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

Рет қаралды 2,980,998

Күн бұрын

Confused 1+2+3+…=-1/12 comments originating from that infamous Numberphile video keep flooding the comment sections of my and other math KZfaqrs 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".

Пікірлер: 9 700

@nivednewalit8117 5 жыл бұрын

This is the math equivalent of a diss track.

@goyonman9655 5 жыл бұрын

Math Battle 😂😂

@bilalkhares9337 5 жыл бұрын

loooooooooool

@jaytan531 5 жыл бұрын

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

@nowonmetube 5 жыл бұрын

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 жыл бұрын

@Multorum Unum 😐

@Josh-zu8cr 4 жыл бұрын

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

@chrisven899 4 жыл бұрын

@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 жыл бұрын

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

@ElectroMathExp 4 жыл бұрын

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 жыл бұрын

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 жыл бұрын

@@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.

@DemitriMorgan 2 жыл бұрын

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 жыл бұрын

how did that go?

@praharmitra 2 жыл бұрын

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

@scinary7052 2 жыл бұрын

@@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 жыл бұрын

@@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 жыл бұрын

@@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.

@PC_Simo Жыл бұрын

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 ай бұрын

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

@PC_Simo 8 ай бұрын

@@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.

@CoryMck 6 жыл бұрын

Things are heating up in the Math community of KZfaq.

@pentacles__ 6 жыл бұрын

Things about to get lukewarm up in this piece

@proghostbusters1627 6 жыл бұрын

Waiting for Numberphile's response.

@turtle7562 6 жыл бұрын

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

@CoryMck 6 жыл бұрын

I'm waiting for the disstrack

@doubtfulguest5450 6 жыл бұрын

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

@dustein4221 3 жыл бұрын

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 жыл бұрын

Mainly sum and equals but yeah

@KRYMauL 3 жыл бұрын

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

@baruchben-david4196 3 жыл бұрын

Also, 1/12

@jensrenders4994 3 жыл бұрын

No, only sum.

@90800905675 3 жыл бұрын

Very much agree with this one, context is everything

@monkerud2108 2 жыл бұрын

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 ай бұрын

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 ай бұрын

@@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 ай бұрын

@@misanthrophex Arguing for uncaused causes.

@scott1564 9 ай бұрын

@@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 ай бұрын

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.

@charlesje1966 Жыл бұрын

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 Жыл бұрын

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 KZfaq... I was fortunate to immediately go into the numberphile comment section and see someone recommend this video.

@osmarfreitas8646 Жыл бұрын

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 Жыл бұрын

@@candylover6419 search for "sum of convergent series"

@anomaliecosmos 7 ай бұрын

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 ай бұрын

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_*

@smith22969 5 жыл бұрын

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

@Mathologer 5 жыл бұрын

@AbhijitZimare1 5 жыл бұрын

If it was Asian, it would be +100

@schrodinger6991 5 жыл бұрын

@@AbhijitZimare1 i don' belive you

@user-kx7do4fh2j 5 жыл бұрын

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 жыл бұрын

I thought it was Indian

@dk6024 4 жыл бұрын

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

@otoyana 4 жыл бұрын

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

@poogmaster1 4 жыл бұрын

Minakami Yuki What’s wrong with Mencken?

@sottallu 4 жыл бұрын

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 жыл бұрын

@@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 жыл бұрын

Kinda like the opposite of Occam's razor

@joshuastucky 6 ай бұрын

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 ай бұрын

Can I ask you something?

@joshuastucky 6 ай бұрын

@@user-yi5cc9wn5c sure

@anhhoanginh4763 3 ай бұрын

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

@DavidSmyth666 6 жыл бұрын

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

@steliostoulis1875 6 жыл бұрын

There is no drama just mistakes

@alephbunchofnumbers 6 жыл бұрын

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

@carbrickscity 6 жыл бұрын

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

@frankschneider6156 6 жыл бұрын

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 жыл бұрын

DavidSmyth666 so this is what future arguments look like

@Daspied 4 жыл бұрын

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

@MrOllitheOne 4 жыл бұрын

i^2

@aaronleperspicace1704 4 жыл бұрын

@@MrOllitheOne = -1

@MrOllitheOne 4 жыл бұрын

shit just became real

@AlgyCuber 4 жыл бұрын

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

@balsoft01 4 жыл бұрын

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.

