The Return of -1/12 - Numberphile
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- Опубліковано 14 лют 2024
- Featuring Tony Feng from UC Berkeley. See our other new video today about -1/12 with Tony Padilla at • Does -1/12 Protect Us ...
More links & stuff in full description below ↓↓↓
See our full -1/12 playlist: • -1/12
Tony Feng's website: math.berkeley.edu/~fengt/
Note: As explained by Brady at the end of this video, Tony Feng did not know about our previous -1/12 videos when we recorded this - we decided to go ahead so you could compare his take with other ones! :)
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We've got ANOTHER new video about -1/12 also out today - see it at: ua-cam.com/video/beakj767uG4/v-deo.html
With another Tony, no less! :)
Squares of 0.999... at varying decimal precision.
0.9^2 = 0.81 trunc to original precision = 0.8
[keep original bounds of infinity / do not increase domain space of infinity length.]
0.99^2 = 0.9801 [0.98]
0.999^2 = 0.998001 [0.998]
0.9999^2 = 0.99980001 [0.9998]
0.99999^2 = 0.9999800001 [0.99998]
0.999999^2 = 0.999998000001 [0.999998]
Follow this growth to infinity , 0.999... ^2 will never = 1.000...^2
you can see 2 infinite patterns grow here, the 9s and 0s
0.999... will never function the same as 1.000....
thus no ... 0.999... is not equal to 1.000...
ending at 0.xxxx8 is due to 2 counts of missing 0.xxxx1 from 0.99999 to equate to 1.00000. at any decimal precision that 0.xxxx1 is the smallest possible portion.
p.s. 10 * infinity in an open set is incalculable as it would mean the original infinity was not infinity but 1/10th of infinity.
10 * infinity of a closed set is 10 copies of that one infinity bound by its closed set.
heh. beak.
No 0.99... is equal to 0.99.... it is never one. Those infinitesimals matter.
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.
Yeah that is probably the biggest reason numberphile becamse what it is, him _not_ being a mathematician and therefore having a better understanding of what is interesting or needs further explanation
Yes - Brady’s question on uniqueness (9:55) really drilled into a central “big idea” of the theory, and a “no, but” answer *could* have led to a lecture series on sheaves
Nothing personal but I think you saved this comment and simply pasted it here cuz you were early
@@canyoupoopand so what if they did?
When you compare and contrast Brady's questions from his much earlier Numberphile videos to now, you see how much appreciation and understanding of mathematics he has grown from this project. His questions in this video were absolutely as on point as I've ever seen them!
Thanks for the explanation tonytonytonytonytonytonytonytonytonytony….
Ha ha
coolcoolcoolcoolcoolcoolcoolcoolcoolcoolcoolcool…
@@harriehausenman8623 Ooooh, brilliant, and this can in some sense or conditions equal Jake Peralta, or in some ways or others be equivalent to Abed Nadir, right?
@@StopHammerTime226868 Infinite-Abed! 😄 (Dont know who the other person is)
@@numberphile
ha ha ha ha ha ha ha ha ha ha ha ha ha ha ha ha ha ha ha ha ha ha ha ha ....
"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"
pretty soon the loot will be within arm's reach.
The version I heard replaced "a million dollars" with whoever was the beautiful buxom actress of the time.
@@darrennew8211 yeah, I heard that too, but it's funnier with money :D
I thought the punchline was going to be "Yes but I'll always be closer than you."
yeah, but the engineer should know, if you're working with matter and you shrink the distance, your arm's length also shrinks.
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 :)
It is amazing how often we hear from people who got into mathematics - even studied mathematics - because of that old -1/12 video... Almost worth all the shouting! :)
@@numberphile i actually did study math in my bachelors :) i'm now in my last semester as a statistics student. I think this video shows exactly, why the world of mathematics is so astonishing
The shouting was just because the video lacked essential qualifiers and therefore contained LIES.
