What does it feel like to invent math?

  Рет қаралды 4,217,881

3Blue1Brown

3Blue1Brown

Күн бұрын

Пікірлер: 5 900
@scottanderson8167
@scottanderson8167 6 жыл бұрын
Whenever I invent math, my teacher marks it wrong.
@briansmith5391
@briansmith5391 4 жыл бұрын
You Too!!!
@nyanrome
@nyanrome 4 жыл бұрын
Lol!
@TechToppers
@TechToppers 4 жыл бұрын
Yeah! I wrote 5+5=15. She marked it wrong. I said that that it is in another base system...
@TechToppers
@TechToppers 4 жыл бұрын
@@রাফি-হ৭ঘ 😅😶🤦🏻‍♂️
@TechToppers
@TechToppers 4 жыл бұрын
@@রাফি-হ৭ঘ What is a bijective base?
@lucaslzt
@lucaslzt 6 жыл бұрын
"You are a mathematician [...], so you don't let the fact that something is nonsensical stop you" A true mathematician spirit
@angelmendez-rivera351
@angelmendez-rivera351 6 жыл бұрын
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
@子淇李-d2e
@子淇李-d2e 5 жыл бұрын
The three dots are where philosophers came in and driven mathematician crazy to dead
@taylorsmurphy
@taylorsmurphy 5 жыл бұрын
@ki kus Math fight!!
@arunjosephshadrach9539
@arunjosephshadrach9539 5 жыл бұрын
Yeah right,lose yourself into catchy words
@kyrlics6515
@kyrlics6515 5 жыл бұрын
@@angelmendez-rivera351 no
@nitinnilesh
@nitinnilesh 3 жыл бұрын
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.
@nenmaster5218
@nenmaster5218 3 жыл бұрын
My silly hobby is to recommend science-channel to my fellow science-fans. Mind?
@JuniperHatesTwitterlikeHandles
@JuniperHatesTwitterlikeHandles 2 жыл бұрын
It's true, I invented a perpetual motion machine last night while listening to this. I'm working on faster than light communication now.
@BadChess56
@BadChess56 2 жыл бұрын
@@JuniperHatesTwitterlikeHandles lol
@sloosh2188
@sloosh2188 Жыл бұрын
Geometric r=2> or = to 1 therefore diverges to +inf
@DefinitelyDrained
@DefinitelyDrained Жыл бұрын
@@JuniperHatesTwitterlikeHandles i know this might be a 'joke' (though I didn't laugh) but it's such a stupidity to compare anything 'faster' than the speed of light.
@P-7
@P-7 2 жыл бұрын
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 😊
@naturegirl1999
@naturegirl1999 2 жыл бұрын
Do you remember the problem? What was the paper called? Can you link it?
@naturegirl1999
@naturegirl1999 2 жыл бұрын
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
@P-7
@P-7 2 жыл бұрын
@@naturegirl1999 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.
@P-7
@P-7 2 жыл бұрын
@@naturegirl1999 It was about finding the shortest way to connect n points together. The paper is “Shortest Road Network Connecting Cities” by Université de Genève
@P-7
@P-7 2 жыл бұрын
@@hike8932 go ahead… we’re listening
@Azmidium
@Azmidium 6 жыл бұрын
Even the universe has integer overflow :o
@nicejungle
@nicejungle 6 жыл бұрын
gold comment :D
@ConnorR.mp3
@ConnorR.mp3 6 жыл бұрын
But with an infinite number of bits, this overflow can never occur!
@waltercomunello121
@waltercomunello121 6 жыл бұрын
Yet it never crashes - or softlocks.
@NoorquackerInd
@NoorquackerInd 6 жыл бұрын
Walter Comunello black hole = garbage collection of the universe that's why
@waltercomunello121
@waltercomunello121 6 жыл бұрын
@@NoorquackerInd never asked why. Nonetheless, Hawking radiation return part of the garbage they collect to the universe. Also, the concept of "garbage" relies on the concept of "mass" which could assume different levels of sense outside of our dimensions - black holes might peek in these extra dimensions.
@Mathologer
@Mathologer 9 жыл бұрын
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 :)
@3blue1brown
@3blue1brown 9 жыл бұрын
+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.
@Mathologer
@Mathologer 9 жыл бұрын
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.
@jacobkantor3886
@jacobkantor3886 9 жыл бұрын
+Mathologer Hey Its Mathologer! You guys should collaborate on videos.
@capjus
@capjus 5 жыл бұрын
+Mathologer I am happy you finally cleared this in your video!
@ijarbis187
@ijarbis187 5 жыл бұрын
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.
@pennrogers4963
@pennrogers4963 6 жыл бұрын
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.
@MuffinsAPlenty
@MuffinsAPlenty 6 жыл бұрын
Very good point, and I agree that this has tripped up a lot of commenters.
@StillnessMoving
@StillnessMoving 6 жыл бұрын
Thanks for that. An important point.
@GAPIntoTheGame
@GAPIntoTheGame 6 жыл бұрын
This is a great explanation that should clear many problems for ppl who have trouble understanding the video
@roberthardie6119
@roberthardie6119 5 жыл бұрын
Oh sugar,I'm negatively stupid!
@AlcyonEldara
@AlcyonEldara 5 жыл бұрын
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.