@tomaszberent801 Жыл бұрын

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 Жыл бұрын

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 ай бұрын

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 ай бұрын

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.

@DanielKRui 3 жыл бұрын

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.

@vaneck4438 4 жыл бұрын

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

@lokithecat7225 4 жыл бұрын

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

@RalfsBalodis 4 жыл бұрын

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

@alexandren.9346 4 жыл бұрын

@- 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 жыл бұрын

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

@mahmoodemami7466 4 жыл бұрын

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

@kristoferkoessel4354 4 жыл бұрын

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

@leonhardeuler6811 4 жыл бұрын

*-1/12th

@MattixHQ 4 жыл бұрын

It's '' halt mein Bier''*

@kristoferkoessel4354 4 жыл бұрын

MattixHQ Sorry guys 😂 you get the point...

@kristoferkoessel4354 4 жыл бұрын

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

@M3tag 4 жыл бұрын

@@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

@foreverkurome 9 ай бұрын

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.

@Jonathan-xb8yf 3 жыл бұрын

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 жыл бұрын

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

@mcjon5477 Жыл бұрын

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

@vgautamkrishna5197 11 ай бұрын

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

@viktorsmets29 2 ай бұрын

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

@martint1775 5 жыл бұрын

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

@blizzbee 5 жыл бұрын

poor cat

@Dondala 5 жыл бұрын

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

@nichitacruceanu9540 4 жыл бұрын

Lmao

@Alex-hj2jd 4 жыл бұрын

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 жыл бұрын

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

@Dreams_Of_Lavender 3 жыл бұрын

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

@amogorkon 3 жыл бұрын

...and then the real fun stuff starts!

@user-dg9eb4mc9t 2 жыл бұрын

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

@anshumanagrawal346 2 жыл бұрын

@@user-dg9eb4mc9t lol

@RichConnerGMN 2 жыл бұрын

nice pfp

@jakeenvelopes9561 3 ай бұрын

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.

@Tekay37 3 ай бұрын

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

@ArnavTHR 3 ай бұрын

which new vid

@Tekay37 3 ай бұрын

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

@v2ike6udik 3 ай бұрын

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 ай бұрын

More data after contact. Cant share. ReptileAI deletes.

@v2ike6udik 3 ай бұрын

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

@MrPLC999 3 жыл бұрын

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 жыл бұрын

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 жыл бұрын

I don't think you did.

@andreicecold4379 2 жыл бұрын

@@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 жыл бұрын

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 Жыл бұрын

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

@markgearhart1606 5 жыл бұрын

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

@matthewboyea3860 5 жыл бұрын

Thats a quality evaluation, Fonn the Human

@alexwang982 5 жыл бұрын

Quali-tea

@user9287p 4 жыл бұрын

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

@torontobud8902 4 жыл бұрын

Omg sisterrrrrr

@ashierapreston 4 жыл бұрын

Jason -e^(pi•i)

@mayaq8324 5 жыл бұрын

You killed my party trick

@christianrasmussen1 4 жыл бұрын

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

@bhavikshankar3235 4 жыл бұрын

Your part trick is still alive see from 41:15

@RedRad1990 4 жыл бұрын

Matt Parker's card trick, my friend :)

@cdavis7693 4 жыл бұрын

What kind of parties have you been going to?

@kristoferkoessel4354 3 жыл бұрын

Do 1=2 proof

@juancarlosortiz6756 9 ай бұрын

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

@madlad4206 3 ай бұрын

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

@Doeff8 3 ай бұрын

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

@yiutungwong315 Ай бұрын

41:20

@AmorLucisPhotography 2 жыл бұрын

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 Жыл бұрын

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 ай бұрын

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 ай бұрын

@@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 ай бұрын

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 ай бұрын

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

@benmcdaniel 6 жыл бұрын

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

@C1Ansy 6 жыл бұрын

Ben McDaniel And that is?