That video actually delayed me enjoying this channel for years. I told UA-cam to stop showing me numberphile.
@@tristanridley1601Don't abandon your house just because there's one cockroach there
I love that he ran with the name analogy and explained it succintly
Definitely a great talent for maths communication.
Not again 💀
☠️☠️☠️
YESSSSS
Just don't watch this channel
I was surprised he did it again after mathologer's video who debunked this big time! Also everyone should watch a proper video for analytic continuation and 3B1B is a great start. People should stop trying to understand analytic number theory in five minutes. There are some things you can't learn in a video. Read a textbook boys and girls....
Yes again
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.
It's always Euler :D
🤣 "Euler did it" just hilarious 😂
and Gauss
@@pedrosauneYes, and Gauss. I'm thinking back to the FFT episode of Veritasium.
This mf died and still his papers were being published for *47 years!*
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…
Nice touch.
The thing that helps me understand it, and which I've not heard enough people use when explaining it, is the question "What do you have to add to it to get to 1?" Which of course is 0.000... I think it's pretty intuitive that 0.000... is zero. And if adding zero to something makes it equal to one, surely it must have already been equal to one.
One more way is to think about thirds. I think everyone knows the decimal expansion of a third is 0.33333... so... What's three times a third?
I've always had a problem with that proof.
I think the reason people are confused by the fact that 0.999...=1 is that they assume that the place-value decimal system we use to represent real numbers has a unique representation for each and every number. However this assumption is false. Some numbers have more than one representation and one is an example of such a number.
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.
Yeah, and that lame-assed bogus "proof" is why I quit watching the Numberphile.
@@Skank_and_Gutterboy not just you, that video really tanked Numerphile's reputation as a whole, lots of other math youtubers and mathematicians came out slamming that video in just a few days after its release.
@@Skank_and_Gutterboy🤡🤡🤡
I have been waiting this moment since… -1/12 year ago ❤
So, you’ll start waiting for it a month from now?
@@drdca8263 Bravo!
brady casually inventing zeno's paradox when asking about the convergence of the number
I'm sure he knows about it, I think he made a video about it, but maybe he forgot
Poor zeno
I was begging Tao would recall Zeno's paradox to justify that the limit is EQUAL to the number. The arrow clearly reaches the target just as 1/2 + 1/4 + 1/8... reaches 1
"Convergence of the number" doesn't make sense. Convergence is a term for sequences or series.
@@samueldeandrade8535 Isn't the point that a number with an infinite amount of digits is a sequence? I mean every real number is equivalent to some Cauchy sequence but that is what we define to be a real number. It feels like one huge semantic game and I know mathematics definitions are all about being careful with definitions but this still seems like a contrived criticism.
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)
or early lunch 😄
@@harriehausenman8623i dont think 18 mins is quite enough for lunch and changing paper😅 you would have to be an incinerator
Wtf do they manufacture the paper how does changing paper take 18 minutes 💀
@@VivekYadav-ds8oz "footage we don't need to see LIKE paper changes", not ONLY paper changes
or bloopers
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.
Yeah I thought that Tony would mention it as a rebuke. If you get infinitely close to the finish line then you reach it.
I have to say I never quite understood what was the paradox in Zeno's paradox. It's like watching the video of someone running towards a goal but halving the replay speed every now and then. Obviously it will take an infinite time to see the person reach the goal but that changes nothing about the fact that the person indeed reached it.
@@WatchingTokyo It's more that if you do that, you basically stop time. And if you allow yourself to stop time in an exercise, it's probably clear that it's not one that works in reality.
@@jajohnek Not stopping time, slowing it down and get infinitely close to 0, hence we never see the person reach the line even though they did
@@WatchingTokyo Zeno's Paradox is asking how something that we know completes in finite time could possibly happen in an infinite number of steps. Any motion can be broken into an infinite number of steps, and each step takes time. No matter how small a time a step takes, an infinite number of such steps clearly takes infinite time. So how can motion take finite time?