@TemperThetaDelta
@TemperThetaDelta Жыл бұрын
As a programmer the 2^n example is easy to answer: the infinite-precision integer storing whole numbers overflowed into the negatives
@devsquares
@devsquares Жыл бұрын
yeah
@damonpalovaara4211
@damonpalovaara4211 Жыл бұрын
The sum of 2^n written out in binary form is (11111111...) which is the twos-complement version of the number -1
@TemperThetaDelta
@TemperThetaDelta Жыл бұрын
@@damonpalovaara4211 exactly!
@JosueMartinez-ww1vj
@JosueMartinez-ww1vj Жыл бұрын
And 9 in binary is 1001 your index and small finger which makes a set of horns 🎸🤘
@jackposiedonforever7774
@jackposiedonforever7774 Жыл бұрын
This needs more likes 😂
@soniczdawun1
@soniczdawun1 9 жыл бұрын
This is undoubtedly becoming my favorite mathematics channel on KZbin. 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!
@imnecessaryevil3879
@imnecessaryevil3879 9 жыл бұрын
+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
@roygalaasen
@roygalaasen 9 жыл бұрын
+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"!
@OonHan
@OonHan 7 жыл бұрын
Agree
@locutusdborg126
@locutusdborg126 7 жыл бұрын
Carbon: Good one. I'm in recovery.
@JoelDowdell
@JoelDowdell 7 жыл бұрын
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.
@fqidz
@fqidz 5 жыл бұрын
How it feels to invent math 5 math, stimulate your senses
@paper2222
@paper2222 5 жыл бұрын
*simulate your equations
@Bleepbleepblorbus
@Bleepbleepblorbus 4 жыл бұрын
Math is the base of science do... being a expert mathematician and scientist means understanding everything.
@Bleepbleepblorbus
@Bleepbleepblorbus 4 жыл бұрын
Not saying I'm smart but that's what it feels like
@limepop340
@limepop340 4 жыл бұрын
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.
@Bleepbleepblorbus
@Bleepbleepblorbus 4 жыл бұрын
@@limepop340 well no doy.
@srinivasaprasannaa915
@srinivasaprasannaa915 4 жыл бұрын
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.
@bayurukmanajati1224
@bayurukmanajati1224 4 жыл бұрын
Teacher: What is 7 + 6? Me: 15. Teacher: What. It should be 13. Me: But my calculator says so. *Calculator set to Octal*
@madkirk7431
@madkirk7431 4 жыл бұрын
@@Aph2773 that's just not understanding that it's a joke
@AlexandrBorschchev
@AlexandrBorschchev 4 жыл бұрын
I understand its a joke but you have to say the base system u use otherwise people are gonna assume its base 10
@RockBrentwood
@RockBrentwood 4 жыл бұрын
*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?
@RockBrentwood
@RockBrentwood 4 жыл бұрын
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.
@shawon265
@shawon265 4 жыл бұрын
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!
@NazriB
@NazriB 2 жыл бұрын
Lies again? Support Indonesia Malaysia
@PolitictalDipsit
@PolitictalDipsit Жыл бұрын
​@@NazriBwho cares.
@valwaeselynck4529
@valwaeselynck4529 4 жыл бұрын
"You decide to humour the universe, ...", maybe the best phrase describing theoretical research.
@morantNO1
@morantNO1 3 жыл бұрын
And the best approach to life in general.
@avikdas4055
@avikdas4055 5 жыл бұрын
Expectation: Determined to fully understand a 3b1b video Reality: Facepalm
@subhrajitroy1477
@subhrajitroy1477 5 жыл бұрын
lmao haha....go and watch his video on conics...u will understand that
@Joe11Blue
@Joe11Blue 5 жыл бұрын
The video simply explains the reason why we use limits.
@aadityabhetuwal5990
@aadityabhetuwal5990 5 жыл бұрын
IDk 3b1b explains it pretty basic without going really advanced
@SYFTV1
@SYFTV1 4 жыл бұрын
×2 Ight imma head out to watch some MHJHB instead
@bacicinvatteneaca
@bacicinvatteneaca 4 жыл бұрын
@@SYFTV1 ok coomer
@RammusTheArmordillo
@RammusTheArmordillo 5 жыл бұрын
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
@valovanonym
@valovanonym 4 жыл бұрын
@Ron Maimon r/woooosh
@xumingyu2948
@xumingyu2948 4 жыл бұрын
@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.
@debblez
@debblez 4 жыл бұрын
@Ron Maimon 😳
@adamuhaddadi5332
@adamuhaddadi5332 3 жыл бұрын
@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
@osotanuki3359
@osotanuki3359 Жыл бұрын
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.
@leonardoabate2799
@leonardoabate2799 Жыл бұрын
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!
@sonialucy1
@sonialucy1 Жыл бұрын
Dang........ I feel dumb
@danishd5366
@danishd5366 Жыл бұрын
​@@leonardoabate2799qell a question in precalc probably means it was a geometric progression
@MakeMakeMake245
@MakeMakeMake245 Жыл бұрын
@@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
@leonardoabate2799
@leonardoabate2799 Жыл бұрын
@@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
@qqleq
@qqleq 5 жыл бұрын
I’ll stick with Crystal Math.
@akimobst8021
@akimobst8021 4 жыл бұрын
Adderall would be better
@azertyuiop432
@azertyuiop432 4 жыл бұрын
crystal meth
@mukeshbhatta7993
@mukeshbhatta7993 4 жыл бұрын
Love this comment
@willywalter6366
@willywalter6366 3 жыл бұрын
😂 LOL - thats how this felt! 👍🏻
@bigchungus2047
@bigchungus2047 3 жыл бұрын
@@azertyuiop432 woosh
@kcz6865
@kcz6865 4 жыл бұрын
Everybody: What was first chicken or egg? Mathematics: 1-1+1-1+...=1/2 so it was half egg and half chicken.