@minerscale 6 жыл бұрын

A funny joke: kzfaq.info/get/bejne/l7WEksV4kty7qZs.html

@benmcdaniel 6 жыл бұрын

When something in math isn't quite right, you name it after Matt Parker: kzfaq.info/get/bejne/l7WEksV4kty7qZs.html

@C1Ansy 6 жыл бұрын

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

@Tymon0000 6 жыл бұрын

I LOLed :D

@leonlu3147 6 жыл бұрын

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

@danildmitriev5884 6 жыл бұрын

Ohhhhhh yesssss, Star Wars references ^_^

@deadaccount4221 6 жыл бұрын

Mr Banana808 What is wrong with you

@NinjaoftheEnd 6 жыл бұрын

Mr Banana808 Are you an actual banana?

@jesuslovespee 6 жыл бұрын

Francesco Santi his pharmaceutical clock has dilated.

@Jotakumon 6 жыл бұрын

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...

@Owlrrex Жыл бұрын

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.

@dave6012 2 жыл бұрын

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.

@j03man44 3 жыл бұрын

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 жыл бұрын

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 жыл бұрын

Not satisfying at all

@davidr1138 Жыл бұрын

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

@thewatchman_returns Жыл бұрын

Physicists being physicists

@PC_Simo Жыл бұрын

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

@jessers1712 4 жыл бұрын

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

@A_Box 3 жыл бұрын

what is your line of work tho?

@jessers1712 3 жыл бұрын

@@A_Box Physicist, sadly ;'(

@kotarojujo6365 3 жыл бұрын

Jesse Kucharek he should meet me.

@DrCorndog1 3 жыл бұрын

Emphasis on "should."

@segmentsAndCurves 2 жыл бұрын

@@jessers1712 Remember to blink twice.

@mattsgamingstuff5867 2 жыл бұрын

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.

@re5o28 Жыл бұрын

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?

@jacfac9969 4 жыл бұрын

Everybody gangsta till there’s math KZfaqr drama.

@elasiduo108 4 жыл бұрын

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 жыл бұрын

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

@elasiduo108 4 жыл бұрын

@@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 жыл бұрын

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

@Wyverald 3 жыл бұрын

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

@jstodd4398 2 жыл бұрын

This is the best counterexample ive seen

@WMHinsch 2 жыл бұрын

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.

@JusticeBackstrom 13 күн бұрын

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

@Purin1023 6 жыл бұрын

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

@Mathologer 6 жыл бұрын

-1/12 time worse :)

@skhumbuzocele1330 6 жыл бұрын

😂😂😂😂😂😂😂

@metacylinder 6 жыл бұрын

All you do is math problems there...chilling

@TheLK641 6 жыл бұрын

I would have said pi time worse.

@ilpinto4925 6 жыл бұрын

it is the analytical extension of regular hell

@markstgeorge405 4 жыл бұрын

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 жыл бұрын

lol

@thelickpolice1210 3 жыл бұрын

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

@jedinxf7 3 жыл бұрын

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 жыл бұрын

PFFFFFTTTT dang!

@russell2952 3 жыл бұрын

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.

@king_noah_2692 2 жыл бұрын

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 ай бұрын

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 ай бұрын

@@PC_Simo I did it just for you

@PC_Simo 11 ай бұрын

@@king_noah_2692 Thank you 😌👍🏻.

@NolimitsNinja 2 жыл бұрын

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!

