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

Embed

**Published on Jan 13, 2018**- Confused 1+2+3+…=-1/12 comments originating from that infamous Numberphile video keep flooding the comment sections of my and other math TheXvidrs 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.

The original Numberphile video is here

thexvid.com/video/w-I6XTVZXww/video.html . 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/NoteBooks/NoteBook1/chapterVIII/page3.htm

This notebook entry was also one of the starting points of my last video on this topic: thexvid.com/video/jcKRGpMiVTw/video.html

Other good videos that deal with this strange “identity” include the following:

thexvid.com/video/0Oazb7IWzbA/video.html (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.)

thexvid.com/video/sD0NjbwqlYw/video.html (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/PPN378970429).

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/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/

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 :)

Burkard

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_theory_book.jpg

For today's maths t-shirts google: "zombie addition math t-shirt", "label your axes math t-shirt".

Vincent Tonkes44 minutes agoThe world makes sense again! :-)

DeepDive10 hours agoThe problem lies in trying to assign a value to a valueless equation.

Also why would assign -1/2 to be 1+1+1... That's just making things fit your answer

Manuz54Day agoIm waiting for numberphiles counter disstrack

unknowning unknown3 days ago^{+1}he sounds like he is going to be busted by fbi for child related crimes

Vulpes Ignitare3 days agoOk the earth is round again.

Jirikberky7 days agoGreat video, but Jesus those chucklehead comments from off camera are infuriating

Zelos Photizo7 days ago^{+1}This guy is in trouble when James Bond see’s this!

Sillius Soddus8 days ago^{+1}Does anyone else find it weird how often they crack themselves up over nothing?

delhigod7 days agoNot really, it's cultural and I didn't have any issues with it.

Reynaldo Matos Hortensi8 days agoThanks! You are my hero!

Stevan Miladinovic8 days ago^{+1}(1-2)+(3-4)+(5-6)+(7-8)+(9-10)+...=

-1+-1+-1+-1+-1+..... =

-Infinitys

Alex Yi9 days agoprobably just mad that you weren't in a numberphile vid

JepZ10 days agoWhy do all mathology videos have such low like ratings?

Trapper10 days ago^{+3}I didn't even feel these 42 minutes passing by. Great work.

KuroiRayman10 days agoSo the numberphile video more or less proofed by contradiction that at least one of the three series has to be divergent, because otherwise the sum 1+2+3... = -1/12 would be correct... Which it obviously cannot be

Kevin Kragenbrink11 days agoHas anyone pinged @numberphile about this? I'd love Brody to release a response video. :D

lutexz200511 days ago1+0-1+0+1+0-1

0--1/2--1

-1---1/2--0

1/2-1/2+1/2-1/2

-1/2--0--1-2

0,0,0,0,0,0,0,0,0,

La súper suma te daría 0

Kaczankuku12 days ago0:30 So why you, Mathologer guys, need so much time (13th January 2018) to published correction of Numberphile mistake???

Kaczankuku11 days ago@slycordinator Me too.

slycordinator11 days ago@Kaczankuku Or maybe I thought your comment was nonsense...

Nah, couldn't possibly be that. There has to be a nefarious plot by people who live Mathlogger and hate Numberphile.

I'm a fan of both channels...

Kaczankuku11 days ago@slycordinator Faithful viewer was just provoked and wants to save the honour of the Mathologer. I wish better methods are developed to hide the truth.

slycordinator11 days ago@Kaczankuku Your question was flawed.

It conflates "waiting to do something" with "needing so much time" for it.

And now that I pointed this flaw out, you again go all conflationary. You conflate "pointing out a flaw" with "pushing back an attack."

Kaczankuku11 days ago@slycordinator My question was to hard for Mathologer guys. That's why immediately some faithful viewer tries to push back the attack. Ha ha ha. Congratulation especially for you.

neko mata12 days agoThe positive whole number series not being -1/12 should be obvious. No numbers are negative so the answer cant be negative. All numbers are whole numbers so it cant be a fraction. The first number in the series, 1, exceeds the sum.

Jimmye Romana Winburn12 days ago^{+2}Ramanugen once told us for fun

that the sum of the numbers from one

was not aleph nought

as we all might have thought

but one twelfth times negative one.

