What does it feel like to invent math?
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- čas přidán 12. 08. 2015
- An exploration of infinite sums, from convergent to divergent, including a brief introduction to the 2-adic metric, all themed on that cycle between discovery and invention in math.
Home page: www.3blue1brown.com/
Music: Legions (Reverie) by Zoe Keating
Thanks to these viewers for their contributions to translations
Italian: Marco Fantozzi
Thai: @korakot, own doggoV●ᴥ●V
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3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with CZcams, if you want to stay posted about new videos, subscribe, and click the bell to receive notifications (if you're into that).
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Whenever I invent math, my teacher marks it wrong.
You Too!!!
Lol!
Yeah! I wrote 5+5=15. She marked it wrong. I said that that it is in another base system...
@@user-lp1md4de7m 😅😶🤦🏻♂️
@@user-lp1md4de7m
What is a bijective base?
"You are a mathematician [...], so you don't let the fact that something is nonsensical stop you" A true mathematician spirit
Rubbish. Something counterintuitive yes, not something nonsensical.
RaniaIsAwesome That means nothing unless you categorize nonsense and distinguish it from counterintuitive results in a rigorous manner. That is precisely the problem. People think the result in this video is nonsensical when in reality it is counterintuitive
The three dots are where philosophers came in and driven mathematician crazy to dead
@ki kus Math fight!!
Yeah right,lose yourself into catchy words
Warning - This background music with his voice can lead to a state of mind where you can invent anything. Thank you 3b1b for this high-quality introspection of math.
My silly hobby is to recommend science-channel
to my fellow science-fans.
Mind?
It's true, I invented a perpetual motion machine last night while listening to this. I'm working on faster than light communication now.
@@JuniperHatesTwitterlikeHandles lol
Geometric r=2> or = to 1 therefore diverges to +inf
@@nav6496 Sorry you didn't laugh, let me try again:
It's true, I invented a toaster last night while watching this. I'm working on perfectly reasonable slower that light communication now.
I hope this joke was realistic enough for you.
As a programmer the 2^n example is easy to answer: the infinite-precision integer storing whole numbers overflowed into the negatives
yeah
The sum of 2^n written out in binary form is (11111111...) which is the twos-complement version of the number -1
@@damonpalovaara4211 exactly!
And 9 in binary is 1001 your index and small finger which makes a set of horns 🎸🤘
This needs more likes 😂
Even the universe has integer overflow :o
gold comment :D
But with an infinite number of bits, this overflow can never occur!
Yet it never crashes - or softlocks.
Walter Comunello black hole = garbage collection of the universe
that's why
Nice
Love this video. I keep getting asked by (numberphiled) students why 1+2+3+... = -1/12 and I usually end up telling them about analytic continuation, etc. From now on I'll also refer them to your video to expand their minds in a different direction :)
+Mathologer Can you think of a way to explain 1+2+3+... = -1/12 in the context of p-adics? You would have to use all of the p-adics, meaning using the coarsest topology over the rationals such that all open sets in all p-adics are open sets in your topology. One way to go would be to say that after each prime tells you that 1+p+p^2+... = 1/(1-p), we can factor 1+2+3+4+... as 1/((1-2)(1-3)(1-5)(1-7)...), hence maybe there's a way to think about why (1-2)(1-3)(1-5)...=-12. This translates to the fact that the sum of all positive integers, when weighted by the mobius mu function evaluated on them, is -12, but I cannot think of a nice way under a p-adic light to think about why that is true.
In the first instance I wasn't thinking of trying to give a p-adic interpretation of 1+2+3+... = -1/12. When explaining to students in what sense 1+2+3+... equals -1/12 I think it is best to talk about analytic continuation and the sort of things that Ramanujan & Co. were trying to capture by writing down this identity. I'd then refer those among the kids who can handle this sort of material to your video for yet another way in which these sort of paradoxical identities can arise naturally. Having said that it would be great if one could come up with a nice way of explaining 1+2+3+... = -1/12 in the context of p-adic numbers.
+Mathologer Hey Its Mathologer! You guys should collaborate on videos.
+Mathologer I am happy you finally cleared this in your video!
Mathologer yeah thank you for correcting numberphile because they used a lot of illegal math in an attempt to simplify a problem in order to make it easier to understand. But their video was just misleading.
I remember when I was trying to solve a problem for a while, and had an epiphany when I was trying to fall asleep one night. I started writing down some ideas until I came to a conclusion about the problem. Not a full solution, but a big step. Later on, I found a paper published in 2008 about the problem, and halfway through the paper they used the same process I did. So I can say that it did feel awesome to come up with that in my own 😊
Do you remember the problem? What was the paper called? Can you link it?
I remember when I was trying to think of a way to remember the perfect squares when I realized that the next perfect square is the previous one plus the next 0dd number, like 1+3=4, 4+5=9, 9+7=16, I know the perfect squares normally up to 144 which is 12*12, Using the rule zi found out, I could either go 13*13 or if I don’t want do do multiplication, I could go 144+25 and get the same result, 169, I’m fairly sure someone else found this out before but I don’t know how to find out what this is called official;y
@@daniellewilson8527 search up Galileo’s odd number rule. Also another fun fact: if you take the difference of those odd numbers (2), then divide by 2, you will get the A value (the coefficient of x^2 in a quadratic equation). For example, 3x^2 + 5x + 7 will give 7, 15, 29, 49. The difference between those is 8, 14, 20, and the difference between those is 6. Divide 6/2 and you get 3, the coefficient of x^2.
@@daniellewilson8527 It was about finding the shortest way to connect n points together. The paper is “Shortest Road Network Connecting Cities” by Université de Genève
@@hike8932 go ahead… we’re listening
literally a few hours ago i forgot the formula to find the infinite sum of a converging series for a precalculus test, but it was the last question and i still had 40 minutes left, so i basically reinvented the formula exactly like this and got the right answer. this is what growing up on 3b1b does to you.