@shantanubadve4668 5 жыл бұрын

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

@XavierDesroches 5 жыл бұрын

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

@Caribbeanmax 5 жыл бұрын

@@XavierDesroches

@realdragon 5 жыл бұрын

This is math war, very brutal war

@crabsynth3480 5 жыл бұрын

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 жыл бұрын

Crab Synth whoosh

@trevorperkins4585 4 жыл бұрын

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 жыл бұрын

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

@nolann2382 3 жыл бұрын

@@irongolem5539 and you're losing

@markopolic9964 2 жыл бұрын

@@nolann2382 You are always losing a game of graphs

@sahiltrivedi69 2 жыл бұрын

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 👍

@drsolo7 8 ай бұрын

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

@swerasnym 6 жыл бұрын

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 жыл бұрын

Cool :)

@swerasnym 6 жыл бұрын

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

@GSandSDS 6 жыл бұрын

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

@Stefan1of3 6 жыл бұрын

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

@DanielBeecham 6 жыл бұрын

Heyo, cool!

@rcb3921 6 жыл бұрын

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?" kzfaq.info/get/bejne/ZrWRrcVorr3eknU.html

@Mathologer 6 жыл бұрын

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 жыл бұрын

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

@TomJacobW 6 жыл бұрын

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

@ExpIohd 6 жыл бұрын

There is also the 'extra footage' video on Numberphile 2 which goes into greater depth of the math on the original- kzfaq.info/get/bejne/e5OUbNCY1J6qd58.html

@AzCcc 6 жыл бұрын

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.

@several9286 2 жыл бұрын

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

@signorellil 3 жыл бұрын

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 жыл бұрын

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 жыл бұрын

@@marshallsweatherhiking1820 thank uou

@alvarogoenaga3965 2 жыл бұрын

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

@samueldeandrade8535 8 ай бұрын

... not really.

@Blananas2 5 жыл бұрын

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

@srimaryati337 4 жыл бұрын

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

@srimaryati337 4 жыл бұрын

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

@hypehuman 4 жыл бұрын

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

@jkellyk7920 4 жыл бұрын

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

@pavanato 4 жыл бұрын

OMG 314 LIKES

@JayWez 4 жыл бұрын

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 жыл бұрын

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 Жыл бұрын

Same here!

@lanchanoinguyen2914 Жыл бұрын

@@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 Жыл бұрын

same here

@l.w.paradis2108 Жыл бұрын

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

@beelzzebub 2 жыл бұрын

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 жыл бұрын

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

@telaferrum Жыл бұрын

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 ай бұрын

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 ай бұрын

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 ай бұрын

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

@69k_gold 2 жыл бұрын

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

@MathManMcGreal 6 жыл бұрын

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

@mheermance 6 жыл бұрын

I think they would prefer a maths off.

@playscirox2129 6 жыл бұрын

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

@awsomebot1 6 жыл бұрын

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

@alexanderf8451 6 жыл бұрын

*sharpens division symbols*

@IllumTheMessage 6 жыл бұрын

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

@eyepatch2696 6 жыл бұрын

Mathematics equivalent of a diss video

@jasonbucy 6 жыл бұрын

haha yes! Mathologer is basically Eminem

@88michaelandersen 6 жыл бұрын

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 жыл бұрын

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 жыл бұрын

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 жыл бұрын

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 KZfaq. Assuming you'd have Internet to open KZfaq. 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!

@jacobbabcock8943 2 жыл бұрын

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

@SunnyKimDev Жыл бұрын

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."

@DamianReloaded 6 жыл бұрын

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

@DanJan09 6 жыл бұрын

unpants? ok, you do you ;P

@AndreiNeacsu 6 жыл бұрын

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 жыл бұрын

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

@VeteranVandal 6 жыл бұрын

This is hardcore math.

@JLConawayII 6 жыл бұрын

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

@ugopinho2121 6 жыл бұрын

TOP 10 ANIME FIGHTS OF ALL TIME

@1stPCFerret 6 жыл бұрын

Anime?

@MrPointness 6 жыл бұрын

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

@glowingdawn9179 6 жыл бұрын

respect

@hernandojosedeavilapereira511 6 жыл бұрын

jajajajajaaj

@solarisone1082 5 жыл бұрын

Vegetto vs Buuhan: Mathematics Edition.