PossiblyKale12 days agoThe argument is less about the sum being equal to the value and more about the strange way algebra can be manipulated.

chu Harry12 days agoWhat is so special at April first

George Simpson13 days agoSeems obvious that an average is not a limit. 010101010101 is NEVER 1/2. That's a mean.

VolcannXd13 days agoWho tf is even this guy?

Abdur-Rahmaan Mohabbat13 days agoAnywhere you can buy the t-shirt?

Bob King13 days agoThe Integral of xdx between the limits of 0 to infinity does not result in a negative fraction.

Cristian L. P.13 days agoHi, I have a question.

In this calcs, where is the mistake?

e^(i*pi) = -1

(-1) * (e^(i*pi))= (-1) * (-1)

(-1) * (e^(i*pi))= 1

(e^(i*pi)) * (e^(i*pi))= 1

e^(2*i*pi) = 1

ln[e^(2*i*pi)] = ln[1]

2*i*pi = 0

2*i*pi/(2*pi) = 0/(2*pi)

i = 0 ???

i means imaginary number

pi = 3.1415...

e = 2.71828...

ln = Natural logarithm

Geoffrey Pingree14 days ago^{+2}does 1+0-1+0+1+0-1.... = 0 - or rather converge to 0? (i didn't see it in the comments). Thanks for this video!

Ian8 days agoTBH I don't get his point. The partial sum goes

1, 1, 0, 0, 1, 1, 0, 0, ...

So the averaging process produces

1, 1, 2/3, 1/2, 3/5, 2/3, 4/7, 1/2, ...

Overall oscillates between (2n+2)/(4n+2) and 1/2. This converges to 1/2

Jianbo Xie11 days agoWant to ask the same question

Josh Low14 days ago^{+2}I'm not disappointed about *wasting my time for watching this video* bec.

*Now we all the TRUTH ABOUT IT!*

zakarunro14 days agolove his laugh.... evil genious style XD

TRIKEN14 days agoIf the limit of the sequence An=n for n -> infinity is not equal to 0, the series diverges.

Douglas Strother14 days agoA really understandable presentation of analytic continuation!

Douglas Strother14 days agoThis kind of stuff is why I decided to skate through college with a Physics Major instead.

Seymour11 days agoSpeaking as an engineer that also had physics and math minors in university, nobody "skates through" a physics degree.

Yo wut14 days agoWhat

D.J.15 days ago^{+2}What a coincidence... I already had the popcorn and hot chocolate before he mentioned it.

The Proof Architect15 days agoDon't be so negative about Numberphile - they just made an honest mistake :)

Not Broihon13 days agoAnd can't be arsed to fix it by removing a video which still misinformes people on a daily basis.

Xeni Rzh15 days ago^{+1}Actually, the sum of all natural numbers equals.... 1/2 × oo(oo+1) ( if 'oo' means infinity).. I even have a proof for it

jason rebello15 days agoWhat about string theory?

YUGzed Natsu15 days agoHe is serving some tea, a hot one.

J Flow15 days ago^{+2}Not maths, math

Asad Naqib16 days ago^{+2}All the three sums used in proving are paradox and the constitution of maths says not to use paradox valued statements

Vitor Bowen16 days agoi hate those psedomath chanels

Frank Ansari17 days agoSometimes you only need to look closely and you see that something is odd.

One may be tempted to believe that the more numbers we add the closer

we come to the "real" result. In our case this is completely wrong.

Here is why.

Let's just look at our sum and then at two cases:

1. we multiply each number with 4

2. we take every forth number

In the first case if we stop this at any finite point (reagrdless whether we

stop at 5 or a googleplex) it will be always true that 4S > S.

In the second case we will add up only one forth of our numbers and our

result will be smaller than S.

In both cases S and our other sums will always be positive values.

Let's do an example with 12.

S = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 = 78

4S = 4 + 8 + 12 + 16 + 20 + 24 + 28 + 32 + 36 + 40 + 44 + 48 = 312

4 + 8 + 12 = 24

As you can clearly see taking every forth number gives a much smaller

result than 4S and also smaller than S.