This is not really correct... Being able to deduce the sum of a converging series is quite hard, way harder than proving that the series converges, it is possible in very special cases using a formula. I think you are referring to a geometric progression but if you did find a general formula for any series in under 40 minutes you are a prodigy and you should publish it!
Dang........
I feel dumb
@@leonardoabate2799qell a question in precalc probably means it was a geometric progression
@@leonardoabate2799 A geometric series is still a series.... And finding a formula for the sum of any converging geometric series definitely doesn't make you a prodigy, but it is still much more than most precalculus students can do
Idk if English isn't your first language but you need to work on your reading comprehension
@@MakeMakeMake245 I meant to say that the way it was written is ambigous.. you cannot find a formula for any converging series, and surely not in 40 minutes. I was trying to be sarcastic btw.
Yeah im italian i try my best with english as you can see, being rude to a random guy on the internet doesnt make you smart
How it feels to invent math
5 math, stimulate your senses
*simulate your equations
Math is the base of science do... being a expert mathematician and scientist means understanding everything.
Not saying I'm smart but that's what it feels like
Aidan Woodward
1 - It’s a 5 Gum joke
2 - No, that’s not even remotely close to what “expert” in these fields means.
The concept that it is physically impossible for a human brain to “understand everything” should be enough. The notion that an expert in mathematics and/or science “understands everything” stems from a misunderstanding somewhere, be it of the field or one’s own competence (that is, arrogance). To think that mathematicians have zero nagging questions and zero new ideas to explore is nonsensical and doesn’t align with reality. To think that scientists excel on their own is likewise.
@@limepop340 well no doy.
"You decide to humour the universe, ...", maybe the best phrase describing theoretical research.
And the best approach to life in general.
I love how you can tell how good grant has gotten with his videos. The voice over, the designs and what not... kudos to you!
Lies again? Support Indonesia Malaysia
@@NazriBwho cares.
Ever since I took calc II, I basically treated “approaching” and “equalling” as the same thing. It’s honestly made things seem less ridiculous. For example, I essentially treat 0 and infinity as reciprocals because of how y = 1/x looks on a graph. It doesn’t entirely work because the limits don’t technically exist, but it still makes the universe seem less ridiculous.
Sometimes it's useful to consider infinity as "arbitrarily [large/far]" and equality as "indistinguishability." For example .99999... doesn't equal 1 (to the eye) but it IS indistinguishable from 1. There is no meaningful method by which .9... can be separated from 1, so we claim they are the same.
An infinite convergent sum doesnt contain an infinite number of terms, but it does have an arbitrarily large number of terms such that its sum is indistinguishable from what it approaches.
Lookup “hyperfinite” and “hyperreal numbers” for more
@@insouciantFox I believe this is basically how floating point numbers work. Any number bigger than some very large cutoff point is treated as infinity, and "equalities" are really just checking that the two numbers are really really close together.
@@insouciantFoxbut .999… does equal one. they are the same. not just indistinguishable, not just effectively the same.
Strictly... no.
When I think about the limit of something, I prefer to think of it as the parameter that takes the "hypothetical lowest difference to the given number". I call it "hypothetical" because technically there isn't any real increase lower than other.
Nevertheless, math rules allow us to work with any number as we want as long as it performs like a "number" (even if it doesn't even exist). Which allows us to make a legal move in which we pass a number for the giving one but following previous or subsequent numbers' rules (in a nutshell, making the limit).
Teacher: What is 2+2?
Me: Of course, it is 7.
Teacher: You do not know any math.
Me: Yeah, you do not understand that I chose a different metric.
Teacher: What is 7 + 6?
Me: 15.
Teacher: What. It should be 13.
Me: But my calculator says so.
*Calculator set to Octal*
@@Aph2773 that's just not understanding that it's a joke
I understand its a joke but you have to say the base system u use otherwise people are gonna assume its base 10
*Teacher* (if the teacher is *really* doing their job correctly):
Then you're actually defining a *different* operation, which we'll call A ⊕ B, for which it happens to be true that 2 ⊕ 2 = 7. So ... now: your idea, your project. Do so. That's your homework.
The operation should have the usual properties that we rely on to do algebra:
(A ⊕ B) ⊕ C = A ⊕ (B ⊕ C), A ⊕ B = B ⊕ A, A ⊕ 0 = A = 0 ⊕ A, A ⊕ -A = 0 = -A ⊕ A,
ideally with the same *negative* operator -A rather than some other, new, one ⊖A; and the same *zero* 0, rather than a new one ⊙, or things will get really messy.
The simplest way to make that all happen is to *define* the operation by A ⊕ B = f⁻¹(f(A) + f(B)), where f(x) is a *strictly increasing* function that is odd (i.e. one in which f(-x) = -f(x), and so f(0) = 0), and f⁻¹(y) is its inverse (i.e. y = f(x) ⇔ x = f⁻¹(y)). So ... your homework assignment is to find a function with these properties such that f(2) + f(2) = f(7). If you can do that, then we'll take *that* as the function to use in the definition of your operation A ⊕ B and I'll accept the answer 2 ⊕ 2 = 7.
As extra credit, define a function f(α,x) such that A ⊕ B = (A + B)/(1 + αAB), where ⊕ is defined with this function and α is a parameter; i.e., find a function f(α,x) such that f(α,A) + f(α,B) = f(α, (A + B)/(1 + αAB)). Describe an application in physics, where α > 0, A, B are interpreted as speeds. What speed does 1/√α correspond to, then?
By the way, there is a solution f(±|x|) = ±|x|^k, where k = (ln 2)/(ln 7/2), with inverse f⁻¹(±|y|) = ±|y|^{1/k}. For A, B ≥ 0 that works out to the definition A ⊕ B = (A^k + B^k)^{1/k}. And you may verify that (under this definition): 2 ⊕ 2 = 7.
It’s important to understand that the theories of p-adic numbers (for each p) and the theory of real numbers are distinct theories. Otherwise, such statements lead to obvious ambiguity. So, the statement “1+2+4+... diverges” is true in the theory of the real numbers, while, independently of this fact, the statement “1+2+4+...=-1” is true in the theory of the 2-adic numbers.