@stevekeiretsu 3 жыл бұрын

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 жыл бұрын

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 Жыл бұрын

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

@Fallkhar 2 жыл бұрын

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.

@FriedChckn13 5 жыл бұрын

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

@fblio7146 6 жыл бұрын

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 жыл бұрын

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

@vacuumdiagrams652 6 жыл бұрын

"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 жыл бұрын

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 жыл бұрын

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

@vacuumdiagrams652 6 жыл бұрын

"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.

@brianfriis4784 Жыл бұрын

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.

@collegemathematics6698 3 жыл бұрын

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

@kueist8952 5 жыл бұрын

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

@constantly-confused5736 4 жыл бұрын

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

@jamest3828 4 жыл бұрын

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

@alexandrubragari1537 4 жыл бұрын

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 жыл бұрын

@@alexandrubragari1537 rip bro

@1992WLK 4 жыл бұрын

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."

@Loonce 6 жыл бұрын

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 жыл бұрын

Yes, a very nice video :)

@MDelorean 6 жыл бұрын

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 жыл бұрын

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 жыл бұрын

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 жыл бұрын

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.

@DrOmar11 2 жыл бұрын

Hi Mathologer, the last part of how the shaded part equals -1/12 is amazing and as you said no coincidence. Is there some PhD you know that connects the analytic continuation of zeta function with the shaded parts in the negative area of quadratics or polynomials? I am not a mathematician so my question might not be all that coherent. But I hope there is something you can investigate this in a PhD dissertation or even general idea of the complex plan area's connection with the negative part of polynomials or the are of the negative component of symmetrical functions such as quadratics. What do you think Mathologer?

@Mathologer 2 жыл бұрын

Actually this connection is not that hard to explain if you are familiar with some more advanced mathematical tools than the ones I allow myself to use in these videos :)

@kosttavmalhotra5899 5 ай бұрын

@@Mathologer please tell the connection i am gone compleley mad before finding this, but unable to any hint

@justinthejerkoff Жыл бұрын

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

@SmileyMPV 6 жыл бұрын

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 жыл бұрын

Very nice summary of most important points. Should be stickied

@ikaro342 6 жыл бұрын

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

@SmileyMPV 6 жыл бұрын

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

@drewkavi6327 6 жыл бұрын

Mathematical équivalent of a diss track

@jgallantyt 2 жыл бұрын

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 жыл бұрын

I think you have an issue ... :)

@sentzeu 2 жыл бұрын

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 жыл бұрын

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 жыл бұрын

@@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.

@Mathologer 6 жыл бұрын

Confused 1+2+3+…=-1/12 comments originating from that infamous 2014 Numberphile video keep flooding the comment sections of my and other math KZfaqrs 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 жыл бұрын

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 жыл бұрын

Mathologer what happened to the plain black shirt at start 😁

@Nmmoinn 6 жыл бұрын

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

@frederickm9823 6 жыл бұрын

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

@alejandrolopeztobon1643 6 жыл бұрын

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.

@papalyosha 4 жыл бұрын

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 жыл бұрын

Well spotted :)

@davidgould9431 3 жыл бұрын

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

@interestedparty00 3 жыл бұрын

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

@captainhd9741 3 жыл бұрын

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

@captainhd9741 3 жыл бұрын

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

@kevinaustin6971 2 жыл бұрын

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

@professoralphane3209 2 жыл бұрын

OK so I've been trying hard to understand this concept . Now my level of maths understanding is low , I'll admit but working on the formula mathologer gave using simple graphs and set substitutions I've tried to understand the subject matter . Visually graphically and substitution wise assessing the raw data of the sets step by step my first real problem occurs with the assumption 0,4,0,8 0, is the same as 4a . If you asses the sets graphically you can see the sets S minus S2 step by step creates a line that varies around the S average line in the same way set S2 varies around the 0 line as should be . The set 4S when plotted does not equal the sets S minus S2 the highest possible value for which is 2S and the low points always 0 . So if S - S2 doesn't equal 4S then the calculation falls down . Unless anyone can describe why this should be .

