3Blue1Brown - 2015-08-14
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: https://www.3blue1brown.com/ Music: Legions (Reverie) by Zoe Keating ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted about new videos, subscribe, and click the bell to receive notifications (if you're into that). If you are new to this channel and want to see more, a good place to start is this playlist: https://goo.gl/WmnCQZ Various social media stuffs: Patreon: https://www.patreon.com/3blue1brown Twitter: https://twitter.com/3Blue1Brown Facebook: https://www.facebook.com/3blue1brown/ Reddit: https://www.reddit.com/r/3Blue1Brown
You lost me at 1+2
1+2 = 11
1+2≠π
ThatOneGuy Genius
it's obviously 12, idiot
How it feels to invent math
5 math, stimulate your senses
*simulate your equations
"You are a mathematician [...], so you don't let the fact that something is nonsensical stop you" A true mathematician spirit
@elcidbob Try using equilaterals as planar units as Buckminster Fuller did, and see what happens watch?v=N3QzD8QC4ko&t=4h59m28s
@TrokutX I hadn't thought of debt, that's surely one way to apply it. Though as you said, only as an idea to simplify bookkeeping.
In all other cases, negative numbers, "negative existence" can only be imagined in the abstract space of mathematics. A little ironic that we call i the "imaginary unit", isn't it?
Just as no amount of dour ontological scepticism will ever persuade a mathematician to forsake the unearthly paradise of tranfinite numbers.
@larry baby Whatever those are, they must be keeping someone busy.
I can see the fabric of space-time now.
@Just Cause silence before i bring Not Cause in here
@Kyrlics bring it on
@Cailou Quality Animations get that pill, you need it
@Cailou Quality Animations Heh say that again bucko.
It's the spice Navigator.
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
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
but unlike other math youtubes he specified the zeta function doesnt work on the 'default' math system every non-mathematician/math entuthiast knows
Dude this didnt feel like he was doing math, it felt like he was doing meth
learn helpful maths from my maths videos.
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.
So if I have a physical system where I need to find the value of "1+2+4...+2^n+.." then should I flip a coin or something? Please...
And more importantly and away from the boring physical world, can the same thing mean two opposite things at the same time under the very same conditions (sum to inf in this case)?
in such case 1+2+3 should be written different to avoid confusion of the numbers we all know.
Generally agree, but a small counterpoint for this case if we extend real analysis (to which infinite sums of integers, rationals, and reals belong) into complex analysis, we actually can give the "same" answer as going to the 2-adic theory.
The geometric sum is an analytic function in an open subset of the complex plane, the interior of the unit disk in this case. Within that disk, it converges and has a closed form, as shown in the video. With the exception of a single pole, that closed form is analytic and trivially convergent everywhere. Since we have two expressions that are analytic and are equal in an open subset of the complex plane, the interior of the unit disk, they must both have the same analytic continuation. Since the closed form is defined everywhere except for p=1, the only reasonable value to assign to the function defined by the geometric sum at p=2 is -1, likewise for 1/2 at p=-1.
The fact that these two wildly different theories give the same answer is most fascinating.
yeaaah
Its clearly make me understood
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.
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.
Tycho Photiou um that’s not what this comment was saying. This comment is describing a different way of looking at the problem: rather than looking at it as a new definition of distance, it can be equivalently defined as a remapping of numbers to different definitions of what a number is. That of course enters into abstract algebra, where what a number is is whatever you want a number to be as long as it’s interesting. Both ideas are valid.
Wow I learnt linear transformation in linear algebra but I didn’t know it can be used his way...even though I still don’t understand the logic but math is incredible wow ~
Does this mean there is a particular form/visible pattern to linear transformations that will satisfy his definition of a "useful" distance function? How will the matrices look in higher dimensions? Are there infinitely many in each dimension?
‘linear’ is not the correct word. In fact, even the word ‘algebraical’ couldn’t do the job. It’s impossible to write down the connection between the ‘realistic’ numbers and the p-adic number without any of number theory or abstract algebra.
Mikayla Merna Nope. Linear transformation can only multiply the distance with some constant.
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.
10:40 that random volume increase was weird
When you apply this to the atomic universe it's got everything to do with a lot of things. Pressure for an example.
The sum of all powers of two also equals - 1 in signed binary numbers
Fun fact, in languages like Python where integers are not fixed length, you can still do 'bitshift' operations on them and it will simulate infinite 0s (or 1s) out to the left where appropriate.
@nupanick cool
@nupanick Python is a Godsend
@nupanick Most other major languages have incorporated similar implementations, such as BigInteger in Java
@Logan Russell don't forget BigInt in JS
I cant understand, when you sad "p must be 0<p1"
And then you equalise p =2
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
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.