On the other hand, field extensions lead to extended theories. For instance, the theory of complex numbers is an extension of the theory of real numbers, or, similarly, for any field extension of some p-adic field. So, in other words, every statement of equality that holds in the theory of real numbers still holds in the theory of complex numbers.
These two concepts, along with the distinction between them, seem to be lost on a good deal of commenters. The first creates a distinct theory with a distinct metric, while the second creates an extended theory with an extended metric.
Very good point, and I agree that this has tripped up a lot of commenters.
Thanks for that. An important point.
This is a great explanation that should clear many problems for ppl who have trouble understanding the video
Oh sugar,I'm negatively stupid!
You need to be careful. If I am studying the galois extension of Q, I might need to use both theories. And adele rings are born.
On the other hand, 3blue1brown made a small mistake at the end of the video. He defined the p-adics in terms of distance, and in that case he gets a group (and not a ring) but he can do with it with any number, not only primes. In fact, the p-adic distance is a kind of "reverse alphabetical distance" of the dictionnary.
This channel's production quality is better than Netflix
Gud grief
Hey 😊 indian guy
Netflix was founded by Marc Bernays Randolph - grand-nephew of Edward Bernays. Edward published the first book on how to create Propaganda in 1928.
@@maverickstclare3756 ok and?
Actually everything is better than Netflix
13:58 "Now this sum makes totally sense"
Me, still stuck on why the powers of 2 are approaching to zero: O_o
Add up the first 15 or 20 elements of the series individually. You'll see that the sums fast approach 1, eventually going up to 0.99999..., which, as he pointed-out, thanks to how mathematicians define a 'limit', is equal to 1.
@@kcnl2522 That's a different part of the video
I’ll stick with Crystal Math.
Adderall would be better
crystal meth
Love this comment
😂 LOL - thats how this felt! 👍🏻
@@azertyuiop432 woosh
Expectation: Determined to fully understand a 3b1b video
Reality: Facepalm
lmao haha....go and watch his video on conics...u will understand that
The video simply explains the reason why we use limits.
IDk 3b1b explains it pretty basic without going really advanced
×2
Ight imma head out to watch some MHJHB instead
@@SYFTV1 ok coomer
I'm grateful that I found your channel ! It makes math ideas look so beautiful and elegant. Especially linear algebra series.
For people who want to read more I suggest “a course in p-adic analysis” by Robert and “P-adic Numbers, P-adic Analysis, and Zeta-Functions” by Koblitz. Both good books that helped me understand p-adic numbers a lot better.
I can see the fabric of space-time now.
My profile picture is better ahahhahahahahah
@Just Cause silence before i bring Not Cause in here
@@matthewtodd9854 Heh say that again bucko.
It's the spice Navigator.
Ok
This is undoubtedly becoming my favorite mathematics channel on CZcams. While I love Numberphile a lot, you give your viewers' level of understanding a lot more credit, and you explain these concepts beautifully. I remember Vi Hart mentioning the p-adics briefly in one of her videos, but you took on the task of actually explaining them in a way that makes sense, and tied it all into the arbitrary (although consistent) notions behind metrics, and how we use them to think of an "organization" to the rational numbers. You just flipped the idea of "closeness" on its head, and I love it!
+Guy Edwards but still this is pretty hard to understand for me because I am not used to this new type of math presented here
+Guy Edwards Indeed a brilliant idea of "closeness". When I saw that part, I was all "Wow brilliant way to introduce the epsilon delta proof for limits"!
Agree
Carbon: Good one. I'm in recovery.
Yeah, it seems that the depth a youtube channel goes into its topic is inversely related to how well known it is. At least for math and science topics.
For the 1/2^n infinite series, imagine them as binary.
1/2 = 0.1,
1/4 = 0.01 and so on so forth.
The sum would equal 0.111111... (let's define this as S)
2S = 1.1111....
2S - S = 1
How 2S = 1.1111.... ?
@@codingforest7442 in binary, multiplying by 2 (represented as 10 in binary) is the same as shifting every digit in the number to the left by 1, just like how multiplying by 10 in decimal (which is base 10) is the same as shifting every digit in the number to the left by 1. so when you multiply 0.111111... by 2 (represented as 10 in binary), you just move every digit one to the left, so it becomes 1.11111...
@@codingforest7442 In binary, multiplying by 2 shifts everything a digit left, like how multiplying by 10 does so in decimal.
@@theuser810 you should have multiplied by 10 (2 in binary)
@@theuser810 ok I got you now, thx.
Thank you for this brilliant illustration. My first instinct was this is completely wrong but I never thought about that I had been constrained in think about distance between numbers in the traditional linear fashion and that if we change the notion of distance, some very counterintuitive results make sense.
Me: Ah nice, a video about inventing math
Me 2 minutes later: OH NO HE'S TRYING TO INTRODUCE US TO THE ZETA FUNCTION BY SUMS AND INTUITION, ALL HOPE IS LOST
@Ron Maimon r/woooosh
@Ron Maimon haha ur so smart like ur so big brained, do you go to harvard? do you think you could coach me on math some time since you know any math? you're so fucking smart dude, you're great as hell.
@Ron Maimon 😳
@Certyfikowany Przewracacz Hulajnóg Elektrycznych actuly it is, even i jumped of my chair like:
8:28 : is zeta(-2) and its 0 so well well well
Dude this didnt feel like he was doing math, it felt like he was doing meth
learn helpful maths from my maths videos.
but that's precisely how doing math feels
Easily mistaken
@@anymaths learning how to spell "COVID-19" with mathematical signs is not math.
Real math in a nutshell
I was fine until he started on about rooms 😭
You're not alone.
