We've got ANOTHER new video about -1/12 also out today - see it at: kzfaq.info/get/bejne/mMuRns1om53Zd2g.html

I'm always blown away by Brady's ability to ask questions. He's really got a talent for it, and I feel like I'm learning more just from him being there to challenge whoever he's talking to.

Thanks for the explanation tonytonytonytonytonytonytonytonytonytony….

I love that he ran with the name analogy and explained it succintly

It's crazy that its already 10 years ago. I remember that video like it was yesterday. It was one of the first videos of this channel that I watched, and it was also one of the reasons to get me hooked to mathematics :)

"If it was a race, I would never finish" reminds me of a joke Mathematician and engineer are set on a line one meter away from a million dollars. Judge says "every minute, you are able to half the distance to money" Mathematician immedietly gives up, but the engineer takes the first step. Mathematician tells him "why do you bother? You will never be able to reach it, you can't halve to zero" Engineer answers "yes, but at some point I will be close enough for practical use"

What helps me a lot in these kind of situations is to keep in mind that “the representation of something is not the something”. In other words, both 1 and 0.9999… represent the same something which is not the symbols 1 or 0.9999…

Man. I was here for the original video in 2014 and I'm here for it now. Gave me throwbacks of being a ninth-grader, fascinated with math, binge-watching Numberphile. Good times.

I have been waiting this moment since… -1/12 year ago ❤

17:09 of *course* Euler did it. Half of maths is basically the "Simpsons did it" episode of South Park, with Euler in place of the Simpsons.

brady casually inventing zeno's paradox when asking about the convergence of the number

What a great video. The logic and clarity of Tony Feng's answers to Brady's sharp questions. It's just really fun to watch.

Random useless fact: If you look at the clock in the background it went from 10:22 AM to 11:03 AM or 41 minutes. The video is 23 minutes so 18 minutes cut footage. (probably footage we don't need to see like paper changes)

Brady has an uncanny knack of asking a simple question (say, about an infinite number of steps), which opens a door to complex problems (such as Zeno's paradox). This makes the problem more accessible to many people, who may be put off by more formal approaches. It's such a valuable way to communicate ideas!

Tony! Toni! Toné! has done it again.

Brady’s question about not crossing the finish line was a perfect moment to bring up Zeno’s paradox as a everyday example where we do have an everyday experience with infinity.

The spotlessly clean blackboard brings me right back to my undergraduate maths lecturers.

I feel really validated that he calls this playing with numbers "mathematical doodling" xD

The poetics, analogies, how Brady's asking questions we viewers might have. Love how these things never change ❤

Tony was great. Thanks for featuring him

"I broke rules when I wrote the equal sign." Love it!

your "finish line" analogy, the 0.9999 one, it makes sense as "stopping exactly at the finish line" rather than "as crossing the line".

Man!!!! More than 10 years watching your wonderful videos. You had make me love Math even more, for decades.

Best explanation yet! Tops the “golden nugget” video and actually easily explains the basics of what analytic continuation is rather than it being shrouded. First time I’ve watched one of these and not left so confused.

This made me realize that I've watched this channel for about a third of my life now. Brady's questions in this video were exceptional by the way.

LETS GOOOO ROUND -1/12!!!

You're 2014 video is one of my favourites! I have come back to it many times over the years, so I'm thrilled that it's back!

Not again 💀

This guy is really well spoken. Keep him coming back!

I had just rewatched your -1/12 videos yesterday its such a coincidence that you posted this after

I just have to get this out of the way: Numberphile is my favorite KZfaq channel and has been for more than 10 years.

This was the best explanation of analytic continuation for me who's not a mathematician. I finally understood the intuition of it! Thanks!