@elvisk6632
@elvisk6632 4 жыл бұрын
Actually yes? They both appeared???
@rotorblade9508
@rotorblade9508 4 жыл бұрын
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
@JaveLester
@JaveLester 4 жыл бұрын
The bird came into existence to lay egg.
@manjulakadali3996
@manjulakadali3996 4 жыл бұрын
@@rotorblade9508 exaclty, like, neither just popped out of thin air, it was a slow process
@luciproductions3294
@luciproductions3294 4 жыл бұрын
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
@cyclingcycles7953
@cyclingcycles7953 6 жыл бұрын
I can see the fabric of space-time now.
@matthewtodd9854
@matthewtodd9854 5 жыл бұрын
My profile picture is better ahahhahahahahah
@kyrlics6515
@kyrlics6515 5 жыл бұрын
@Just Cause silence before i bring Not Cause in here
@pixelpatter01
@pixelpatter01 4 жыл бұрын
It's the spice Navigator.
@behemoth2887
@behemoth2887 4 жыл бұрын
Ok
@Bleepbleepblorbus
@Bleepbleepblorbus 4 жыл бұрын
Oh yeah?I can see the entire beta-verse now
@Jaymac720
@Jaymac720 2 жыл бұрын
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.
@insouciantFox
@insouciantFox Жыл бұрын
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.
@TheRevAlokSingh
@TheRevAlokSingh Жыл бұрын
Lookup “hyperfinite” and “hyperreal numbers” for more
@jukmifggugghposer
@jukmifggugghposer Жыл бұрын
@@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.
@mcmonkey26
@mcmonkey26 Жыл бұрын
@@insouciantFoxbut .999… does equal one. they are the same. not just indistinguishable, not just effectively the same.
@GabriTell
@GabriTell Жыл бұрын
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).
@uumatter_0106
@uumatter_0106 5 жыл бұрын
Dude this didnt feel like he was doing math, it felt like he was doing meth
@anymaths
@anymaths 5 жыл бұрын
learn helpful maths from my maths videos.
@igorstrozzi
@igorstrozzi 4 жыл бұрын
but that's precisely how doing math feels
@chimmychonga4795
@chimmychonga4795 4 жыл бұрын
Easily mistaken
@SStupendous
@SStupendous 4 жыл бұрын
@@anymaths learning how to spell "COVID-19" with mathematical signs is not math.
@NStripleseven
@NStripleseven 4 жыл бұрын
Real math in a nutshell
@anaskarkout8871
@anaskarkout8871 6 жыл бұрын
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.
@missbond7345
@missbond7345 5 жыл бұрын
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.
@kosatochca
@kosatochca 5 жыл бұрын
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
@georger.2036
@georger.2036 5 жыл бұрын
This great explanation makes sense and should be incorporated into the video. The video fails to explain that concept which is important.
@5gonza541
@5gonza541 5 жыл бұрын
Jorge R. Agree
@tychophotiou6962
@tychophotiou6962 5 жыл бұрын
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!
@shubhamsharma-cp8te
@shubhamsharma-cp8te 4 жыл бұрын
This channel's production quality is better than Netflix
@ArIyan_yt
@ArIyan_yt 3 жыл бұрын
Gud grief
@harshkale9390
@harshkale9390 3 жыл бұрын
Hey 😊 indian guy
@maverickstclare3756
@maverickstclare3756 3 жыл бұрын
Netflix was founded by Marc Bernays Randolph - grand-nephew of Edward Bernays. Edward published the first book on how to create Propaganda in 1928.
@grossly820
@grossly820 3 жыл бұрын
@@maverickstclare3756 ok and?
@khalilghady2251
@khalilghady2251 3 жыл бұрын
Actually everything is better than Netflix
@pedroivog.s.6870
@pedroivog.s.6870 3 жыл бұрын
13:58 "Now this sum makes totally sense" Me, still stuck on why the powers of 2 are approaching to zero: O_o
@ChristAliveForevermore
@ChristAliveForevermore 2 жыл бұрын
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.
@user-dh8oi2mk4f
@user-dh8oi2mk4f Жыл бұрын
@@kcnl2522 That's a different part of the video
@Lamassu112
@Lamassu112 7 жыл бұрын
My engineering school destroyed my love for maths. KZbin restored it. ❤
@mr.clickable3899
@mr.clickable3899 7 жыл бұрын
was it univeristy or engineer technology school? that is big diffrence
@btdpro752
@btdpro752 6 жыл бұрын
SIMPLIEST COMMERCIAL really? What is it
@benjaminlavigne2272
@benjaminlavigne2272 6 жыл бұрын
it's not just KZbin. it's also the creators ! :D
@bigbox8992
@bigbox8992 6 жыл бұрын
There were math in times of war.
@rg5113
@rg5113 6 жыл бұрын
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?
@GeorgWilde
@GeorgWilde 7 жыл бұрын
10:50 Completely lost it. I have no idea what the rooms mean.
@arunjosephshadrach9539
@arunjosephshadrach9539 5 жыл бұрын
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)
@EebstertheGreat
@EebstertheGreat 5 жыл бұрын
@@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.