Think about it this way. You want a distance function that has all the abstract properties of the regular distance function. Shift invariance, triangle inequality, etc. In a sense, these properties are what defines the distance function as what it is, not the technical details of how it is necessarily defined or how we normally understand it working. If any distance function has these properties it can be used in the exact same way as the distance function in terms of logic and proofs. We are looking for generality, and the we can generalize the distance function as a family of functions with a certain set of properties essentially. In the video is a visualization of a logical system to define a function that has these such properties. It doesnt matter as much if you dont understand the technical details of how this is working, as long as you understand the goal, I'd say.
@@bjordsvennson2726 So did he arbitrarily choose what numbers go into which rooms? I don't understand why he put the numbers where he did.
@@rangerwickett he constructed the left hand side of the "rooms" such that powers of 2 would converge towards zero in the left hand subrooms. He then constructed the right hand side in accordance of the rules of shift invariance. As a consequence the numbers 1 less a power of 2 approach -1 in the right hand subrooms. Then with this sequence of numbers divided into smaller and smaller rooms he uses it to define his distance function. If you watch the video again you will notice that as he's describing the definition and highlighting numbers, given the definition the distance ends up being the inverse of the distance we would normally assign it if both inputs are positive. E.g. dist(5, 7) = 2 normally and 1/2 in this system. This is a complete redefinition of distance, but since it had the same properties of shift invariance by definition, it will behave in the same abstract way. However, in the specific way he constructed this distance function, it makes sense that powers of 2 add to -1.
@@rangerwickett I'd say the choice of arranging the numbers into rooms was an arbitrary choice for the ultimate purpose of making sense the nonsensical equation. but despite this arbitrary choice, the arrangements of numbers in relation to each other is consistent.
This, I find, is the earliest 3b1b video I've seen. It's refreshing to hear that Grant hasn't always been both a math teaching wizard & a master of perfect audio presentation. But he's still and always has been a math teaching wizard. Much appreciated.
Everybody: What was first chicken or egg?
Mathematics: 1-1+1-1+...=1/2 so it was half egg and half chicken.
Actually yes?
They both appeared???
Technically it was an egg from a bird that was not chicken yet but almost a chicken lol
But the transition was so slow at some point it I don’t know where you could call it a chicken or not. And btw wild chickens are exotic beatiful jungle birds from asia
The bird came into existence to lay egg.
@@rotorblade9508 exaclty, like, neither just popped out of thin air, it was a slow process
Its a funny thing when is a bag full of sand a bag full of sand and when is it a sand full of bag if u get what i mean
You lost me at the sub-rooms...
Same here. Everything up to that point was fine then he started playing with the definition of distance...... my brain broke.
Yeah, me too, I have no idea how to make sense of that.
same bro..
I watched that part more than 3 times, then I start to understand a little bit of it. They are re-ordering numbers in a way that is not linear, so that the distant(A,B) has a consistent meaning. instead of 2-1 = 0, they have something like dist(2,1) = ??, something like that.
What's going on here is that, from what I gather, is that 1 and -1 are the same thing. In essence, since infinite sums are so strange their inverse is the same as themselves. it's like saying .999 = 1 or .999 = 2. It's easier to understand if you look at it as .999 = x. instead of a real number.
It feels like, as the inventor said, "OOGA BOOGA"
Is it you Patrick?
It feels pretty good. I came up with tetration (the operation higher than exponentiation) on my own before finding out other people already thought of it
The only understandable thing I learned througt this video is: 'If you think that something doesn't makes any sense, you probably only use the wrong definitions.'
PHScience this actually comes relatively close to what he is actually saying
Then, dividing has the wrong definitions since you can’t divide by zero
@@kjl3080 Or maybe zero has the wrong definition?
My engineering school destroyed my love for maths.
CZcams restored it. ❤
was it univeristy or engineer technology school? that is big diffrence
SIMPLIEST COMMERCIAL really? What is it
it's not just CZcams. it's also the creators ! :D
There were math in times of war.
Can someone explain this to me? At 6:42 you say 1/(1 - p ) = summation p going from 0 to infinity p^n. But isn’t it only true if |p| < 1. That’s what I learned in my math class. Why do you say that we can plug in any number at all?
As always, thanks for presenting this valuable information, great work.
In my opinion the reciprocal sums are so profound and beautiful, that it really makes me to ponder if I do really understand mathematics. For my profession as an experimental particle physicist I have learnt substantial advanced mathematics. But honestly, our courses have brutally killed the core beauty of the mathematics itself. I don't blame the courses as our primary focus were just an application of the subject and use it as a tool. I remember in our post graduate course our professor who was teaching us Riemann Zeta function apologized to us for not being able to demonstrate us its entire beauty. He gave us an example like, we draw certain geometric drawings on a piece of paper for having a perception of physical things, but; those drawings are definitely not piece of art. Although, both are made just the same way; some scratches of a pencil. I don't remember his exact words, but his points were clear.
Congratulations to you for your brilliant effort in spreading the art of mathematics to the world.
I loved this video thoroughly and I understood none of it
wow! very impressive video i guess (but supposed to be educational)
Me three.
If you loved I assume you did not understand it!
Here it is:
There's not always just one way to solve a problem, but it can be hard to know which ways will lead to the most useful mathematical conclusions. Mathematicians try to avoid leaving out any possible solutions by making as few assumptions as possible. (For example, If I *assume* that the only way to mars is by rocket, which is a valid assumption, I have already left out teleportation just by assuming something). In this case, we assumed that there is only one way to find the distance between two numbers, and it turns out that there are multiple ways to do that. He explains one way to define distance at the end, and this way of calculating distance leads to the conclusion that 2+4+8+16...=0.
Henry Parker. Now I feel a little bit better xD
10:50 Completely lost it. I have no idea what the rooms mean.
Confusing indeed. A gist is that you define numbers based on what he says. Meaning you could say that even the addition of 1 + 2 does not give 3(as per the way he defines) You define numbers in an entirely different way. You won't need this nonsense if you are not a mathematician.
Edit: read the reply(long one)
@@arunjosephshadrach9539 1 + 2 = 3 in any p-adic metric. Addition of rational numbers is still done in the normal way. However, the distance between two numbers is no longer given by the absolute difference. That doesn't matter for rational numbers, because they are defined in a way that is independent of their metric, but it does matter for irrational numbers, since they are defined in terms of the limits of sequences, and the limit will of course change if our idea of distance changes.