@macronencer 6 жыл бұрын

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 жыл бұрын

That's the way I look at it :)

@CGoody564 6 жыл бұрын

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

@screwhalunderhill885 6 жыл бұрын

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 жыл бұрын

kzfaq.info/get/bejne/ZrWRrcVorr3eknU.html

@macronencer 6 жыл бұрын

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.

@How-Do-I-Nezzy 5 жыл бұрын

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 жыл бұрын

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

@Dondala 5 жыл бұрын

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 жыл бұрын

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

@inyobill 4 жыл бұрын

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

@tommyvasec5216 4 жыл бұрын

He is contributing, representing you the ignorant public.

@jorgenharmse4752 9 ай бұрын

Analytic continuation is completely determined, _provided_ some conditions are met. The new domain must be specified, it must be a connected open set, and an extension must exist. For example, the logarithm cannot be complex-analytically continued to the complex plane, even after you throw out the obvious singularity at 0. You need a branch cut from 0 to complex (unsigned) infinity, and the values depend on how you choose the branch cut.

@wayneosaur 3 ай бұрын

To be fair, the various sums he says are divergent assume the agreed upon convention of partial sums. One could use different conventions.

@TheMrBlackRaven 6 жыл бұрын

the answer is 42

@tsresc 5 жыл бұрын

That's the answer for everything+nothing. 42=(-1/12)+X. So the value of nothing is 503/12. Yeah, I discovered the value of nothing. I'm starboy mathematician. Yay! Bingo! Allons-y! Eureka! Ola! Yo! THICC!

@samt1705 5 жыл бұрын

What was the question though? 😃

@the_luna_lily6234 5 жыл бұрын

Sam T everything 42 is the answer to life

@aidankhan6194 5 жыл бұрын

@@samt1705 it's a reference to hitchhiker's guide to the galaxy. There's actually people who try to prove this.

@samt1705 5 жыл бұрын

@@aidankhan6194 just what I expected it to be.. Thanks!

@ragnkja 6 жыл бұрын

In an earlier Numberphile video, Dr James Grime described S_1 as PSEUDO-convergent, which I think is the most accurate description, since it doesn't *really* converge to 1/2.

@cameronholt4407 6 жыл бұрын

Gimme a link fam I wanna see Grime :)

@ragnkja 6 жыл бұрын

Here's the relevant video: kzfaq.info/get/bejne/hqmlkqV_s6-ZqGg.html And here are a couple of other videos he's made on his own channel about infinite sums: kzfaq.info/get/bejne/bcx3osyf2J3VY6c.html kzfaq.info/get/bejne/mt2Jg7Kakq7Kl2w.html

@cameronholt4407 6 жыл бұрын

Thanks!

@samus88 6 жыл бұрын

Then the infinite sum doesn't *really* converge to -1/12... because it just doesn't converge at all. It goes to infinity.

@cameronholt4407 6 жыл бұрын

willprogresivo I agree I'm just here for the maths drama ;)

@zacharymesecke9638 Жыл бұрын

This video gave me hope that maybe, just maybe, maths isn't as crazy difficult as I thought it was

@saptarshidebroy7075 Жыл бұрын

nothing is crazy difficult if you practice enough

@Kfruistik 11 ай бұрын

@@saptarshidebroy7075 and "crazy difficult" things require much more practise. Some people wouldn't be able to comprehend it

@DarioVolaric Жыл бұрын

Never doubt someone who explains math in a german accent.

@grantorino2325 7 ай бұрын

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

@faith3174 6 жыл бұрын

Thank you for explaining analytic continuation in an actually good way. I've seen so many math KZfaqrs 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 жыл бұрын

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

@jbiasutti 6 жыл бұрын

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.

@bluthemeth 3 жыл бұрын

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

@grantorino2325 3 жыл бұрын

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 жыл бұрын

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 жыл бұрын

@@grantorino2325 :O

@roseCatcher_ Жыл бұрын

This video proves you wrong too.