Aristotle had the right of it, though he expressed himself in a confusing way. Time and space are not made of atoms (well the other answer is they both must be), such that there is an amount of time a smallest step takes. Whatever amount of time you pick, an infinite number of steps finish in a shorter amount of it. That's what being a continuum means.
Tony! Toni! Toné! has done it again.
Always givin me the blues
Tony did it right, unlike the original video.
Tony factorial?
I feel really validated that he calls this playing with numbers "mathematical doodling" xD
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.
"I broke rules when I wrote the equal sign." Love it!
Why did you love that? It is actually a s1lly perspective about Math.
He did not break any rules when he shifted the numbers.The empty spots are just 0#
@@samueldeandrade8535Silly is not a swear word
@@themathhatter5290 ok.
I prefer not to be taught false nonsense.
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!
your "finish line" analogy, the 0.9999 one, it makes sense as "stopping exactly at the finish line" rather than "as crossing the line".
Yeah I was thinking "if you have the value 'one exactly' that doesn't cross the line either"
The spotlessly clean blackboard brings me right back to my undergraduate maths lecturers.
LETS GOOOO ROUND -1/12!!!
Is there even a generalization of the triple factorial beyond the natural numbers?
@YellowBunny I believe you can
It'll never stop! 😆
That's quite a big number, or a small one, ain't it?
@@JustAnotherCommenter Is "both" an allowed answer 😄
The poetics, analogies, how Brady's asking questions we viewers might have. Love how these things never change ❤
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.
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.
Tony was great. Thanks for featuring him
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!
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.
The err is when they said 10 years ago, s = some infinite series. Therefore everything after is wrong. They should have used the infinity sign. Then everyone would see the error. However, when the series doesn't add to to infinity, you can use variables like x or s.
Think about + and = as specific procedures that have specific requirements and properties. They come up with ways to amend these procedures such that requirements are relaxed but some properties do not hold anymore. It's not the same equality and not the same sum.
The rigorous stuff is that -1/12 is the result of the analytic continuation of the zeta function. Analytic continuation is a technique to expand a function's domain to the entire complex plane, and you can lose some of its original meaning in the process. This means that zeta(-1) = -1/12 is only possible with said continuation.
There are other famous analytic continuations such as the gamma function, which is the continuation of the factorials. Factorials are defined only with positive whole numbers, but gamma is also defined everywhere else. For example, gamma(4) = (4-1)! = 6.
@@njgskgkensidukukibnalt7372 No that's in fact not how we make this rigorous. That's how we show that 1+2+3+... makes no sense. To get the -1/12 you have to do a funny roundtrip through complex analysis. It's not a classical sum in any way and instead uses generalized summation techniques that maintain some properties from the classical one - but they are not what we usually understand as summation (seriously: look up how ramanujan summation works. It's bonkers). It's kinda like taking the cauchy principal value of classically divergent integrals: yes you get a value, but it's not the actual value of the integral.
It’s really not that hard to understand. You have some function that’s only valid for certain inputs. You have _another_ function which gives the same outputs for those inputs, and also works for other inputs. That second function is a continuation of the first one. It’s not right to call them equal or say they’re the same function.
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.
Yes, exactly. Indeed, I'd prefer not to use equality at all without specifying a particular structure or process.
Specifically, what we mean when we say an infinite sum is equal to a value is that the limit of sequence of partial sums has that value. Don't abuse that notation and use it a second incompatible way.
There isn't any problem with just saying the unique analytic continuation of this series is equal to such and such. Now you've been perfectly clear about what you mean.
@@petergerdes1094 I see your point, but isn't that exactly what the three dots at the end of the expression "0.9+.0.9+0.009+...=” conveys? It explicitly means "continue the sum to its infinite limit".
@@petergerdes1094 by Euler, you people are completely ins@ne.
I just have to get this out of the way: Numberphile is my favorite UA-cam channel and has been for more than 10 years.