@scian8929
@scian8929 4 жыл бұрын
@@EebstertheGreat you are a god, I broke my brain a few times but I finally understood, thanks !
@Ochnokama
@Ochnokama 3 жыл бұрын
@@EebstertheGreat I was lost here too, thank you for your explanation !
@phscience797
@phscience797 7 жыл бұрын
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.'
@dieterklaus4717
@dieterklaus4717 5 жыл бұрын
PHScience this actually comes relatively close to what he is actually saying
@kjl3080
@kjl3080 4 жыл бұрын
Then, dividing has the wrong definitions since you can’t divide by zero
@alekisighl7599
@alekisighl7599 3 жыл бұрын
@@kjl3080 Or maybe zero has the wrong definition?
@mpperfidy
@mpperfidy 3 жыл бұрын
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.
@jesperdj
@jesperdj 8 жыл бұрын
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.
@ontoverse
@ontoverse 8 жыл бұрын
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.....
@potatopassingby
@potatopassingby 8 жыл бұрын
that immediately made me think of the theory that we live in a computer simulation.
@MrEvilNES
@MrEvilNES 5 жыл бұрын
It's like an infinite overflow xD
@ineednochannelyoutube5384
@ineednochannelyoutube5384 5 жыл бұрын
@@potatopassingby Of course it would look like we live in a compuiter simulation when you redefine the real number line into the one clmpuiters use...
@sorinpepelea
@sorinpepelea 4 жыл бұрын
Very practical.
@szymek1567
@szymek1567 9 жыл бұрын
I do not usually rate videos, nor taking comments, but this... I haven't seen such inspirational video on youtube for years!
@RedHairdo
@RedHairdo 9 жыл бұрын
+szymek1567 Indeed.
@AdvaitSaravade
@AdvaitSaravade 9 жыл бұрын
Quite true.
@wasp89898989
@wasp89898989 9 жыл бұрын
You're my new favorite KZbin channel. Please don't stop!
@Meow_yj
@Meow_yj 3 жыл бұрын
I'm grateful that I found your channel ! It makes math ideas look so beautiful and elegant. Especially linear algebra series.
@adelarscheidt
@adelarscheidt 8 жыл бұрын
You lost me at the sub-rooms...
@Eyevou
@Eyevou 8 жыл бұрын
Same here. Everything up to that point was fine then he started playing with the definition of distance...... my brain broke.
@NetAndyCz
@NetAndyCz 8 жыл бұрын
Yeah, me too, I have no idea how to make sense of that.
@aguuaaa
@aguuaaa 8 жыл бұрын
same bro..
@maxi0361
@maxi0361 8 жыл бұрын
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.
@Eyevou
@Eyevou 8 жыл бұрын
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.
@outmyskiessmarg460
@outmyskiessmarg460 8 жыл бұрын
I loved this video thoroughly and I understood none of it
@v3le
@v3le 7 жыл бұрын
wow! very impressive video i guess (but supposed to be educational)
@locutusdborg126
@locutusdborg126 7 жыл бұрын
Me three.
@CGoldthorpe
@CGoldthorpe 7 жыл бұрын
If you loved I assume you did not understand it!
@henryparker3420
@henryparker3420 7 жыл бұрын
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.
@csm5040
@csm5040 7 жыл бұрын
Henry Parker. Now I feel a little bit better xD
@philiphunt-bull5817
@philiphunt-bull5817 5 жыл бұрын
I don't get it... Like, any thing from the "rooms" part onwards.
@atimholt
@atimholt 5 жыл бұрын
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.
@madhououinkyoma
@madhououinkyoma 5 жыл бұрын
@@atimholt this is not helpful..
@nin10dorox
@nin10dorox 5 жыл бұрын
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.
@LynX2161
@LynX2161 4 жыл бұрын
I feel you bro, Here I am looking at the comments after he started talking about rooms
@user-en5vj6vr2u
@user-en5vj6vr2u 4 жыл бұрын
I guess I could have understood but I didn’t know the point
@theuser810
@theuser810 2 жыл бұрын
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
@codingforest7442
@codingforest7442 2 жыл бұрын
How 2S = 1.1111.... ?
@lmfao6125
@lmfao6125 2 жыл бұрын
@@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...
@theuser810
@theuser810 2 жыл бұрын
@@codingforest7442 In binary, multiplying by 2 shifts everything a digit left, like how multiplying by 10 does so in decimal.
@aksharasbhat6480
@aksharasbhat6480 2 жыл бұрын
@@theuser810 you should have multiplied by 10 (2 in binary)
@codingforest7442
@codingforest7442 2 жыл бұрын
@@theuser810 ok I got you now, thx.
@Stopitpls
@Stopitpls 6 жыл бұрын
You lost me at 1+2
@No_handle-_-
@No_handle-_- 6 жыл бұрын
😂😂
@Stopitpls
@Stopitpls 6 жыл бұрын
Topher TheTenth No...
@Magnus_Deus
@Magnus_Deus 6 жыл бұрын
It is 21
@Magnus_Deus
@Magnus_Deus 6 жыл бұрын
@DAVID MELLA no u they are reverse
@rajeshwarsharma1716
@rajeshwarsharma1716 5 жыл бұрын
I have difficulty in 1 plus 1.
@adityasankar274
@adityasankar274 6 жыл бұрын
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.
@Safwan.Hossain
@Safwan.Hossain 6 жыл бұрын
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.
@AyushKumar-oo2zu
@AyushKumar-oo2zu 6 жыл бұрын
The teachers were inspired by him
@fgorn
@fgorn 6 жыл бұрын
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.