He raises a very valid point that is easy to get lost in all the numbers - you actually have to define what you mean by 'equals'. If you're using the sort of conventional, Peano Arithmatic way of thinking about equals, then no, it doesn't make any sense to say a diverging series 'equals' anything; additionally, you have to do a bit of extra work before you're even allowed to say that a *converging* infinite series actually 'equals' something, but you *can* get there without too much trouble. We take the idea of 'equals' for granted in maths, to the point that we don't even say it sometimes, we just say 'is': "What *is* two plus three", when really, the idea of equals is a lot more careful and detailed than that. In an unironic way, it really does depend on what the definition of 'is' is. In other things, on the 0.999... repeating decimal issue, I find the best way I've seen it explained is that there is a difference between 'numbers' an 'notation'. We have to use 'notation' to be able to write down numbers, but there's no guarantee that our notation actually means anything if we aren't careful in following the rules set up for it, many of which become 'assumed' or 'unspoken' rules over time, but nevertheless are still important. It's important to remember what your notation is actually saying when you write it down, and to make sure that using notation to say that actually makes sense. In this case, repeating decimals are a shorthand notation for a very specific mathematical process; that process, when taken to its conclusion, yields equivalence with the number 1, just as surely as saying the process (4 minus 3) yields equivalence with the number 1. There's lots of ways to write things that equal 1, so there's no reason to feel weird that 0.999... is one of them.

wow this video is infinitely better than the one 10 years ago. so glad this has come about the way it has.

I really don't feel like I understand it better than the first time around. When he says "there are ways to make it rigorous", then that way is what I would like to hear about.

Yessss. This topic (from the last video) more than any other stoked my passion for math. Thanks for positively affecting my life Numberphile

So does this explain how even though I am endlessly adding money to my bank account it still ends up with a negative value?

I was in college when that video came out. I was taking calc 2, so we were taking infinite series, and I remember one of my classmates bringing up how it was "proven" that 1+1/2+1/3+... = -1/12. My professor was too old and was like "what are you talking about?" and then dismissed him and continued with the lesson. This was 10 years ago.

whenever learning I NEED multiple perspectives to bring everything into place, so this was very helpful thanks. Tony's general explanation of analytic continuation helped cement the concept for me, previously I had a hard time discerning the generic concept of analytic continuation from the specifics of Reimann.

This is the most well spoken and easy to understand explanation of hard math i have ever seen

One way to think about 1 = 0.9999... is that 1 - 0.9999... = 0.0000..., which will have infinitely many 0s. If x - y = 0, then x = y. This is where it's helpful to remember that Decimal is just a way of representing a value as a power of 10, but the Reals are a continuous number system where a value is defined by it's relative position to another value (so infinitely small change is 0 in Decimal). All we're really say here is that it takes infinitely many Decimal digits to represent all Reals, but there are some Reals that can also be represented by a finite series of digits.

I just attended professor Feng's complex analysis section this morning! Funny to see him on numberfile.

the true final nail in the coffin of whether 0.999~ = 1 is that there is very clearly no number between 1 and 0.999~

I still think Mathologer's video about this subject is the clearest.

That infinite series of "1 - 1 + 1 - 1..." has been bugging me for 10 years, but now I think I finally get it! Great video.

Thank you for your videos Brady!! Have a great day :D

By far the best of the -1/12 shows.

The greatest comeback ever!!!!

The return of the king

It’s always nice when the title alone brings out a chuckle in you that you rarely have anymore!!! 😂

Love this guy. Makes this very approachable

Best thumbnail yet

I LOVE THESE VIDEOS! more of these please!!!!!!

This is the best explanation of this I've seen to date Very natural 🙇 👏

Amazing explanations and replies from Mr Feng, thank you!

Wonderful -1/12 10 year anniversary

Thanks for years of education and entertainment

This guy is awesome some of the best explanations so far imo

I like Tony Feng, his way of explaining doesnt feel like wizardy but like we are just playing a bit and see what happens

Omg ive been watching numerphile for over 10 years 😮😮😮😮

Thanks Tony for your great and inspiring explanation! I am not a mathematician, but I believe I understood it. Thanks. It made me think that the idea of "analytic continuation" is another great counterintuitive breakthrough of our civilization, which provided us with a "tiny" but clear understanding of this wonderful world. The other breakthroughs at this level, for me, are a) the zero (along with negative numbers we had the decimal numeric system), b) irrational numbers, c) imaginary numbers, and finally, d) the invention of Calculus as an effective language to talk with Nature!

This is what you get when your math teacher is a Mathematician. In public school most of us had, as a math teacher, a social studies or PE coach moonlighting as a math teacher. This lead us to college without the foundations needed to understand an exceptional math professor like Tony.