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.
you can't set an infinite series = s, as it's not a variable. it's infinity
1+2+3+4, not 1+1/2+1/3+1/4
I was also in calculus 2 in college at the time lol
Your professor was correct. That original Numberphile video was educational malpractice.
Tony Feng explained correctly and also the people on the comments. We cannot say x = any infinite series. We only can say that x = A CONVERGENT infinite series.
wow this video is infinitely better than the one 10 years ago. so glad this has come about the way it has.
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.
whoa, i love this
The best explanation I've heard for 0.99999... = 1 is that infinite decimals don't do a great job representing values. 1/3 + 1/3 + 1/3 is obviously 1, but 0.3333... + 0.3333... + 0.3333... = 0.9999..., which doesn't look like they add up to 1, even though intuitively we know they do.
Represented in fractions, the sum of the numbers is obvious. It's only because infinite decimals are difficult to grasp that 0.9999... = 1 seems strange.
I had just rewatched your -1/12 videos yesterday its such a coincidence that you posted this after
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.
I just attended professor Feng's complex analysis section this morning! Funny to see him on numberfile.
This was the best explanation of analytic continuation for me who's not a mathematician. I finally understood the intuition of it! Thanks!
This guy is really well spoken. Keep him coming back!
The return of the king
The Desolation of Riemann
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.
It is 0.5((0 + 1)/2).
If you think of it as a digital signal converted to an analog signal that in reality has to be bandwidth limited, then if you would measure the signal with an oscilliscope you would see the signal moving from 1 to 0 and then 0 to 1 units and so on. 0.5 units would make sense as an average value and that would be the measured value on a multimeter perhaps depending on the waveform.
Outside of reality you could theoretical have unlimited bandwidth where the signal would only be 1 or 0 and never a value in between. Assigning the value of 0.5 just seems to be the wrong answer and theoretical oscilliscope with unlimited bandwidth would only measure 1 or 0 and never 0.5. The multimeter average would still be 0.5 I suppose.
Interesting to think about to me anyway. Sorry to derail your comment thread with something outside of mathematics.
IMO; It’s zero if you extrapolate up to infinity in the term length. All of the negatives and positives cancel.
@@vodkacannon But with that answer, you could say 1 - (1 - 1 + 1...) and have 1 - 0 = 1
But the problem is that 1 - (1 - 1 + 1...) is equal to 1 - 1 + 1 - 1... so you end up saying that 0 = 1.
Having the series equal 0.5 solves this issue.
Yessss. This topic (from the last video) more than any other stoked my passion for math. Thanks for positively affecting my life Numberphile
Thank you for your videos Brady!! Have a great day :D
I LOVE THESE VIDEOS! more of these please!!!!!!
Return of the -1/12???
More like The Two Tonys
:-)
Lovely video both! Very interesting and so fun
Love this guy. Makes this very approachable
This is the most well spoken and easy to understand explanation of hard math i have ever seen
The greatest comeback ever!!!!
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.
What's more, this is the expression 1+2+2^2+2^3+..., which, by Tony's formula, "is" -1 ...
Welcome in the 2-adic numbers world.
This is the best explanation of this I've seen to date
Very natural 🙇 👏
Amazing explanations and replies from Mr Feng, thank you!
Thanks for years of education and entertainment
Thank you for sticking with us.
Zeno's paradoxes come up almost straight away with convergence of an infinite series.
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!
Been waiting for this!
Best thumbnail yet
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:
"Hold my zeroes" 😆
Pov: 00000000 vs 80000000
We don't deal with infinities in everyday life, and for most people, that's a relief, but if you're a programmer it's a pretty big darn nuisance.
Thank you for a lovely exploration of math!
tony's quite the explainer! enjoyed this, thank you
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.
Wonderful -1/12 10 year anniversary
This guy is awesome some of the best explanations so far imo
Tony is the best! More Tony!🙏
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!