In this video, you saw that using the normal ("Euclidean") metric, the series 1+2+4+... diverges. But using a different metric called the 2-adic metric, it actually converges to -1. Each partial sum is still the same as what you would expect (1, 3, 7, 15, ...), but under the p-adic metric, these numbers get arbitrarily close to -1. In the Euclidean metric, to find the distance between two numbers, we subtract and then take the absolute value. So for instance, the distance between 3 and 7 is |3-7| = 4. In the p-adic metric, to find the distance between two numbers, we subtract and then take the "p-adic absolute value," where the p-adic absolute value of rational x is |x| = p^-n, whenever x can be expressed as x = p^n(a/b), with a and b integers that are not divisible by p. So for instance, the 2-adic absolute value of 1/6 is 2, because we can write 1/6 = 2^-1 * (1/3). In other words, the largest power of 2 that is a factor of the denominator is 2^1, so the 2-adic absolute value is 2^1. Similarly, the 2-adic absolute value of 20 is 1/4, because we can write 20 = 2^2 * (5/1). Thus the distance between 3 and 7 is not 4 in this case but |3-7|_2 = |-4|_2 = |2^2 * (-1/1)|_2 = 2^-2 = 1/4.
Just as with the rational numbers under the Euclidean metric, we can define when sequences of rational numbers "converge" (or technically, are Cauchy) in the p-adic metric. We can organize these sequences into equivalence classes, where they are equivalent, loosely speaking, if they should converge to the "same number" (though we haven't necessarily defined the value they actually converge to yet). We call each equivalence class a p-adic number, an exact analogy to the real numbers, and apply the same sort of reasoning but using this strange metric.
@@EebstertheGreat you are a god, I broke my brain a few times but I finally understood, thanks !
@@EebstertheGreat I was lost here too, thank you for your explanation !
i just can't stop coming back and appreciating this masterpiece!
Me having a sudden unexplainable urge to watch a math video at 2 am in the morning
i can relate .
Yup
I'm literally reading your comment at 2:03am.
as opposed to 2 am in the afternoon ?
@@diatonicdissonance 3:46
Being a software developer, this immediately made me think of two's complement - how most computers represent integers. An 8-bit byte with all ones (1 + 2 + 4 + 8 + 16 + 32 + 64 + 128) represents -1 in two's complement, likewise a 16-bit or 32-bit word with all ones is -1, etc., you can extend this idea to a word with infinitely many bits that are all 1 to represent -1, so 1 + 2 + 4 + 8 + ... = -1 makes sense from that perspective.
You can extend that idea much further; if we consider "2-complement" to be a multiplicative operator that projects from positive to negative numbers, and take (ω+1) = -2, (ω+2) = -3 and so on, we have an additive subgroup that is precisely Z with addition, ie we have defined negative numbers as a function of infinite sums of positive numbers! Interestingly, in this notion of numbers there is only one type of infinity: uncountable infinity. N can count the elements of R! To be fair, it's counting equivalence classes of equal area, but it's valid. If rather useless.....
that immediately made me think of the theory that we live in a computer simulation.
It's like an infinite overflow xD
@@potatopassingby1148 Of course it would look like we live in a compuiter simulation when you redefine the real number line into the one clmpuiters use...
Very practical.
You're my new favorite CZcams channel. Please don't stop!
Creating new math, being the first to prove something, felt great and exhilarating in my experience, even though I have done so only on a very modest level with a few fringe results during my master and Ph.D. studies, nothing even remotely approaching the level of maths shown in this video.
I have never been so engaged and challenged in a math video like this, legit thought i was in a classroom. There was even a point where i got things confused that I used his room drawing/explanation to understand the wrong formula (1/2+1/4....) And somehow made it make sense as in half room distance+quarter room distance i.e sub-sub rooms all approach one (or it's position). Dunno if that's true or not. But all in all this a very enjoyable and challenging learning experience.
I don't get it...
Like, any thing from the "rooms" part onwards.
You know how 0.99999… equals one, and how, conceptually, any number can be thought of as having an infinite number of leading zeros? It’s *kind of* like the 0.9999… thing, but in the other direction. But it only works with prime bases, like base 2, 3, 5, etc.
@@atimholt this is not helpful..
You're not alone.
I think that this explanation isn't quite as good as his newer videos.
It reminds me of "surreal nunbers", which I heard about from Numberphile. I dont understand them, but they might be what he's talking about.
I feel you bro, Here I am looking at the comments after he started talking about rooms
I guess I could have understood but I didn’t know the point
This is just a linear transformation...
Changing the definition of distance between two numbers actually changes the meaning of the numbers. We are no longer saying that "15 apples are 14 apples more than 1 apple."
The change of distance definition inevitably changes the meaning of addition. So, yes, we can definitely define any distance function and by doing so define a new mathematical dimension where numbers no longer represent real-life quantities, rather quantities that only make sense in that universe, but can be linearly transformed to the universe we understand.
In this video, 1 + 2 + 4 + 8 + ..., is no longer equivalent to the sum of increasing positive numbers on the number line. The way we divided numbers into rooms and sub-rooms and sub-sub-...rooms makes 1+2+4+8+... in this coordinates system equivalent to this:
1 - 1 - 1/2 - 1/4 - 1/8
after doing a linear transformation back to the real-life coordinates system.
We have to define whether going from a number to a number on the right means adding or subtracting the distance, because dist(x,y) = -dist(y,x).
This video assumes that distance from 1 to 0 is -1 (going left means subtracting), which makes this straightforward. Distance between 1 and 2 is -1, distance between 2 and 4 is -1/2, distance between 4 and 8 is -1/4 and so on... so from the starting term of the sum "1" we get:
1 - 1 - 1/2 - 1/4 ... and that's how 1/(1-p) when p_new_coordinates = 2 converges to -1. Because p_new_coordinates = 2 === p = 1/2 where the sum is actually a negative sum, and n starts at 1 not 0.