But if we continue S up to infinity (and only then) a strange thing happens:

4S becomes part of S (every forth number). Taking every forth number and

building 4S becomes the same thing. So in this case it is 4S < S.

This can only be true if S is negative.

Lucas Carman17 days ago^{+1}The longer this video went on the dumber I felt

Douglas Strother15 days agoYou are feeling new folds in your cerebral cortex forming!

Taterz17 days ago^{+6}it just bugs me at a simple level because a divergent series does not converge to an answer. glad to see i'm not crazy.

Ollomont17 days agoIsn't like the first thing you learn on cameraman school "if you are behind the camera not as subject, do not intervene with the subject in front of your camera in any way"?

Nafis Forkan17 days ago0

Avers18 days agoИНСУЛЬТ.... ....... .....

Frank Ansari18 days agoThe way he explained how eta and zeta functions are connected is really great!

So for zeta(-2) = 1 + 4 + 9 + 25 + 36 + 49 + 64 + ... = 0

we can prove easily with this knowledge.

We already know from this video that

zeta(z) = eta(z) / (1 - (2 / 2^z))

This means in our case z = -2 so the denominator becomes -7.

So we can now just calculate the eta function.

1 - 4 + 9 - 16 + 25 - 36 + 49 - ... = ?

To do this we double the value.

1 - 4 + 9 - 16 + 25 - 36 + 49 - 64 + ...

0 + 1 - 4 + 9 - 16 + 25 - 36 + 49 - ...

1 -3 + 5 -7 + 9 - 11 + 13 - 15 + ...

Since we still have no result let's double it again!

1 - 3 + 5 - 7 + 9 - 11 + 13 - 15 + ...

0 + 1 - 3 + 5 - 7 + 9 - 11 + 13 - 15 + ...

1 - 2 + 2 - 2 + 2 - 2 + 2 - 2 + ...

Now we recongize the Grandi series.

1 - 2 * (1 - 1 + 1 - 1 + ...) = 1 - 2 * 1/2 = 0

4 * eta(-2) = 0

eta(-2) = 0

zeta(-2) = eta(-2) / -7 = 0 / -7 = 0

That's it!

Jack Alvarez18 days ago^{+2}The cameraman needs to shut up. Adds exactly 0 value to every video I've heard him interject in.

An Unknown18 days agoPlease watch the last moment of the video. Everybody will pretty much shocked. Don't focus on his Rapping and T-shirts.

Suani Avila18 days agousing

sum = a/(1-r) where

(sigma) ar^k we get that

1-1+1-1+...

is 1/(1-(-1)) or

1/2

idk just ran into this wanted to share

Suani Avila18 days ago^{+3}just a nitpick

zombie + human = (zombie)^2 when it should be zombie + human = zombie + zombie

Stefan Hummert18 days ago^{+1}Hi, first of all, thanks for making this video.

Second: I am no mathematician, but the word supersums seems missleading for me. I would look at it as an average (or even better it goes more to stochastics as an expected value).

Also, if you write the series as geometic graph, one would see that it makes an difference with what value you start and that when you shift the second graph one position, that it has not the same meaning as if you do not shift it. (hope you get what I mean)

Marcelo Resegue19 days ago^{+1}Thanks for making this video. I just used it again to explain for the nth time whats going on with those summations.

Lethal19 days ago^{+2}I think what NP was saying is, that "infinity" isn't a Number, it's an idea and expressed in a series as a limit. I thinks that's it, so Debunked well, not sure.

TommyB10 days agoThey are free to say that, but that doesn't make them correct. Infinities are a VERY well studied, researched and understood concept in mathematics. This the sum over N equals -1/12 business basically says that all these centuries of study have been meaningless and their results wrong, while the proof by obtaining that number is wrong it baffles me.

And rey19 days agoCongratulations you managed to suck all the fun out of it ☹️ By 15:00 it was like squeezing a dry lemon ... by 28:00 “please stop he is already dead 😭

nekad200020 days ago^{+2}Numberphile fails at math but spectacularly succeeds a trolling

Nicolas Bertozzo20 days agoOk, but, what about his tshirt

Gershon Aizikowich20 days agoAs per your recommendation I bought the book of Konrad Knopp, Theory and applications of infinite series, Dover books, 1990 (Kindle edition)

The first paragraph (The SYSTEM of rational numbers) is purely horrible. Errors! Total nonsense!