Omg ive been watching numerphile for over 10 years 😮😮😮😮
Thank you.
It’s always nice when the title alone brings out a chuckle in you that you rarely have anymore!!! 😂
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.
I like Tony Feng, his way of explaining doesnt feel like wizardy but like we are just playing a bit and see what happens
I still think Mathologer's video about this subject is the clearest.
YES!! I remember that. Indeed settled it.
3 Reasons this is absolute garbage.
1: There is NO way to sum ANY number of integers to get anything other than an integer. You can add billions to trillions to quadrillions and still get an integer as a result (assuming you're adding integers to begin with). This is a fundamental rule of how integers work.
2: Summing ANY number of positive numbers together will ALWAYS result in a positive (no negative numbers are added).
3: The infinite series 1/2 + 1/4 + 1/8 + 1/16.... is equal to 1. It converges to 1. Simply "linking" this up to integers shows that they must be larger. For example, the first four terms of the integer sequence (1+2+3+4) is larger than the first 4 terms of the infinite series (1/2 + 1/4 + 1/8 + 1/16).
Therefore, it is obvious that the series 1+2+3+4...... is larger than 1.
@@jazzabighits4473 3 reasons YOU might be the absolute garbage
1: calling other people having fun "absolute garbage"
2: not letting people be artistic with math and break the rules
3: talking like your set of axioms is the only one there
And don't hate on me, don't insult me, because I only said this about you because you're a hater and so I can be a hater too 😔
There's one by 3blue1brown as well
@@jazzabighits4473 Really it comes down to that the value of the Riemann zeta function at -1 is -1/12. But if we look at the valid form of the Riemann zeta function when input x is greater than 1 (gonna ignore complex numbers for now) looks like the summation of 1 + 1^-x + 2^-x + ..., and if we look at the expression (not the value) of that summation at -1, it'd look like 1 + 2 + 3 + ... .
So while the summation 1 + 2 + 3 + ... does not equal -1/12 because that summation expression isn't valid at -1 to begin with, there is clearly some special and non-arbitrary relation between 1 + 2 + 3 + ... and -1/12. You just can't call that relation "equals" as how "equals" is defined in everyday mathematics.
The special relation does serve practical purposes, though. I think some areas of quantum mechanics observe that relation crop up in what is observed.
So, how I've heard it explained is that, there's 2 parts to infinite sums: there's seeing if it converges, and seeing what it converges to. The second step can be completed without the first one, it just won't provide the answer you expect. As I've heard, in the case of a convergent series, we see what it converges towards. In the case of a divergent series, we see what it diverges away from, and in the case of an oscillating series, we see what it oscillates around.
But I also like how he mentions that just because 1/2 was right in the middle, that wasn't guaranteed to be the right answer because of that alone. It shows that our intuition isn't always guaranteed to give the right answer.
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.
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.
What happens is the people in Quantum Phisics are using The Rimmand Z- function whithout knowing the are using the Riemman Z- Function. I mean they are using the analytical continuation of p-series with p = -1.
i believe the solution has implications on encryption mechanisms.
A connection between the Riemman zeta function and quantum mechanics does not settle this debate. Physics often motivates the development of mathematics, but mathematics is not beholden to physics. Math is abstract, based on logic. Results in physics don't deductively prove anything in math, they provide inductive evidence of the workings of our universe, which we use math to describe.
Basically, mathematics reveals that by shifting your framework, it is sometimes possible to make some sense of nonsense.
For example, by extending the exponent rules, we can make sense of non-natural-number exponents where under the first introduction of exponents as repeated multiplication, an exponent of, say, -0.5 is nonsense (okay, so, exactly how many times are we multiplying the number by itself with an exponent of -0.5?)
In physics, we often use mathematics to model reality.
When we see mathematical nonsense in the models in physics, sometimes shifting the mathematical framework it was constructed from refreshes the model to work better with reality.