@NTNscrub Жыл бұрын

@@roseCatcher_How so?

@Gilsao157 3 жыл бұрын

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

@ganzeige 3 жыл бұрын

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

@thorH. Жыл бұрын

If you insert an infinite 0 into the series and calculate the super sum, you can get an average that converges to 0 if the amount of zeros is sufficiently greater than the amount of positive/ negative values of that series, which could converge to a different value.

@jorgenharmse4752 9 ай бұрын

Starting with 1-1+1-1 ..., inserting zeros in the right places can make the super sum come to anything between 0 and 1 (inclusive), since you effectively choose how much the two possible partial sums are repeated. I think you can even arrange to have no super sum by inserting increasingly large blocks of zeros.

@BenDRobinson 5 ай бұрын

I'm scrolling through the comments to see if anyone answered the puzzles he set, because I had a quick think about the one he asked about the super sum of 1+0-1+0+1+0-1... he said the super-sum is not = 1/2, but I'm quite sure that it is. Mind you, he didn't exactly say that the inserted 0s are every second element of the new version of the series, but said "if we insert infinitely many 0s, _like_this_", and the fact that he posed the question as though there was a fixed answer suggests that we can assume that pattern continues.

@BenDRobinson 5 ай бұрын

@@jorgenharmse4752 I think you're right on both counts. (Which is pertinent to my other comment.) I doubted your second conjecture at first, but it does make sense. If you keep adding enough 0s while the partial sum is on 1, you can drag the cumulative average up as close as you want to 1, and then likewise down arbitrarily close to 0 again while the partial sum is on 0. Then again, maybe the fact that you get to keep iterating the averaging step an unlimited number of times defeats that strategy. I suspect that for any finite n you could put in enough zeros for the first n iterations not to converge, but you might not be able to ensure it forever with a fixed choice of inserted zeros.

@jorgenharmse4752 5 ай бұрын

@@BenDRobinson I agree that my conjecture is not obvious. I said 'I think' because I was too lazy to try to write a proof at the time, but here it is. The original sequence of partial sums is just 1,0,1,0,1,0,1,0,... . Padding the summands with zeros gives us blocks with no change, e.g. 1,1,0,1,1,0,1,1,0,..., for which the long-run average is 2/3. Once you have a convergent sequence, repeated averaging just gives you more sequences that converge to the same limit. Can we make the blocks grow fast enough to avoid convergence entirely? Consider an eventually-constant sequence a_1,a_2,...,a_m,c,c,c,c,c,... of real numbers. However large m may be and however big the absolute values of a_1,...a_m, the limit of the averages is c, and any amount of additional averaging gives us the same limit. Averaging slows the convergence, but for any k we can pick n such that for every average up to k-th order the (m+n) position has a number that differs from c by at most 1/k. Now consider any sequence that starts with a_1,a_2,...,a_m and n occurrences of c (in that order). Anything after the (m+n) position has no effect on the repeated averages up to that position, so the (m+n) positions for all averages up to k-th order still have numbers that differ from c by at most 1/k. Now consider a sequence with n_0 occurrences of 1, n_1 occurrences of 0, n_2 occurrences of 1, ..., n_2k occurrences of 1, n_{2k+1} occurrences of 0, ... (in that order), where n_0=1 and n_k is determined recursively so that all averages up to k-th order have in the (n_0+n_1+...+n_k) position numbers that are within 1/k of the constant in that block. For every k, the k-th order average has inferior limit equal to 0 and superior limit equal to 1, so it doesn't converge.

@jlhjlh 4 жыл бұрын

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 жыл бұрын

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 жыл бұрын

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 жыл бұрын

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

@twobob 3 жыл бұрын

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

@JohnRandomness105 3 жыл бұрын

@@twobob I never heard of that one before.