@RieMUisthegoaT
@RieMUisthegoaT 5 жыл бұрын
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
@shreeganesh9962
@shreeganesh9962 5 жыл бұрын
Call him Grant.
@tomasouzaheuert
@tomasouzaheuert 5 жыл бұрын
10:40 that random volume increase was weird
@SYFTV1
@SYFTV1 4 жыл бұрын
it is for you to wake up
@tvboxdoscarvalhoslucascare3477
@tvboxdoscarvalhoslucascare3477 4 жыл бұрын
SYFTV1 mama aqui uiuiuiuiaiaia
@SYFTV1
@SYFTV1 4 жыл бұрын
@@tvboxdoscarvalhoslucascare3477 exacto v:
@fahrenheit2101
@fahrenheit2101 4 жыл бұрын
It's because this is where the video gets intense, so you need to concentrate.
@Monkeuyy
@Monkeuyy 4 жыл бұрын
Me having a sudden unexplainable urge to watch a math video at 2 am in the morning
@Samir_Zouaoui
@Samir_Zouaoui 3 жыл бұрын
i can relate .
@Firefly256
@Firefly256 3 жыл бұрын
Yup
@safwanshahriar4108
@safwanshahriar4108 3 жыл бұрын
I'm literally reading your comment at 2:03am.
@diatonicdissonance
@diatonicdissonance 3 жыл бұрын
as opposed to 2 am in the afternoon ?
@Xayuap
@Xayuap 3 жыл бұрын
@@diatonicdissonance 3:46
@XoIoRouge
@XoIoRouge 4 жыл бұрын
7:37 I read it in iambic -pentameter- trimeter and now I need a modernized Shakespearean play about mathematics.
@isavenewspapers8890
@isavenewspapers8890 Жыл бұрын
Wouldn't it be iambic trimeter?
@XoIoRouge
@XoIoRouge Жыл бұрын
​@@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.
@grainfrizz
@grainfrizz 8 жыл бұрын
Does this mean that as we approach infinity, the size of my laptop's RAM will be -1 gig?
@zekrinealfa1113
@zekrinealfa1113 8 жыл бұрын
no, because it is based on the number line, also talking about physical things makes no sense in this context
@grainfrizz
@grainfrizz 8 жыл бұрын
+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?
@zekrinealfa1113
@zekrinealfa1113 8 жыл бұрын
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
@zekrinealfa1113
@zekrinealfa1113 8 жыл бұрын
Not*, thing*
@jacobkrebs5026
@jacobkrebs5026 8 жыл бұрын
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
@satyenpandita6848
@satyenpandita6848 4 жыл бұрын
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.
@chandradoychatterje
@chandradoychatterje Жыл бұрын
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.
@samirkaushik8863
@samirkaushik8863 5 жыл бұрын
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.
@henryginn7490
@henryginn7490 5 жыл бұрын
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.
@1997CWR
@1997CWR 8 жыл бұрын
I like your Grahams Number reference!
@MarkGingrass
@MarkGingrass 7 жыл бұрын
I saw that too. Nice!
@theriversexitsense
@theriversexitsense 7 жыл бұрын
1997CWR i dont
@SinclairLocke
@SinclairLocke 7 жыл бұрын
Search Graham's Number Numberphile
@UltraLuigi2401
@UltraLuigi2401 7 жыл бұрын
g(g64) ahhh i just don't want to think about that
@apocryphonnill1180
@apocryphonnill1180 6 жыл бұрын
At 3:44
@madkirk7431
@madkirk7431 4 жыл бұрын
It feels like, as the inventor said, "OOGA BOOGA"
@minleyfox5231
@minleyfox5231 4 жыл бұрын
Is it you Patrick?
@mohit6925
@mohit6925 4 жыл бұрын
at 06:55 value of p is taken as -1 but in the derivation, we restricted p to be in between 0 to 1
@adrianordp
@adrianordp 2 жыл бұрын
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.
@Anmol_Sinha
@Anmol_Sinha Жыл бұрын
@@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.
@PaoloPolesana
@PaoloPolesana 7 жыл бұрын
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!!!
@Baseguy100
@Baseguy100 9 жыл бұрын
That g(g64) killed me... I was like oh yeah graham's number. Oh wait thats the number of g's...
@zuzusuperfly8363
@zuzusuperfly8363 9 жыл бұрын
+Kane Angelos Ouch, my mind.
@666unknowndevil666
@666unknowndevil666 9 жыл бұрын
+Zuzu Superfly I know right? The fact that is was in the denominator made it even more unfathomable to me.
@JamesSmith-ek1or
@JamesSmith-ek1or 9 жыл бұрын
+666unknowndevil666 it hurts to even try to comprehend it's vastness
@mike4ty4
@mike4ty4 8 жыл бұрын
+Logan Retamoza Or since it's in a denominator, its _tininess_...
@rbdoppler7841
@rbdoppler7841 8 жыл бұрын
+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.
@madkirk7431
@madkirk7431 4 жыл бұрын
3B1B: rooms Everybody: *visible confusion*
@thatguy3369
@thatguy3369 5 жыл бұрын
Then god said “let there be analysis”
@-Timur1214
@-Timur1214 5 жыл бұрын
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
@YasuoMidOnly
@YasuoMidOnly 5 жыл бұрын
Timur1214 oh no Im starting after christmas break, tgen this video popped up. Should I be scared
@-Timur1214
@-Timur1214 5 жыл бұрын
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
@thatguy3369
@thatguy3369 5 жыл бұрын
Donut be afraid just let the math gods guide you and everything should be trivial....