In this case, the mathematical model of quantum physics that leads to the infinite sum seems to fit reality worse than a similar model that instead leads to the use of the riemann zeta function, as the second model makes predictions that fit observations where the first model cannot.
It's like having units of s^(-0.5) - it doesn't seem to make physical sense in the original definition (try explaining in plain English what s^(-0.5) "means" in seconds); yet, a constant with that in the units can be useful in a mathematical model of the behaviour of ideal pendulums.
One *could* argue then that the new model "better reflects reality" or "is a deeper understanding of reality" but that kind of interpretation is more of a philosophical debate.
@@thes7274473 Well, yeah, but one thing bothers me. Mathematicians are themselves physical objects operating according to physical rules, Mathematics is done my mathematicians, so I conclude Mathematics is indeed beholden to physics. Even if physicists don't know all the rules.
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!
The name analogy is a proper insight. 0,9- converges because it can name (describe) univocally a number, now 10+100+... means different numbers 10, 110, 1110....
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.")
That's only a theoretical paper. It failed to prove the theorem.
Excellent referencing
There is no 'et al'. Shakespeare, W was the sole author.
Here's your medal, now see yourself out🥇
It should be mentioned that although the core thesis of that treatise remains intact, several of the outlying corollaries were disproven by Law, R., Gower, P., 2001, "Effects of introduction of a tertiary anthropomorphic variable on electrochemical interactions post-exposure to _R. cadava."_
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
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.
"That's maybe a bit more philosophical"
He's got you there, Brady
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.
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.
What number could you add to 0.999.... to get 1?
Well it'd be 0.000... forever.
It's the same as 1/2 + 1/4 + 1/8 etc being 1. Literally, because that's the same thing in base 2.
I think it might be easier to explain in the opposite direction, honestly.
Start with a circle. Chop out 9/10ths of it. Chop out 9/10ths of that. On and on forever. There aren't any points on the circle that you won't chop away at some distant point in time, so with infinite chops, you've got the whole thing.
@@Nebukanezzer I really like the parallel with 0.11111... in base 2, very intuitive if you already know that 1/2 + 1/4 + 1/8 + ... = 1
10x - x and similar expressions are equal to ∞ - ∞, which is undefined, or indeterminate at best.
@@MrConverseBut with 0.99999.... x is not infinite so 10x-x is valid.
This was fantastic! Thank you.
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!
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.
Thank you! I knew I had seen some easier (not requiring complex analysis) way of getting this value, using a difference of a sum and an integral, but I couldn’t remember the details
@@drdca8263 I have a video on it on my channel, from 3 years ago or so.
@@NikolajKuntner Cool, I think I’ll take a look, thanks
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.
but he is clarifying that it doesn't justify the answer without the continuation, which isn't present in the older videos, the key difference.
Helllls Yeah. I love this!!!!! We back baby!!!!
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.
So does this explain how even though I am endlessly adding money to my bank account it still ends up with a negative value?
My body is ready
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."
u are all my teachers, thanks
Never clicked so fast
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"
13 is the luckiest number
Which also means that the sum of all positive integers minus the sum of all powers of 13 equals zero.
It must mean that x = 13^x holds for all positive integer values of x!
The sum of all real numbers equals the sum of all the powers of 13. This isn't really a shocking revelation. Both are infinitely large.
But I should note that 1+2+3... doesn't actually equal -1/12. I believe it's called the Ramanujan sum, which is different (it would be like saying that 1+1 = 10 when referring to binary, but then applying that to base 10)
"There aren't enough small numbers to meet the many demands made of them." ~ Richard K. Guy (1988)
Yes! More! We need more, much more!!!
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 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.
Did these guys ever make a reponse video to Mathologer's diss track ?
Not that I am aware of, but of memory serves, "Scouse Tony" once mentioned -1/12 and forcefully/ facetiously added "there's nothing controversial about that, right?" or words to that effect.