I hope that the rest of the book will be better. (

Mathologer20 days ago@Gershon Aizikowich Chatty trash? Well let's see: six editions of the German book with Springer, the last one in 1996, translations into other languages, several editions in those languages, etc. Lots of people, me included, think this is a GREAT book which contains lots of amazing details and results that you won't find anywhere else. Having said that I think most of those Knopp fans would agree that the first couple of chapters dealing with the foundations are outdated and in these chapters the wordiness DOES get in the way of readability :)

Gershon Aizikowich20 days ago@Mathologer It is possible that errors are result of transferring text to Kindle edition, but I omitted the Part I and started to read Part II... It is chatty trash anyway and it couldn't be the result of the same process or even language translation.

The book is haplessly outdated. I am not sure, though, that it was good for even its time.

Sorry, if I have hurt your feelings. It's just my opinion.

The video is very good. Thank you for that.

Mathologer20 days agoActually I did NOT recommend the Kindle edition. In fact, I would never recommend the Kindle edition of any mathematical text :)

Kaan Onay21 day agoThe result is correct. -1/12 is proportional number.

Zero, infinity and indefinable is also a number type.

Whole proportion is already known as infinity. The result is 1/12 proportion of minus infinity and its total -12/12.

But there are another thing reality is relative. When relativity is considered then infinity must be considered as relative fact it means it is not infinite.

There are mathematical facts being used in math. Other than numbers there are facts as zero, infinity and indefinable.

Numbers are apart by reality of zero because each number or thing or particle in universe is separated with emptiness. Numbers are also on the base of zero. There is one single zero and many numbers. Modern math has different zero types such as imaginary numbers but in classical math zero must be known as unified base and mentioned not as number. Zero is a base for numbers.

Why I’m going in to this subject because infinity is not infinite. Infinity is ending by indefinable and it has proportional structure.

Numbers and zero are together bur after last number there is zero then infinity getting started. This part up to zero is relative. Then relativity is infinity, which has dual structure. It is infinite but it is limited such as circle. Circle looks infinite cycle but it is limited object. The circle is base geometrical shape and its relativity is not a surprise.

These are explanations of relativity, being two different things at the same time.

Relative reality has layer and each layer has its own individual relativity. Such as numbers and zeros are relativity and numbers are relative to each other so on. On the infinity part, infinity has Its limit because after limited infinity indefinable must be placed.

Indefinable is the end of infinity.

From this structure we can understand -1/12 has a meaning. if we distinguish the layers and understand the layers in their own structure…

By the way infinity is relative by its limited structure and the continuity of relativity leads us to the understanding of indefinable must be singular and it singular structure must be the source of relativity which makes it relative but singular at the same time. Actually it has another explanation but this is enough for -1/12

Abhishek Singh21 day ago^{+4}Sounds like world war III is about to begin..

Error 40422 days agoAssumption:

1+1+1+1+1+1+1+1+1+1+1+1+1...= infinity

Then

1+2+3+4+5+...=(infinity^2)/2

Frank Ansari18 days ago1+1+1+1+1+1+1+1+1+1+1+1+1+... = zeta(0) = - 1/2

Etinymous22 days ago^{+1}Imagine this dude seing Einstein work when he just finished inventing is equation E=mc^2 :

Dis anser wont get yu ful maks

Alex Zheng22 days agoComments on math and educational videos are arguably one of the most heated and expressive. These are the hottest comment wars

Alex Zheng22 days agoDoesnt -1/2 =-1/2,the same as -2/4,-3/6, and -8/16?

Kueist22 days ago^{+7}"If you've made it this far you know..." I stopped knowing at the 10 minute mark

Zubungo The Best23 days agoNumberphile explains that the sum of an infinite series in Fisics can't result in infinite

Jake Y23 days ago^{+3}I appreciate this video, but they did disclaimer it in the numberphile video as only applying to the math of the physical and observable universe. They contextualized the type of series math they were doing as being needed for physics which doesn't have access to real infinities.