If we assume that distance from 1 to 0 is 1 (going left means adding), then we have to divide numbers between rooms differently, because in this system, distance from 0 to 1 is going right (negative), but from 1 to 2 is going left (positive) which means dist(0,1) =/= dist(1,2).
Side not, this system has no meaning of "infinity". 0 takes out the place of the smallest number, and -1 takes out the place of the largest number. The greatest distance between two numbers is 1 and the smallest distance is dist(x,x) = 0, which really helps imagining it, again, on the number line where all numbers fall between 0 and 1. It's also a spherical system, where each number is the center of the universe.
If these numbers dont really mean 1 and 2 in the transformed room then they should probably tag it with something else. Otherwise it gets confusing since 2 actually means something in the physical word and addition of 2 and 4 means something as well. If 2 and 4 in the distance space could be re-interpreted then prob they should add a symbol. Like saying 1g+2g+4g+8g.... approaches -1 and then the g like complex numbers denote the transformed entity where it belongs.
missbond the interesting thing about these new numbers is that we discovered them very casually without groundless assumptions. So noting them the same way is to show the intricate connection between real numbers, infinite series and p-adic numbers
This great explanation makes sense and should be incorporated into the video. The video fails to explain that concept which is important.
Jorge R. Agree
This is the best comment I have seen. Obviously a hidden deception is going on which most people ignore because they think "I'm not clever enough to understand this and because he is cleverer than me he must be right.".
OBVIOUSLY the conclusion is bullshit and it shows that if you are clever enough you can convince the masses of ANYTHING!
Fun fact (perhaps somebody else has commented this too):
This is actually the way negative integers are stored in your computer. -1 is stored as 2^64-1 when using 64 bits to store an integer.
Pretty sure you are very wrong.
@@pelasgeuspelasgeus4634 You might want to look up "two's complement"
I do not usually rate videos, nor taking comments, but this... I haven't seen such inspirational video on youtube for years!
+szymek1567 Indeed.
Quite true.
You lost me at 1+2
😂😂
Topher TheTenth
No...
It is 21
@DAVID MELLA no u they are reverse
I have difficulty in 1 plus 1.
Grant, this is one of the best videos I've ever watched. It just clicked why 1 + 2 + 3... and so on, = -1, when you imagine the idea of sub rooms (although it was kinda weird to think about). I love it. That in of itself, I find, is the coolest thing I've ever seen. How amazing!
It's already 5 years from when the channel started!
Great content all along!!
10:40 that random volume increase was weird
it is for you to wake up
SYFTV1 mama aqui uiuiuiuiaiaia
@@tvboxdoscarvalhoslucascare3477 exacto v:
It's because this is where the video gets intense, so you need to concentrate.
I like your Grahams Number reference!
I saw that too. Nice!
1997CWR i dont
Search Graham's Number Numberphile
g(g64) ahhh i just don't want to think about that
At 3:44
5:30 that moment felt like a cartoon about mathematicians trying to solidify stuff to beat other mathematicians over the opinions of does that concept make sense.
My comment certainly doesn't but the general idea is here.
Tbh I kinda described real math with a contrast filter.
My silly hobby is to recommend science-channel
to my fellow science-fans.
Mind?
3B1B: rooms
Everybody: *visible confusion*
7:37 I read it in iambic -pentameter- trimeter and now I need a modernized Shakespearean play about mathematics.
Wouldn't it be iambic trimeter?
@@isavenewspapers8890I actually like your necro post. You've corrected my mistake (which I've now edited in) but you've also reminded me of this video that I haven't seen in 3 years and an enjoyable concept of poetic math. Thanks.
Mr. 3Blue1Brown, how do you understand these concepts so deeply and innately? How did you study math and from where did you develop such deep understanding of the subject? We're you inspired by your teachers? Your videos bring me the greatest joy. I am in awe after each of your videos. My eyes are filled with tears to see such beauty unravel out of a seemingly simple idea. Thank you, please keep inspiring.
The fact that he can explain these concepts perfectly to a layman only makes your point stronger. For one to explain complex concepts in simple, concise way, they must have a profound understanding of what they're talking about, which Mr. 3blue1brown clearly demonstrates.
The teachers were inspired by him
True. I feel the exact same way, and I feel love for the subject, and an understanding that I could never even concieve of before, all thanks to Mr. 3Blue1Brown.
1.Go to university
2.study
3.???
4.get a phd in mathematics
5.read a shitton of books
6.???
7.now you are a mathematician
Call him Grant.
I'd never noticed the poem at 7:41. It's lovely! :D
When I saw the equation at 7:56 I thought you were going to explain that the result being -1 meant that the sum would always be 1 less than the next power added to the sum. I didn't expect you to invent a new way to arrange numbers to visually make sense of it.
Does this mean that as we approach infinity, the size of my laptop's RAM will be -1 gig?
no, because it is based on the number line, also talking about physical things makes no sense in this context
+Zekrine Alfa wow. thanks for that simple explanation! make sense to me now. but isn't -1/12 found in physics, which is about physical stuff? how come this is different?
I don't know, I have nit gotten to that yet in college, the only think that I can say is that infinite ram is unlikely, also ram is not an infinite sum it is 2 to the power of something
Not*, thing*
Daniel Astillero no because this video the math is wrong. You can only use the formula 1/(1-n) for any sum->infinity if the value being added falls 0
This was the one video I didn’t really get, but now I’ve covered metric spaces at uni it makes more sense. Most people are lost at the rooms, and to try and explain a bit better, it won’t make sense with the usual way of thinking of distance.
0:44 I have been on this one, when in school we were told on concept of fractions. It was truly fascinating experience to discover concept that in between of 2 unequal numbers there are infinitely many numbers.
One of the best videos on the internet.
This is the best math video ever! That's because you did not just plainly explained a charming math fact, but you guided us to your (interesting!) idea of what's mathematics. Thanks!!!