When I'm about to find answer..I lost myself and appear at the end of this video.
I’ll stick with Crystal Math.
Honestly inventing math feels kinda like landing an airplane successfully. Kinda like being glad you're alive that you were the one who made it happen.
I loved this video thoroughly and I understood none of it
hey no way!
Same here
I was only just able to keep up and I've already completed an analysis course
ha same
I believe that you are being humble, which is pretty cool 😎
lol🤣🤣🤣🤣
4.30 am and I'm on the verge of falling asleep and I've completly lost track of what's going on here so I'll continue tomorrow. Goodnight
Conclusion: "a new type of number":D
Me, just seeing the vid's thumbnail:
Questions my existing math knowledge
After watching it: meh
6:35 how in the universe rhs makes sense for any values of p. You just said 0<p<1..how can someone change the domain of the function and expect the range to be valid on invalid domain.
Damn everytime I watch this video I stop at the "putting numbers in rooms" part, and I always tell myself I'll watch it another day so maybe I can understand it then. Turns out a year has passed and for all the times I watched it I still couldn't understand it :(
You should probably revisit p-adic numbers one day, even as an excuse to talk more about metrics and convergence
I had thought of another theory myself. It’s probably already a thing but i haven’t heard of it. Basically i thought of numbers in an infinitely large circle rather than a line, where approaching infinity is the same as approaching zero from the negative side. Essentially this perspective makes all numbers relative, and also explains that summation equaling -1, as negative one on this circle is the same as one less than the infinitely large power of two, which would equal zero.
Matthew Ryan it’s all relative to your perspective. Just like there’s no specific number assigned to any one spot in the universe. You find a baseline and call it zero. In a way of thinking, yes -1 would be the same as infinity-1, but that’s not an inherent mistake in the system. I should re-emphasize that the circle is of infinite size, too. Some definitions of infinity can have an end. It’s not perfectly applicable but i suggest looking up supertasks
Yeah I know what you mean, its a cool concept. There are some nice extensions in model theory.
Such a line is very similar to Real projective line, but real projective line does not have any metric
@Matthew Ryan imagine it as a clock face. Positive numbers 0 to infinity goes through 6 via 5, 4, 3, 2, 1 to 12 counter clock wise. negatives from 6 via 7, 8, 9, 10,11 to 12. 6 is zero and 12 is overlap of negative and positive infinities.
And complex numbers in an axis 90 degrees to this. Probably we have a third type of number which takes up the z axis too.
"You decide to humour the universe, ...", maybe the best phrase describing theoretical research.
8:32 what did u just do in there?
“A mathematician is a blind man in a dark room looking for a black cat which isn’t there.”
- Charles Darwin
I doubt that Charles Darwin said that, check also: https://en.wikipedia.org/wiki/Black_cat_analogy and "Darwin misquotes"
Once you started to describe the definition of "distance" and sub-rooms ... I got lost. But watched it to the end just for the nice music.
I remember the days when u were at 200K subs, you have come a long way indeed :")
Keep changing lives man!
You lost me at the sub-rooms...
I would say "You lost me at (early point in the video)" but that seems cliche.
This is mathematical deviancy.
Same
😂😂😂
Same
It never reaches one.
7:27 don't these numbers approach infinity?
These "rooms" just reminded me quickly of radix-2 FFT... Math is beautiful!
This has a sense now. I've never looked before at sums from point of view of the non-standard distance function.
:O
00:12 - Also called buffer overflow
12:47 how does shift invariance hold for that though?
Im procrastinating doing my math homework by watching this
same
You have infinite time.
3:43 g sub g sub 64? C'mon now, g sub 64 already breaks my brain. Heck, so does g sub 1.
Oh gosh I just went to revisit this video and it reminded me of the music you used to use back then... feels so good.
this is how calculus should be taught :)
this process feels like an inverse Hilbert's curve :)
That's why I am a physicist.
I was able to follow, but... damn(!) 🤣
...
I know, eventually, this will be useful to me, I just don't see it right now.
...
It humbles you down. To know, that somebody is way smarter than you, and, at the same time, it makes you so happy to realize that there is so much to learn out there, that you don't know where even to begin.
(Takes off his hat, and throws it away, 'cause he knows, he has no use for it anymore, since he's gonna have to take it off here every time...)
The only dread is attending the horrendous education system. But I’m gonna make the most of it.
Young aspiring physicist and mathematician here 😎
@Papa Demon i hope i surpass you. good luck by the way.
I also want to be a Theoretical Physicist .
Maths makes physics readable
@Dr. Doctor "I hope I surpass you" 😂😂🧐
Does this mean that as we approach infinity, the size of my laptop's RAM will be -1 gig?
"As far as laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."