Dan Kelly9 days agoThe simple and crucial point is that Numberphile did not use the = sign as is 'normally' defined and as a result they in effect click baited us. 1+2+3... is not equal to -1/12. That series only works when = is being used defined in a different way.

Mars Truth23 days agoWhat a great presentation! And ya gotta love a mathematician with a sense of humor.

Rhomib Joaar23 days agoWhat a legend

Roronoa Stark23 days agoYou're wrong dude

Rhomib Joaar23 days agoSum how haha

Simon Fischer24 days agoI think you're doing Numberphile as well as their viewers very wrong if you assume there was anything to "debunk" in the first place. I don't think there are many of the 6 million viewers who were really tricked into believing that this sum converges. Numberphile just didn't bother to state the obvious because they know their viewers aren't idiots. They did not neglect the fact that their video was inaccurate and also have another video with more details.

Most importantly, there are things in Numberphile's video that are 100% missing from this one: passion, love, and humor.

The "Numberphile guys" have names btw.

Fonn The Human25 days ago^{+58}Y'all so focused on James vs Tati vs Jeffrey while this right here is some high quality tea

Matthew Boyea9 days agoThats a quality evaluation, Fonn the Human

Collectio25 days agomuggle

Penny Lane26 days ago^{+4}"Now I'm gonna kill myself." Then goes on to talk about the "next video." Now the zombie shirt that suddenly appears at 5:30 becomes ominous foreshadowing!

GMP Studios27 days ago^{+1}I love how the other guy yells "or less" when this guy says "0 marks".

GMP Studios27 days ago^{+1}5:49, in my institute they give -1 marks for getting answers wrong.

Chris Tose27 days ago^{+3}ye right off the bat S is oscillating between 0 and 1 it isn't 1/2

Anuj Chitale8 days agoAnd yet Quantum mechanics explains things better by considering the average of something that oscillates between 2 values!

#LegendInBusiness #00728 days ago^{+11}while watching this, i just remember the shakespear’s ANTONY SPEECH where Antony blames but praises brutus to stay away from a quarrel

here mathologer says the guys at numberphile are smart just like Antony says brutus is a honorable man

#LegendInBusiness #00728 days agowhile watching this, i just remember the shakespear’s ANTONY SPEECH where Antony blames but also gives a gist of honor

here mathologer says the guys at numberphile are smart just like Antony says brutus is a honorable man

Avery28 days ago^{+1}A quiet Indian genius vs. a TheXvidr with their own background diss crowd XDD

Mathologer28 days ago^{+1}vs. ???? you did not watch the video, right? thexvid.com/video/YuIIjLr6vUA/video.html

eduardo soto rojasMonth agoSo, why the answer is not -1/12?

Mathologer: Yes

Petr BarbořákMonth ago^{+1}He gave a clear anwser - it cant have an anwser

Sonny De SmitMonth agoare you doofenshmirtz's uncle?

robin malikMonth ago^{+2}Second shirt is all about supersum!

shourya ranjanMonth ago^{+1}Either he is dumb or all the other mathematicians are dumb

MuffinsAPlentyMonth ago^{+2}Or there's a third option, which is that you don't know what "all other" mathematicians say about the topic.

Two BrothersMonth ago^{+1}Numberphile's aren't fool they are intelligent

MuffinsAPlentyMonth ago^{+1}Yes, and Mathologer mentions this at the beginning of the video. They are smart people! It's just that, in trying to make a short video that is easily accessible to a general audience, things got really messed up.

Kloseven RoyaleMonth ago^{+1}*It’s so easy when someone else is doing it* :(

savcobMonth agoSolve this:

$10 x $10 = $100

but $10= 1000 cents thus

1000 cents x 1000 cents = 1000000 cents which equals to $10000

The Football FreakMonth ago^{+1}$10 * $10 = 100 $^2

1000 cents * 1000 cents = 1000000 cents^2 = 1000000/10000 $^2 = 100 $^2

Lasse Liang HansenMonth ago^{+3}something key for me, learning maths, which i rarely hear emphasised outside the math-world, is the clarification of existence and uniqueness.

for the series, Numberphile determine uniqueness, but fail to prove existence (in this case convergent)

NobyMonth agoHmmm... this reminds me of ppl denying the square root of negative 1. Maybe they are ahead of our time? And maybe -1/12 is an answer?