I cant understand, when you sad "p must be 0
I agree. That is where I lost confidence in this "proof"
I mean. That's the entire point of the video. I would suggest rewatching the video, keeping in mind the point of the video (the title tells you the point), and paying careful attention to what Grant says.
That's the point -- what if it *did* make sense for p > 1 or p < 0 ?
and that's the rigor he was referring to. Certainly, in our image of numbers it doesn't make sense. So how do we /make it/ make sense? And there we go.
He just goes through the different cases.
Even if they don't apply, leading to a convergence
The case of apparently leading to 1/2 or 0.5 is interesting, because you can group the elements of that sum into 0, and 1.
And if you would try to see the * average * of all these present elements
Its 0.5
We can use formula a/(1-r) for sum of infinite gp series when r1
I thought you would talk about residual class rings, but p-adic numbers seem even more exciting.
I had difficulty distinguishing between the colors of the boxes. I found the topic interesting, and will be reading more.
Contentwise a good video - if you could improve your mic quality it would be perfect
+Ubermensch Man, I don't know why sound quality wasn't something I cared enough about back then. Trust me, all future videos will be made with a good mic.
Your doing a great work buddy !
+3Blue1Brown
I would also recommend some acting lessons and/or voice training.
An interesting thing about this is that the idea that 1+2+4+8+16+...=-1 is used to represent negative numbers by computers. If you have 8-bit signed number, you assume the that the most significant but continues forever (when you convert 8-bit number to 16-bit number you fill missing bits with the most significant bit of the 8-bit number) do for positive numbers most significant bit is 0 and repeating it forever doesn't change the value, but for negative numbers if you have for example 11110000, it's 16+32+64+..., which is -1-2-4-8+1+2+4+8+16+... which is -1-2-4-8-1, which is -16 any that's the number that is represented by 11110000.
I love that I found this during an unrelated search learn invent cycle. Thanks for your hard work
Then god said “let there be analysis”
And I hate it. It started so easy and like the next week I have to proof the rational numbers and the week after prove that the complex numbers consist of some Cauchy sequence and body/ring rooms
AND I DONT EVEN STUDY MATH
Timur1214 oh no Im starting after christmas break, tgen this video popped up. Should I be scared
Yea you should know that already at the beginning you have to study a lot of new math. But if you already know physics (assuming you study physics) then you can atleast focus on learning the new math while physics is so easy that you can neglect it at the beginning. Also right now, after 2 months it became way more chill. Though for analysis I have to learn in the holidays now ^^'
If I could go back I would have focused more right at the beginning and made sure I understood everything from week 1 and not thought "ah I'm gonna learn it with the time anyways", thats true but now it's kinda unpleasent to ask stuff from 1-2 months ago xd
Donut be afraid just let the math gods guide you and everything should be trivial....
watch my maths videos to learn something.
The sum of all powers of two also equals - 1 in signed binary numbers
@@nycki93 cool
@@nycki93 Python is a Godsend
@@nycki93 Most other major languages have incorporated similar implementations, such as BigInteger in Java
@@loganrussell48 don't forget BigInt in JS
@@parabirb COBOL had that in 50s . It is called the decimal data type.
Fantastic video. Very good. Keep doing videos like that.
this is a cool video but im glad your audio setup has improved
at 06:55 value of p is taken as -1 but in the derivation, we restricted p to be in between 0 to 1
I don't get why people just ignored that restriction. No wonder why math gone wrong after that. Any minute after ignoring that restriction is just for fun and cannot be taken seriously, at least as far as I understand how math works.
@@adrianordp I am not a mathematician so take it by a grain of salt.
p was indeed given a restriction that it must be between 0 and 1. But, as said in the video, we simply arbitrarily chose the numbers in form of a line where the numbers 2,4,8 etc cannot be in-between 0 and 1. For generalization, we openly accepted other possibilities and cases such that the powers of 2 actually fall between the values of 0 and 1. This case, depicted by rooms is known as 2-adic systems which differ from our normal number system(line). We didn't violate anything as we have followed the fundamental rules used to construct the system of representation which was the distance function.
The formula still holds as in this new system, all powers of 2 are between 0 and 1. (You will be correct if you argue that this doesn't approach -1 in the conventional system. We only claimed that it's true for a different system.) and we did all this because we are supposed to think as a mathematician here and must always remove arbitration and generalize our findings.
P.S. I agree with you that most people(non mathematicians) who would work with it would do it for fun and do not take it seriously.
What a beautiful video.
I came here one year ago and I thought I was getting it.
Then, coming back here now with more proof-based math knowledge and having seen some of the concepts already, it makes so much more sense.
I am curious to see what I'll get from this video one year from now :)
What did u get now
Intuitively, it seems to express the fact that when you add the powers of 2 from 2^0 to 2^n, you always fall short by 1 of the next power of 2.
my pea sized brain can't
That g(g64) killed me... I was like oh yeah graham's number. Oh wait thats the number of g's...
+Kane Angelos Ouch, my mind.
+Zuzu Superfly I know right? The fact that is was in the denominator made it even more unfathomable to me.
+666unknowndevil666 it hurts to even try to comprehend it's vastness
+Logan Retamoza Or since it's in a denominator, its _tininess_...
+Kane Angelos I couldn't really comprehend it, so I just saved my brain and said, "Yeah that's basically 0."
1/(g(g64)) is so ridiculously tiny.
It is one the greatest pleasures to derive stuff which are mentioned in books as formulae without any background. Whenever I do it, I feel more confident in mathematics.
I remember sitting on my couch and thinking of a similar form of 1/2n approaching 1 which was adding 10, then 5, then 2.5, then 1.25, and so on, and thinking if it would ever reach 20, then i had learned about 1/2n approaching 1 about 5 years later and was ecstatic that I'd thought of something similar myself and that that's how mathematical concepts are discovered.
This whole video summarized in one sentence
'I don't need sleep, I need answers.'