@anymaths
@anymaths 5 жыл бұрын
watch my maths videos to learn something.
@medunco
@medunco 6 жыл бұрын
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
@ptr_does_music7042
@ptr_does_music7042 2 жыл бұрын
God finally somebody tried to make sense of this. I'm looking at your comment again after I digest the video lol
@YuvrajBachira
@YuvrajBachira 2 жыл бұрын
Does this means we can't put p=2 as it was said that 0
@toolbgtools
@toolbgtools 2 жыл бұрын
actually point of video is, we got (1-p)+p(1-p)+...+p^n = 1 for 0
@Domo3000
@Domo3000 2 жыл бұрын
That's the whole point of this video. He himself says that the function only makes sense for values 0
@greenybeany8001
@greenybeany8001 2 жыл бұрын
In a way, once they reshuffle things into that 2-adic system, 0
@baltakatei
@baltakatei 6 жыл бұрын
Just casually dividing by (1-p), officer. Perfectly legitimate, I swear.
@YellowJelly13
@YellowJelly13 5 жыл бұрын
As long as p isn't 1 it's all fine!
@lorenzosarria3359
@lorenzosarria3359 5 жыл бұрын
5:49 Smaller than 1 Si its ok
@TheFerdi265
@TheFerdi265 6 жыл бұрын
I really love the way this "Generalizes" the Two's Complement to an "infinite" number of bits.
@NicholasMarshall
@NicholasMarshall 4 жыл бұрын
I had difficulty distinguishing between the colors of the boxes. I found the topic interesting, and will be reading more.
@marksmod
@marksmod 8 жыл бұрын
scratches the surface of the tip of the iceberg floating in the sea of secrets on an alien planet
@Ricocossa1
@Ricocossa1 3 жыл бұрын
I'd never noticed the poem at 7:41. It's lovely! :D
@AntiCitizenX
@AntiCitizenX 8 жыл бұрын
Math is all about making up rules and definitions, then following them to their logical conclusions.
@DuckRiceFarmer
@DuckRiceFarmer 7 жыл бұрын
AntiCitizenX Beautifully expressed
@thecomicindex4673
@thecomicindex4673 7 жыл бұрын
Calculus can help innovate! here's a simple derivatives vid (easy) kzbin.info/www/bejne/epibeZqBf9msn68
@OH2az2
@OH2az2 7 жыл бұрын
Philosophy, too.
@Drigger95
@Drigger95 6 жыл бұрын
Nice, you you're an anti-realist? So that means the KCA is correct? Checkmate.
@SupremeError
@SupremeError 6 жыл бұрын
And the absolute most beautiful part of it is those logical conclusions NEVER end up wrong when we fact check them in real life. That is in no way trivial and tells me that there is some solid, grounding logic governing our universe.
@GlorifiedTruth
@GlorifiedTruth 9 жыл бұрын
This popped up a zillion times in my "recommended for you" list. Finally I relented and watched. I am very glad I did. You have a great way of explaining... who knows, with profs like you, I might have gone a little further in higher math.
@LARAUJO_0
@LARAUJO_0 3 жыл бұрын
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.
@xot9897
@xot9897 4 жыл бұрын
I was fine until he started on about rooms 😭
@pedroivog.s.6870
@pedroivog.s.6870 3 жыл бұрын
You're not alone.
@bjordsvennson2726
@bjordsvennson2726 3 жыл бұрын
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.
@rangerwickett
@rangerwickett 3 жыл бұрын
@@bjordsvennson2726 So did he arbitrarily choose what numbers go into which rooms? I don't understand why he put the numbers where he did.
@bjordsvennson2726
@bjordsvennson2726 3 жыл бұрын
@@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.
@limenpepper
@limenpepper 3 жыл бұрын
@@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.
@chrvberg
@chrvberg 5 жыл бұрын
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.
@Pablo360able
@Pablo360able 4 жыл бұрын
But as the number of bits approaches infinity, the summation which adds to -1 approaches 1+2+…+2^∞
@ptr_does_music7042
@ptr_does_music7042 2 жыл бұрын
No..in two's complement the largest term is NEGATIVE (it represents -2^(n-1)) that's why the total sum can be -1.
@mandolinic
@mandolinic 9 жыл бұрын
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.
@3blue1brown
@3blue1brown 9 жыл бұрын
+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.
@RedHairdo
@RedHairdo 9 жыл бұрын
Mind = Blown
@kikones34
@kikones34 9 жыл бұрын
+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.
@ArnabAnimeshDas
@ArnabAnimeshDas 9 жыл бұрын
Nice observation you got there
@douggwyn9656
@douggwyn9656 7 жыл бұрын
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.
@ДенисКолесник-й6ш
@ДенисКолесник-й6ш 5 жыл бұрын
I cant understand, when you sad "p must be 0
@wessydafall
@wessydafall 5 жыл бұрын
I agree. That is where I lost confidence in this "proof"
@MuffinsAPlenty
@MuffinsAPlenty 5 жыл бұрын
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.
@VikeingBlade
@VikeingBlade 5 жыл бұрын
That's the point -- what if it *did* make sense for p > 1 or p < 0 ?
@illyon1092
@illyon1092 5 жыл бұрын
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.