As a programmer, this is like my worst floating-point fears come true 😰 "These two kind of equal numbers are actually very equal"-problem can't hurt me. The "these two kind of equal numbers are actually very equal"-problem:

Zeno's paradoxes come up almost straight away with convergence of an infinite series.

tony's quite the explainer! enjoyed this, thank you

This video is MUCH better than the last one.

When they say "running a race and getting closer and closer," that's a confusion between "unbounded" with "infinite." It's one of the more important distinctions when discussing this sort of thing.

Return of the -1/12??? More like The Two Tonys :-) Lovely video both! Very interesting and so fun

Thank you for a lovely exploration of math!

It's like you're messing with the machine code of the universe, learning it's quirks, like how you can use +(base-n) as a stand in for (-n). I love it.

0.999999999999….. is equal to the sum of the geometric sequence 0.9, 0.09, 0.009, 0.00009 etc where T(n)=0.9*0.1^(n-1) Thus the first term a=0.9, ratio of the sequence=0.1 Thus, the Summation of the sequence S(n)= a(1-r^n)/(1-r) (In case you’re wondering, this is because (1-r^n)/(1-r)=1+r+r^2+r^3….+r^(n-1)) Thus, for S(infinity)=a(1-r^(infinity))/(1-r) =0.9(1-0.1^infinity)/(1-0.1) =0.9(1)/(0.9) =0.9/0.9 =1

My body is ready

Been waiting for this!

My calc prof in college always liked the little story of putting two kids in a room, one on ether side. They each move half way toward each, and halfway again, and again. The mathematician says "They never get close enough to kiss." The physicist says, "they get close enough."

In computer programming, there are many ways to represent numbers in binary. In one very common approach which allows representing negative numbers is called two's complement. In that scheme, the 32-bit signed binary integer 11111111111111111111111111111111 represents negative one. Now here, although the number of bits is finite, one still might notice a bit of vague similarity to some of the infinite cases discussed in the Numberphile videos.

22:10 I can imagine the quantum physics was kind of skimmed over because it’s incredibly complicated but having a real life connection to the zeta function seems like it would put this whole debate to an end. Would love to hear even a little more info on how this function is useful in real world situations.

This was fantastic! Thank you.

This guy was probably winning high school math olympiads when the first -1/12 video came out

Hello Brady!! Been watching your videos for a long time. I just wanted to say thank you for the awesome content that you have delivered since the past 1 decade!

9:00 it's easy for me to think that 0.999... = 1 the same way as we think 0.333... = 1/3, that is, the repeating decimals are just an artifact of the base we choose to count with.

Tony is the best! More Tony!🙏

Helllls Yeah. I love this!!!!! We back baby!!!!

Here’s a nice variant that works in basic analysis: Consider the perturbed number N(n) := n·q^n. It represents n in the sense that the limit of q to 1 of N(n) equals just n. Further consider the sum (S) from n=0 to n=k of N(n). Also consider the integral (T) from 0 to k to over N(n)·dn. Now then the limit of q to 1, of the limit of k to infinity, of the difference S-T equals -1/12.

Brady's name argument actually proves, rather than disproves, Tony's point. You can indeed use Tony in place of the infinite-letter-name, so long as the person you're referring to doesn't change. (see Shakespeare, et. al., 1597, "Independence of reference labels and olfactory receptor stimulatory effect of blossoming plants.")

Tony explains himself very well despite waving his hands at some of the rigour. Excellent video!

I like the finish line analogy. You may never arrive at the finish line, but at some point your position will be infinitely indistinguishable from being at the finish line.