I watched this yesterday and came back to it, trying to work out the part where 1 is split up into p and 1-p, so on. Then I realized why the sum of 2^n = -1 is so strange. The original and only sensible assumption is that 0
God finally somebody tried to make sense of this. I'm looking at your comment again after I digest the video lol
Does this means we can't put p=2 as it was said that 0
actually point of video is, we got (1-p)+p(1-p)+...+p^n = 1 for 0
That's the whole point of this video. He himself says that the function only makes sense for values 0
In a way, once they reshuffle things into that 2-adic system, 0
In Two's Complement representation of signed integers, this equation becomes somehow clear: E.g. the binary number 11111111 represents -1 in signed 8-bit integers. The only difference is, that summation is not infinite.
But as the number of bits approaches infinity, the summation which adds to -1 approaches 1+2+…+2^∞
No..in two's complement the largest term is NEGATIVE (it represents -2^(n-1)) that's why the total sum can be -1.
Now i want a video about these neither real nor complex numbers
i don't wanna forget this dream when i wake up in the morning.
BEST
CHANNEL
EVER
ditto
ditto
Pikachu
Yes
scratches the surface of the tip of the iceberg floating in the sea of secrets on an alien planet
I WAS LOOKING FOR THIS VIDEO SINCE LIKE 2 MONTHS AND BRUH THIS VIDEO WAS ALREADY MADE LIKE 6 YEARS AGO , thanks 3blue1brown for the video :))
It's important to notice that those weird "super" sum results can be seen as values of the (complex) analytic continuation of given series. In your example values of the analytic continuation of the geometric series (z) => 1/(1-z) valid in the complex plane \ {1}. Famous are also crazy values of the sum form of the zeta function only valid in {Re(z)>1), by replacing it by the known values of the analytic continuation of zeta using its functional equation => zeta(-1)=-1/12 "=" 1+2+3 ....
In a computer integer value, each value of 2^n is stored by setting bit n to one (counting from the right and regarding the least significant bit as bit 0). So adding together all the powers of two you get a word which has all bits set. But, in the two's complement system used by computers, a word with all bits set has the value of -1.
Presumably a weird coincidence.
+Mandolinic Great comment, it's actually not a coincidence! When you're representing integers with n bits, in a sense you are working not with integers but with integers mod 2^n. This is because as you increment from 0 upwards you will be forced to roll the meter back to 0, so to speak, once you hit 2^n. The reason the word with all bits set to 1 nicely represents -1 is that -1 and 1+2+4+8+...+2^{n-1}=(2^n)-1 are congruent mod 2^n. Notice, this means they are very close to each other in a 2-adic sense. As you let n tend to infinity, the words with all 1's are essentially representing -1 in a more and more encompassing representation of the integers, which makes the infinite sum feel a bit more reasonable.
Mind = Blown
+Mandolinic While trying to understand the representation of p-adic numbers, I realized that too. For some reason it made me so happy to see such a strong relation between two matters which I thought had nothing to do with each other! Math never ceases to fascinate me with the level of abstraction it manages to accomplish.
Nice observation you got there
Actually two's-complement is not the only representation that has been used in computers. That's why the standard for the C programming language allows also for ones-complement and signed-magnitude. I've encountered all three architectures; there are pros and cons for any of them.
I really love the way this "Generalizes" the Two's Complement to an "infinite" number of bits.
Sum of an infinite GP is defined as a/1-r for -1 < r < 1.
a is the first term, r is the common ratio.
the background fiddle music, sounds good.
Sounds like backstage before start.
Trying to watch video. Keep getting distracted by the lovely cello music.
I come back and watch this video every so often. It's inspirational to someone that sits just beyond the accepted truths of mathematics. Someone that sees beyond Russel, Gödel, Hilbert, Church, and Turing's works.
Just a note that might make the geometric sum formula more palatable (probably not), is that functions of the complex plane that are analytic in an open set (like the complex numbers with magnitude less than 1), have a unique analytic continuation. It is extremely convenient, in this case, that the sum has a mostly well behaved closed form, and thus the analytic function must coincide with the closed form.
What's nuts is that swapping the sum index and function parameter gives you the zeta function rotated by 180 degrees, and all the nice properties fall apart.
Some Mathematical Fields like Prime numbers and Number theory, are Infinite Computations and Are Outside the Realms of Most Mathematicians, like Hilbert , Godel, Russell, are masters of Formalism Some Number theorist like the India Mathematician Ramajuan and Cantor,Euler Gauss touched on this. It's mostly a Intuitive Approach rather than a Formal method
9:42 This property means that the distance has the form dist(a, b)=f(a-b). For example, for 2-adic distance f(d)=1/(the smallest power of 2 that you have to use to write d as a sum of unique powers of 2)
The distance formula in terms of rooms could also be described as "the number of least significant bits that match"
one way you could show your friends 0.99... = 1 is
1/3 = 0.33...
2/3 = 0.66...
3/3 = 1
0.33... + 0.66... = 0.99... = 3/3 = 1
1/3 + 2/3 = 3/3 = 1 ...right?
+Altus Boren Right.
+Altus Boren don't you always stay 0.0000...1 short? 0.33 isn't the same as 1/3. So how does it ever become 1? I really don't know, would love a simple explanation.
+KedraIrke right, 0.33 is not the same as 1/3. Instead, I was writing "0.33..." to signify "0.33 reoccurring".
+KedraIrke What does 0.000...1 mean? What does 0.9999... mean? When you answer these questions, you find that, for the usual meanings of the expressions, the first one doesn't really mean anything, and the second one means the same number as 1. If you define the result of an infinite sum as being the limit of the sequence of the partial sums, then you find that 0.9+0.09+0.009+... = 1, so that seems like the only reasonable meaning for 0.999... under that definition of an infinite sum, for the usual meaning of a limit. (you could probably define other sorts of "limits" where its different, but those aren't the most useful ones for most cases?)
+KedraIrke
1 - 0,(9) = 1 - 0,999... = 0, 000... = 0
1 - 0,(9) = 0
1 = 0,(9)