@florinburian7291
@florinburian7291 5 жыл бұрын
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
@grapesalt
@grapesalt 3 жыл бұрын
This whole video summarized in one sentence 'I don't need sleep, I need answers.'
@frenzscivola3099
@frenzscivola3099 8 жыл бұрын
BEST CHANNEL EVER
@muzammilhalim3188
@muzammilhalim3188 8 жыл бұрын
ditto
@BobbyAlter
@BobbyAlter 8 жыл бұрын
ditto
@quarkyquasar893
@quarkyquasar893 8 жыл бұрын
Pikachu
@zacksima8333
@zacksima8333 5 жыл бұрын
Yes
@lorenzosarria3359
@lorenzosarria3359 5 жыл бұрын
When I saw 1+2+4...=0, my idea was thinking of powers of 2 as being how many times you can divide by a power of 2 and get an integer, and the equation simplifies to 2^infinity=0, which makes sense, because 0 is the only number you can divide by 2 infinitely and always get an integer
@no-body-nobody
@no-body-nobody 4 жыл бұрын
woah big pp confirmed
@Mathologer
@Mathologer 9 жыл бұрын
Great video!
@kurumi394
@kurumi394 7 жыл бұрын
Oh hi Mathologer! I watched your video about this topic as well, great video!
@Liridiona
@Liridiona 6 жыл бұрын
Lol you said 1-1+1.... does Not equal 1/2
@kairostimeYT
@kairostimeYT 6 жыл бұрын
1. Well, he is right in that case. However, it's super-sum is 1/2 though. 2. How does that make you laugh out loud? @Lirie Aliu
@Liridiona
@Liridiona 6 жыл бұрын
lol it is just a habit sorry
@Liridiona
@Liridiona 6 жыл бұрын
I man't to say that saying "lol" is just a habit
@kelvin31272
@kelvin31272 9 ай бұрын
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!
@wildbillgaming6596
@wildbillgaming6596 9 жыл бұрын
This is the best math video I have ever seen on the internet!
@jaafars.mahdawi6911
@jaafars.mahdawi6911 Жыл бұрын
i just can't stop coming back and appreciating this masterpiece!
@sketch4363
@sketch4363 2 жыл бұрын
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
@Т1000-м1и
@Т1000-м1и 3 жыл бұрын
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.
@nenmaster5218
@nenmaster5218 3 жыл бұрын
My silly hobby is to recommend science-channel to my fellow science-fans. Mind?
@Tomyb15
@Tomyb15 8 жыл бұрын
I wonder how many other useful things in math are waiting to be discovered/invented but may be never found because our common sense doesn't let us see in these very abstract ways.
@macpr0c
@macpr0c 6 жыл бұрын
I think you leave your common sense when you first enter a calculus class :P I wonder how mathematicians actually approach creating new theorems where do you draw the line of absolute absurdity and brilliant creativity
@EraLily
@EraLily 6 жыл бұрын
Common sense? Normal people have that?
@PanDiaxik
@PanDiaxik 2 жыл бұрын
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.
@Alfetto8
@Alfetto8 6 жыл бұрын
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 :)
@AshrafAlHaqq
@AshrafAlHaqq 5 жыл бұрын
What did u get now
@vko7059
@vko7059 2 жыл бұрын
One of the best videos on the internet.
@ilanpi
@ilanpi 5 жыл бұрын
I think you have missed a most important point: You missed the physical application, what I call the non-Archimedean Zeno Paradox. In the Zeno paradox, Achilles chases a turtle going 10 times slower than him, and catches him by summing the infinite series. In the non-Archimedean version, it is the turtle that is chasing Achilles and the turtle also catches him, but summing the infinite series as in the video. What happens is that by "going through the singularity" time is reversed. See www.lix.polytechnique.fr/Labo/Ilan.Vardi/zeno.html This shows that such formal methods are physically useful. An example is the "functional determinant", where the definition is done by analytic continuation. This is applied to String Theory.
@anymaths
@anymaths 5 жыл бұрын
watch my maths tricks.
@olbluelips
@olbluelips Жыл бұрын
I've never seen that, pretty cool
@ishaanlohani5326
@ishaanlohani5326 4 жыл бұрын
3:57 As a first time endeavour, is actually incredibly clever... -3blue1brown
@Abhi-jd5tn
@Abhi-jd5tn 4 жыл бұрын
holy shots i gotta note that down
@harshkale9390
@harshkale9390 3 жыл бұрын
Holy shitt 😂 You should j
@samuelgarbergs8901
@samuelgarbergs8901 5 жыл бұрын
The sum of all powers of two also equals - 1 in signed binary numbers
@samuelgarbergs8901
@samuelgarbergs8901 5 жыл бұрын
@@nycki93 cool
@Onaterdem
@Onaterdem 5 жыл бұрын
@@nycki93 Python is a Godsend
@loganrussell48
@loganrussell48 5 жыл бұрын
@@nycki93 Most other major languages have incorporated similar implementations, such as BigInteger in Java
@parabirb
@parabirb 4 жыл бұрын
@@loganrussell48 don't forget BigInt in JS
@smyrnianlink
@smyrnianlink 4 жыл бұрын
@@parabirb COBOL had that in 50s . It is called the decimal data type.
@RH-ro3sg
@RH-ro3sg 4 жыл бұрын
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.
@Jellylamps
@Jellylamps 5 жыл бұрын
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.
@ku5h5
@ku5h5 5 жыл бұрын
Dude no joke I was thinking the same way.
