9741 3 года назад

ohthis Shiny but the sum depends on how you add the terms. If you add 1 and then -1 etc you get a different result (if any really) which will be different if you add them in groups

@knocknockify 6 лет назад

I graduated college a while ago, haven’t done any math in a while, but I really love this video. You are so concise and clear with your explanations. You make me miss math class lol

@stevenhs8821 5 лет назад

Right on! I love math too. It is great, like music. So concise so clear. Nothing greater.

@inyobill 5 лет назад

This comment summarizes my feelings also.

@Gilsao157 3 года назад

mathlogger: "if you made this far you also heard of riemmans hypothesis" me: i didnt even knew that zheta functions existed...

@ganzeige 3 года назад

I read this comment exactly when he said it in the video LOL

@samsibbens8164 6 месяцев назад

Unlike the previous times I watched this video, I managed to understand convergent vs divergent series, and basically understood everything up to the Zeta function. Now I'm watching just trying to understand xD