MathologerMonth agoHmm, as far as maths is concerned the VERY interesting and meaningful connection between 1+2+3+... and -1/12 is over one hundred years old ! What's important here is to be very clear that this connection is NOT an equality :)

Prasanna TMonth agoThe last minute of the video where you find the area under the graph intrigued me a lot. I feel like an idiot now but without any shame i want to ask you this: what is the meaning of addition? What does it really mean to add numbers?

Finn JensenMonth agothanks I protested, when it came out , and thought it was a joke 1/4 april fool but my brain got fried as I tried to understand the reasoning

Ali GatorMonth agoTo most commenters :

We have a sorry state of affairs at this comment stream. I posted an incredibly simple and as far as I can see unassailable proof why the 1+2+3+...= -1/12 is nonsense.

I have not received even a single comment agreeing or disagreeing.

What I get is supersums, Cesaro sums, evasion, prevarication and plain hostility or contempt. What's the matter with the lot of you ?

=======================================================

Here is my argument :

There is absolutely no reason to try to debunk the particular steps he used to come up with a bogus proof and an outrageous conclusion.

The bottom line is this :

1. There is no way the sums of positive values can turn into a negative quantity.

(The sum of positive values is always located further away to the right

from the origin than any of the components. )

2. There is no way that the sums of whole numbers can produce fractions.

(Whole numbers have no fractional parts to begin with.

Summing them will not create any either, since all activity takes place at

whole number intervals using integer arithmetic. )

Ali Gator27 days agoThank you for agreeing with me. Three comments down 10 million to go ( to undo the damage by the two professionals. )

P.S. I liked the video Teaching an idiot basic maths | Blackadder - BBC I thought computer programming courses at a university and algebra and geometry classes at high school. The BBC video brings back not so fond memories

MuffinsAPlenty28 days agoI shouldn't make statements about whether or not you like something, that's fair. But it's also incorrect to say that infinity does not exist in mathematics. You mention that in math, we prove or disprove. Well, how do you prove something? You need _assumptions_ in order to prove something - a starting point - something you take to be true without proof. Such statements are called _axioms._ The existence of infinity is taken as an axiom in mathematics (typically). You are welcome to work in a different version of mathematics which does not allow the existence of anything infinite (some mathematicians do that), but it boils down to which axioms you take, and the axioms one takes are a matter of preference.

You are welcome to disagree with statements in this video and on other videos on finitist grounds (rejection of anything infinite in math), but that's not what your original argument was. Your original argument was talking about how adding positive integers never results in a negative fraction. This argument is _not quite_ good enough if you allow for infinity in mathematics. If you reject infinity, then there's no point in making this argument, since "the sum of all positive integers" is a completely nonsensical concept to begin with.

But at least now I know where you stand. You reject anything infinite in math. Working under this philosophy of mathematics, then yes, I agree with your statements. :)

Михаил СибиревMonth ago@Ali Gator Have a nice day. I think you are absolutely right in your reasoning. thexvid.com/video/g4IQjUpTNVU/video.html

Михаил СибиревMonth ago@Ali Gator Sorry, I agree with You completely. Sorry for the layman's opinion plus is the action operator? In the amount of 1.2.3.4... we can drop the numbers and go to the action of the plus operator. The introduction of minus and divide operators must be justified. Attention is also drawn to the fact that there is no zero in the original sequence 1.2.3.4...

MuffinsAPlentyMonth ago"P.S. Why switch to a different sum ? Were you hoping that the result would be negative ? Even worse why switch to some convoluted 3-adic topology ? "

I don't know enough to answer that question. The p-adic number systems show up in number theory, but I don't know if they have any useful applications. I have read that some physicists tend to think that p-adics may model some quantum physics better than the real number system, but I don't know how widespread that hypothesis is or how much data there is to suggest it is true.

"I have a couple of examples : If I stated that 8 = 1000, everybody

would scream "stupid". (after 6 million YTube hits ).