@@adamnealis I see. It was probably best to just ignore it then.
One thing that I wish schools taught about mathematics is the flexibility and creativity of it. It is highly rigorous and grounded in logic yes but the logic itself can be pretty much anything you care to dream of, the only important thing really is that it’s internally consistent and you justify what you are saying. This is actually the exact opposite of the image so many of us grow up with about math. So many of us learn 1 + 1 = 2 because of some unspoken fundamental property of the universe and we as teachers are here to tell you to just know that’s the right answer. In reality it’s closer to 1 + 1 = 2 because we have chosen to define it that way and it has lots of useful properties. But hey if you want to try to say 1 + 1 = 10 that might be something you can do if you know how to define it logically and are consistent and it might lead to extremely useful math! Again, creative!
This is so much better than the original video on this. Proper explanation, as opposed to the handwaving previously
That reminds me the Veritasium video about infinity, with the one with the infinite hotel rooms example: there are infinite suite that you can “counted” and some that cannot be “counted”.
Yeah, loved that one. Infinity is just wierd. "countable" infinity?! "uncountable" infinity? Seems theres a whole set of separate rules when it comes to the concept of 'inifinity'.
This is called the hotel of Hilbert, not the hotel of Veritasium 😉
Not sure if this was the same video, but someone asked how you would get back from moving the last patron from the last room.
@@theevermindYou don't you send instructions with the guess. For example guess at room 1 will tell the guess of room 2 what to do. This instruction must be very clear and finite for example "If you are a guess in room X then move to room 2X and tell the guess in room X+1 the same instruction along the way.
@@mikefochtman7164The difference between countable and uncountable infinities only started making sense to me after seeing how rationals differ from irrationals. Like say you're tasked with writing all of each type of number between 0 and 1.
Rational numbers have methods which can write all of them if given an infinite number of steps. Irrationals don't have that; even given _infinite time,_ you can't even begin to make a dent.
15:00 for 1-1+1-1+1-... the extension makes intuitive sense. Of course it should be halfway between them. But for 1+10+100+1000+..., -1/9 is just bananas. Is this a case where I should check my intuition at the door and just keep going forward? Or is there some difference between why it "makes sense" for one but not the other?
Does it have to do with convergence? Or maybe that instead of "equals" we should think of it as "functions just like"? That is, 1+2+3+4+... functions just like -1/12 when you manipulate it?
1-1+1-1… is definitely NOT equal to 1/2. It makes intuitive sense, but our intuition is often wrong. The sum simply oscillates between 1 and 0, so the infinite sum just doesn’t exist.
So what does the equals sign mean? Is it some hand wavy thing that means "the analytic continuation of this equals"?
@@LetsGetIntoItMediaFor infinite series, the equals sign means something very specific.
If, as you add more terms, the sum of the sequence can get arbitrarily close to some number, the infinite series is equal to that number.
For example, if I take 1 + 1/2 + 1/4 …, if you pick any arbitrarily small distance from the number 2, I can find some number of terms of the sequence that sum to a number within that distance from 2. If instead we try checking 3, the sum does get closer to 3 as you add more terms, but it will never get within, for example, 0.5 away from 3.
If this was confusing I can explain again.
@@insertcreativenamehere492I personally took a long time to grasp this concept of “arbitrarily close” so I will put here what made it finally click for posterity. The sequence 0.9, 0.99, 0.999 etc. is *never* bigger than 1. However, *no matter what number you pick* that is smaller than 1, you can go far enough along the sequence so that the sequence is closer to 1 than that number you picked. Thus, 0.999… is closer to 1 than *any other* real number, and because it itself is a real number, we conclude that it in fact is 1.
This also uses what’s called the Archimedean property, where you can always find a number between 2 numbers, for instance by taking the larger one and subtracting 1/n from it with a large enough n. Since we can always find a 3rd real number between any 2 real numbers, if 2 real numbers are closer together than *any* real number, then they must be equal.