- Albert Einstein
@Yosef MacGruber Real number are not reality you know. There are more intuitive, that's all
@miggaz elquez
Isn't there the implication that the real numbers are in some sense, more real or more useful than the "imaginary" or complex numbers are?, although obviously, this is debatable. I think that the complex numbers and the quaternions have been shown to exist, to my satisfaction. But obviously at a higher level of abstraction than what primitive tribes might find useful. And I still want to know how many is i apples. I think we know how many is -1 apples. It means when you get an apple, you really have 0 apples, because 1 is owed to somebody else.
What about the saying that only the natural numbers actually exist? We never actually see an actual "half apple" in nature. And we can never be certain that both halves are equal, unless we carefully weigh them, which tells us that they are slightly unequal. Is "slightly unequal" even a thing in mathematics?
@Yosef MacGruber even the natural number doesn't all exists : all number greater than 10**85 is greater than the number of particule of universe.
For me, the best way to think about that is to say that number are an abstraction, that we have built in top of our intuition (at least for natural number, reals, ...). We have built these abstractions because the majority are useful for our understanding of the physic world, but there are not "real". I'm not sure that the concept of a number is actually real.
@miggaz elquez
If you want to talk of ultimate truth, or what is true, then we should look to the Bible. However, the Bible doesn't really delve very deeply into the mathematics of the "idle rich" because why should the productive peasants need to know much about advanced mathematics? According to the Bible, we have the tithe, which suggests if you want to look to the Bible as to what mathematics concepts much surely exist, that the numbers 1 through 10 must exist, along possibly with the rational numbers. Maybe not so much the irrational numbers, because who really needs to measure the diagonal distance of a 2 meter long square to very many decimal places? According to the Bible, numbers at least up to a million must exist. The KJV curiously says "thousands of millions" of descendants within Genesis 24 : 60, suggesting that numbers as large as the current "billions" of people of the world population, the word "billions" wasn't in the common usage of the people several centuries ago at the time of the King James Version translation. Also, we didn't have so much governments "printing" "trillions" of dollars in debt back then of their paper fiat "funny money". According to the Constitution and the Bible, gold and silver is what constitutes valid money.
Now sad to say, I think we still have a few problems in mathematics, sad to say, that most calculators seem to think that the answer to √(-1) = E which is a rather bizarre answer, because the set of reals is obviously incomplete, because mere normal algebraic operations can produce numbers outside the set of reals. We need to go to the complex numbers in order to have proper closure. Yet but only one calculator on my smart-phone understands that the complex numbers even exist? What gives?
BTW, I don't recall even one reference to complex numbers in the Bible. How "unreal" is that? Well perhaps so. Why would the peasants need to know about complex numbers? You would need to have more than one day a week of rest, to contemplate mathematics that advanced. Anyway, kind of what I am saying, is to be thankful that mathematics has more opened up to the common people, because not long ago, one would have to be kind of wealthy in order to justify spending much time with exploring mathematics. And so most calculators seem to be relegated to all the old boring stuff of basic arithmetic. Hey, if you can add, subtract, multiply, and divide, what more do you need? Maybe a 0? Really? Did many cultures not even have a 0 until several centuries ago? I find that hard to believe. How would you program a computer program instruction loop to do something 0 times? What if your list has 0 elements to sort? How then do you display or process it? Oh well, they didn't have much for computers back then anyway. And humans were smart enough to figure out, that if you have 0 money, it shouldn't take long to count it.
Save god “Dalamber test” for convergence
Whatching for the tenth time
The video is soooo relaxing..
Can you make korean sub?
I want to see this!! T^T
Azmidium - 2018-05-28
Even the universe has integer overflow :o
Tu rf - 2019-09-30
1 + 2 + 4 + 8 + ...
In binary it means
1 + 10 + 100 + 1000 + ...
Summing it in binary would yield 1111...
All ones
In 2's complement, all ones is considered as -1
So yeah, it's an overflow with an infinite bits integer
Harry R - 2019-10-14
1000th like!
badjonatan - 2019-10-19
An integer overflow would defy infinity. As far I imagine it is more like an underflow and for some reason universe does not allow the diapason of numbers between 0 and h(6.62607015×10^−34) It is maybe reserved for something else.
Shreerang Vaidya - 2019-12-10
2³² + 1 = -1
larry baby - 2020-01-09
...and here we see how many confuse matters of logic with matters of metaphysics and ontology.
Who told you people that numbers must or do actually number things?
Who told you that the existence of things somehow in need of being numbered is a sine qua non for number theory?
Why, in any possible worl;d, would logic, or that attenuation of logic called mathematics, be in any way contrained by the nature, essence or quantity of what there may happen to be?
Your confusions genuinely cause puzzlement.