@louiskohnke2343 6 жыл бұрын

*3* *2* *1* *intro music* "What is up DramaAlert Nation?! I'm your host Killer Keemstar! Let's get roooiiight into the news! This week something crazy happened. The KZfaqr Mathologer actually uploaded a video calling out Numberphile! That's right, he actually disproved the claims in their old video 'ASTOUNDING: 1 + 2 + 3 + 4 + 5 + ... = -1/12' by calling it "completely wrong"! Watch this! 0:20 *dramatically looks into the camera* Immediately I contacted Mathologer and Brady Haran, the host of Numberphile asking for an Interview. But both of them haven't responded yet! This is the first time we have seen such drama in the education part of KZfaq, but unfortunately it seems like the maths war has only just started! The comment section of the original Numberphile video is currently full of comments calling out the false maths. We will have to wait and see Numberphile's reaction, but I'm all for presenting correct maths! I don't get why Numberphile would upload such a video, I don't get it... Also in the news: Logan Paul..."

@Mathologer 6 жыл бұрын

@Roescoe 6 жыл бұрын

This This... is incredible.

@X_Baron 6 жыл бұрын

Logan Paul asks: "What are 'maths'?"

@dlwatib 6 жыл бұрын

Logan Paul is American so he would never ask "What are 'maths'?" To Americans, mathematics is singular, not plural, just like physics, and so is abbreviated to math. Therefore, "What is math?" is correct.

@X_Baron 6 жыл бұрын

Don't you think that would be the whole reason he'd ask that question, given that Keemstar is also American and, in the transcription by the starter of this thread, seems to use the plural spelling? :D

@samsibbens8164 6 ай бұрын

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

@NerdWithLaptop 2 жыл бұрын

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

@ScottBogert 4 жыл бұрын

You should invite the professor over at Numberphile to a discussion of the topic. You could live stream a hangout, or something. It could be interesting.

@bikedawg 4 жыл бұрын

But armed with a sharpie, knife and a cleaver.

@koalasquare2145 3 жыл бұрын

I don't know if they can overcome this beef

@jamirimaj6880 3 жыл бұрын

@@koalasquare2145 sad that both those two guys are from Australia

@The1DistantFl4pjack 3 жыл бұрын

Not likely to happen. When he was called out in the comments/on twitter, he got incredibly defensive and wrote a whole blog post on how “actually this is totally allowed and you’re all wrong”

@jamirimaj6880 3 жыл бұрын

@@The1DistantFl4pjack who, Brady?

@teukkaboy 5 жыл бұрын

I get scared everytime he laughs :(

@inyobill 5 жыл бұрын

Funny, I find his laugh charming. Different strokes, and all that.

@Spathephoros 5 жыл бұрын

Hilarious

@teukkaboy 5 жыл бұрын

@@Spathephoros Seems like to some people it was

@chrisprilloisebola 5 жыл бұрын

lol

@dannygjk 5 жыл бұрын

Don't worry, (unless he is holding a big knife).

@spearmintlatios9047 Жыл бұрын

Calc 2 moment: There’s a more in depth way to prove that the 3 used series for the false equivalence: Representing each sum using n notation you get this: 1 + 1 - 1…. This sum is the series from n = 0 to n = inf of (-1)^n. 1 - 2 + 3…. is the series (n+1)* (-1)^n (first term is 0 can we can just treat is as n * (-1)^n) 1 + 2 + 3 … is simply the sum of n + 1 (but first term is 0 so we can ignore it) For any series, if the limit of the term as n approaches infinity is not equal to 0, it diverges to infinity. This is why sum 3 is infinite. The same can be said about the first oscillating sum. The first and second sums, you can go into a little bit more detail. For an alternating series (one multiplied by (-1)^n), it can be written as (-)^n * (Sum B). In this case we have (-1)^n * 1, and (-1)^n * (n). If the limit of this second term is equal to 0 and the term is decreasing, then the series converges. The neither of these facts are true for either, so they are divergent and oscillating.

@ricardoguzman5014 Жыл бұрын

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 Жыл бұрын

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 Жыл бұрын

@@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. KZfaq 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.