@matthewryan9283
@matthewryan9283 5 жыл бұрын
But where would negative numbers be? Would they be in the same spot as infinity-1 infinity-2 ect? Because then you're saying that there's an end to your circle? It's a cool idea though.
@Jellylamps
@Jellylamps 5 жыл бұрын
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
@matthewryan9283
@matthewryan9283 5 жыл бұрын
Yeah I know what you mean, its a cool concept. There are some nice extensions in model theory.
@shashwat1330
@shashwat1330 5 жыл бұрын
Such a line is very similar to Real projective line, but real projective line does not have any metric
@altus1253
@altus1253 9 жыл бұрын
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?
@descore7578
@descore7578 9 жыл бұрын
+Altus Boren Right.
@KedraIrke
@KedraIrke 9 жыл бұрын
+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.
@altus1253
@altus1253 9 жыл бұрын
+KedraIrke right, 0.33 is not the same as 1/3. Instead, I was writing "0.33..." to signify "0.33 reoccurring".
@drdca8263
@drdca8263 9 жыл бұрын
+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?)
@ugurcansayan
@ugurcansayan 8 жыл бұрын
+KedraIrke 1 - 0,(9) = 1 - 0,999... = 0, 000... = 0 1 - 0,(9) = 0 1 = 0,(9)
@mr.cheese5697
@mr.cheese5697 3 жыл бұрын
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.
@AkshatSinghania
@AkshatSinghania 3 жыл бұрын
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 :))
@lagduck2209
@lagduck2209 8 жыл бұрын
using p-adic numpers, I've discovered there actually exists cardinality between N0 (aleph-null) and N0pow2(aleph-one) (and it's actually (phi)pow2< clearifying it's connetction to ultimate phibonacci's pattern). I have really elegant proof for it, but it's too large to fit in this comment.
@lagduck2209
@lagduck2209 8 жыл бұрын
(actually, I meant N0, 2 pow N0 and phi pow N0)
@lagduck2209
@lagduck2209 8 жыл бұрын
(so Continuum hypothesis is false)
@FraktaleFatalitaet
@FraktaleFatalitaet 8 жыл бұрын
Could you please somehow put that proof online?
@lagduck2209
@lagduck2209 8 жыл бұрын
I'll try; that proof is actually in russian and in form of 7 hand-written draft-like pages; need to formalize it and bring to well-founded form.
@lagduck2209
@lagduck2209 8 жыл бұрын
until then, it's no more than a Fermat's joke about his great theorem
@kephalopod3054
@kephalopod3054 3 жыл бұрын
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.
@Spy_Complex
@Spy_Complex 8 жыл бұрын
Contentwise a good video - if you could improve your mic quality it would be perfect
@3blue1brown
@3blue1brown 8 жыл бұрын
+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.
@aryann2525
@aryann2525 8 жыл бұрын
Your doing a great work buddy !
@bored_person
@bored_person 8 жыл бұрын
+3Blue1Brown I would also recommend some acting lessons and/or voice training.
@pgc6290
@pgc6290 10 ай бұрын
At 9:35 but i liked his animation of numbers changing just a shot before than just putting changed numbers directly but a bang//ie cut.
@GabriTell
@GabriTell Жыл бұрын
I love that there're people who can't assimilate that there are infinite sums but got no problem assimilating that we can multiply a number "π" times itself.
@guycomments
@guycomments 5 жыл бұрын
This isn't just about math. Your diagram at the end is fundamentally how we interface with reality as such, in all fields and practices.
@nenmaster5218
@nenmaster5218 3 жыл бұрын
Hello and Aloha! My silly hobby is to recommend science-channel to my fellow science-fans. Mind?
@ASLUHLUHC3
@ASLUHLUHC3 6 жыл бұрын
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
@ABera-bm7ns
@ABera-bm7ns 2 жыл бұрын
This needs way more likes. Most underrated video in 3Blue1Brown
@harikishore2514
@harikishore2514 6 жыл бұрын
6:23 you can not substitute all integers when derive it using 1>p>0.
@dankazmarek1259
@dankazmarek1259 4 жыл бұрын
why not? p means a fraction and at the base of this guy, there can be any no. 2, 4, 5, 6, pi, tau everything is it not so?
@csvegso
@csvegso 6 жыл бұрын
I am just wondering how did this "room based" distance definition come to his mind. Human mind is so amazing ...
@antoniusnies-komponistpian2172
@antoniusnies-komponistpian2172 2 жыл бұрын
I thought you would talk about residual class rings, but p-adic numbers seem even more exciting.
@johnchessant3012
@johnchessant3012 3 жыл бұрын
4:45 Answers to the infinite series problems: geometric series: sum(1/3^n) = 3/2 subject of this video: sum(2^n) = undefined (-1 if working in 2-adic) Basel problem: sum(1/n^2) = pi^2/6 log(1+x) Maclaurin series: sum((-1)^n/n) = -log(2) cos(x) Maclaurin series: sum((-1)^n/(2n)!) = cos(1) Ramanujan's pi formula: sum((2sqrt(2)/99^2)((4n)!/(n!)^4)((26390n+1103)/396^(4n))) = 1/pi [Notes: Starting index should be 1 for the -log(2) one. There was a "k" instead of "n" on the Ramanujan one.]
@reetomghosh2611
@reetomghosh2611 6 жыл бұрын
@7:02 how can u replace 2 in that function? It's domain is limited within 0&1!
Hilbert's Curve: Is infinite math useful?
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