But if I added the binary system qualifier then the equality is correct."

Now you're finally getting to the point of this video.

I know you don't like infinite series. That's fine. It's true that infinite series are _not_ the same as finite series. You can't "compute" the sum of an infinite series in the same way you compute the sum of a finite series, adding one term at a time (or coming up with a formula and plugging in a "last term"). Instead, mathematicians _generalize_ the notion of a finite series to allow for infinitely many addends. You're right this isn't the same as having finitely many addends. And you may think it's a mistake to refer to the generalization as a "sum", and that would be a very interesting debate to have! The issue at hand is that there is a standard meaning for the sum of an infinite series as the limit of the sequence of partial sums. So if anyone has studied anything about infinite series before, they should have an understanding of what it means to find the sum of an infinite series (using the standard definition). So whether or not you agree with the terminology used, it is the standard terminology, and most people involved in the discussions are aware of this and have accepted the terminology and moved onto discussing the actual mathematics at play.

But, right in line with your criticism of infinite series, you can have alternative definitions for the sum of an infinite series. And you can generalize even the standard definition for the sum of an infinite series. But if you're using a nonstandard definition, the onus is on you to _state what that is._ Just like how people would scream stupid if you stated "1000 = 8" without any quantifiers, the same thing applies to Numberphile when they stated that 1+2+3+4+... = -1/12. There are alternative definitions and meaningful connections between the infinite series 1+2+3+4+... and the value -1/12, but it's certainly not the standard definition for the sum of an infinite series. So that's actually one of the points Mathologer makes in this video... It's one of the points you just made. If you're going to do something nonstandard, then you have to make that clear!

But, as I already stated, there are meaningful connections between the infinite series 1+2+3+4+... and -1/12. So the rest of the video, Mathologer shows that, even under alternative definitions of infinite summation, Numberphile's method for linking the series 1+2+3+4+... and -1/12 is invalid, and then Mathologer goes on to show the actual meaningful connection between the series and the value.

So your criticism that there's no reason to say anything other than the arguments you made in this comment thread really does miss the point of the video. It's to explain that Numberphile's video got it *very* wrong (to the extent that _even if_ you are using nonstandard definitions, their explanation cannot be made correct), and then he establishes the *actual* connection.

Commander ValerMonth ago*just said that we can't do anything with infinity number* *respect yollo*

inyobillMonth agoThat's not what he said. For instance, look at his 9.999999 … = 10.0 video.

Sayan BhattacharyaMonth agoHow did you take the area under the curve and prove it at the end?! Please explain a bit.... I have to understand that..

inyobillMonth agoThe area under a curve is given by taking the integral of the function on the bounded region. If you can give me a time tag, I can give a more explicit answer.

aryamanWMonth agoBut if they use it in physics, how can it be wrong?

Cong SuMonth ago^{+1}It's because of how they use it in physics. First, physicists formulated some theory and when they used mathematics to calculate based on it, they obtain a bunch of infinities. Because there are obviously no infinite energies and such in the real-world, they have to "fix" the theory and somehow get rid of the infinite terms in their equations. This supersum is a handy tool for them to do precisely that, and allows physicists to tweak their theory to match experimental observations.

In other words, this supersum is used in theoretical physics, not observed physics. The theoretical physics can be modified in any number of ways to match the observed physics. It is more likely that those infinities exist because the theory is not good enough rather than because nature has infinities in them.

Paul ShinMonth agoYou should check out the comment written by Reinaldo García García (assuming he hasn't changed his username by now). It's a bit esoteric, but you might get the gist of what he is trying to say.

Fields and TreesMonth agoThere were no adverts on this video.

Ali GatorMonth agoThere is absolutely no reason to try to debunk the particular steps he used to come up with a bogus proof and an outrageous conclusion.

The bottom line is this :

1. There is no way the sums of positive values can turn into a negative quantity.

(The sum of positive values is always located further away to the right

from the origin than any of the components. )

2. There is no way that the sums of whole numbers can produce fractions.

(Whole numbers have no fractional parts to begin with.

Summing them will not create any either, since all activity takes place at

whole number intervals using integer arithmetic. )

jalbert425Month agoThanks math broke me for a sec