One thing I took away from this video is that 1 + 2 + 3 + 4 + ... = 1 + 13 + 13^2 + 13^3 + ... , as they both resolve to -1/12. It feels like there should be interesting implications of this "equality"

I think the problem with this is purely the equals sign. Make it something else like "=>" because what we are actually doing is TRANSLATING series into something else. My intiution is this would be useful for comparing series in a more digestable way. His different langauges comparison with his name I think was the best point in this.

u are all my teachers, thanks

This is so much better than the original video on this. Proper explanation, as opposed to the handwaving previously

The thing with 0.999…=1 is that the infinite 9s is just a quirk of our positional notation syntax in decimal. Similarly in binary 0.111…=1, in octal 0.777…=1, in hexadecimal 0.FFF…=1, etc. The positional notation allows us to write a number like 0.999… and under the rules of the syntax it has to have a unique meaning (one expression cannot have multiple different meanings). That meaning actually emerges from those same rules, and when investigating the emergent meaning of the expression "0.999…" more closely it turns out that it has to mean the same mathematical object as the expression "1". Thinking that the different syntactic expressions of "1" and "0.999…" would _have_ to mean different things is just a cognitive bias (I don't know what to call that bias, or if it even has an established name). One rigorous proof of the equality of the numbers expressed with 0.999… and 1 is based on the fact that between any two real numbers there always exists another real number: if a

I can not pin down why but this video made sooo much sense to me. I don't think I have seen a justification of abstract math that made so much intuitive sense to me. The video demonstrates beautifully how math can be incredibly pedantic and rigit in the rules it works under but at the same time its this infinitely flexible tool we invented to make sense of things by extending logical relationships we can not intuitively grasp by abstracting them into these pure math constructs that don't really make any sense on their own but are incredibly powerful if we can bring back their results into the real world.

coming from the follow-up video at 13:50 one can immediately grasp why a regulating function would be neccessary (and much more important: reasonable) to get 1/2. Shows that cutting of when going to infinity can't be the naturally correct handling. Outstanding combo of videos!

I like to think of analytic continuation as "expanding the definition" of a function the same way you learned to expand the definition of operations as you learned types of numbers in school. If we can see subtraction as removing one amount from another as kids, then as we got older we see it as adding by a number of the opposite sign when we found out about integers, a seemingly incomplete function, like the Reimann zeta function under its original definition, can get its definition expanded.

Never clicked so fast

i feel like we are just repeating the same errors again, at around the 17 min mark the guy references euler to explain the shifting of terms around in which he tries to justify by saying it is still the same infinite sum. that is true if you are dealing with a series that absolutely converges and clearly the sum of natural numbers is divergent if its an infinite sum. for example the alternating harmonic series coverges to log(2) but the series doesnt absolutely converge since the sum of the absolute values of the terms is just the regular divergent harmonic series. the specific arrangement of 1-1/2+1/3-1/4.....will converge to log(2), but rearrange the terms and it will converge to a different value. although the shifting of terms the way he does, does get to -1/12 , it relies on faulty thinking and its only correct due to an anayltic contiuatiion of the riemann zeta function for when s =-1 for which btw is not defined for Re(s) = -1, hence the need for the anayltic contiuation to extend the domain in which the riemann zeta function is defined to include Re(s) = -1.

Around 9 years ago I saw the first numberphile video about the -1/12 magic and that video convinced me to do a bachelor in maths because I simply loved how math is creative. After all this time, coming back to a video about this problem gave me a lot of flashbacks

I was inspired by almost all the mathematicians and their ideas, concepts, and theories. I used to be scared of the subject called "Mathematics". I even recognized it as a Demon that will drag me down on class grades, and it did. But I got my comeback with my deep curiosity in the heart of mathematics. Gotta say Numberphile also added extra curiosity in mathematics. Long journey ahead of me with exciting mysteries!

I just thought of this. What do you get when you divide 1 by 3? 0.333333..... So logically, if you did the reverse, you should get 1 back, right? And yet, when you multiply 0.3333..... by 3, you get 0.99999...... But both of those should be the same thing according to our rules. Which is why 0.99999... must be the same as 1.

14:39 actually there is! if you have an infinite series whose partial sums are s_1, s_2, s_3, ..., whenever the partial sums converge to a limit (so that the infinite series makes sense) the sequence of running averages of the partial sums will also converge to the same limit. this allows you to define a summation (called Cesàro summation) which assigns values to more general classes of sums but agrees with the regular one wherever the regular one is defined; and Cesàro summation indeed assigns 1/2 to the sum 1 - 1 + 1 - 1 + ... so in fact, if your partial sums are 0 half the time and 1 half the time, then your sum will "equal" 1/2