i will now write all my texts and articles with the following notation: :) signifies the end of a proof :/ signifies a question :( signifies a limitation in the experiment edit: :o signifies a theorem or lemma
his tones are off though, but i never spoke mandarin in a colloquial setting so maybe there's something about when it's ok to change tones in certain contexts.
12:00 this is why Grant is among the best maths educators out there. He talks about the emotion in maths, about stressing what's important vs the detail. It explains why the first time I properly intuited linear independence was after looking at his Linear algebra series. Some people will argue you can't visualise all maths but I think there's a happy medium between slavishly following symbols on a page and characterising what they represent more fully. This is why Grant is a bit of a hero for me.
Well just to tell as a fact , here in India , textbooks do have these kind of emojis , more like characters as whole in textbooks interacting with children and trying to tell them the concepts like a story or something through the textbook . I remember here in my state board textbook (local educational board's published textbook ) , there was a Rubik's cube interacting with us asking us questions or giving us a recap of the previous chapter at the start of a new chapter and also giving us some tips , they put him into corners and it was interesting.
I would recommend Jiří Matoušek's "Thirty-three miniatures". It's about 33 beautiful applications of linear algebra. Also, all textbooks by him that I checked out are gold.
When talking about natural talent, I think about crafting. When I have just started teaching someone to knit or crochet, at some point, after they've gotten just good enough to reliably make stitches, they'll watch me do the same craft and say something like "gosh you're so fast and I'm so slow, I wish I was fast". And when it's about crafting, the response to this is actually really intuitive. I say "well, I've been doing this about two decades longer than you have, so I should be faster; if I wasn't, there would be something going rather wrong".
Being a practicing engineer, I'm really interested in theory of nonlinear dynamics and mathematics behind. The Nonlinear Dynamics Book will be a real help. Thanks very much, Grant! ❤
I am really enjoying these videos on this second channel. Great to hear conversations and interviews that give more depth behind the characters in mathematics. On the nature vs nurture aspect. I really do feel that there is an in-built ability to engage with mathematics in some people. I know that I saw friends with whom I performed on level with in undergrad really outpace me in graduate courses. It really showed me there was a difference in ability. There is a spectrum of human ability and Tao and Von Neumann are really at the peak of this. If there are people at the top and the bottom, there are going to be people at every point in between. Of course, this is really hard to quantify, and I don’t think there is value in approaching education this way. I think most topics taught in school should be approachable to everyone, but after a point, you see real differentiation between students.
@@GrantSanderson yeah, it really is hard to characterize and identify. Though it is undeniable with people like Gauss, where they comprehended arithmetic at 3 years old. But that may be apocryphal too.
that question about not being gifted enough was hard to hear. coming from a competition math background, i often felt that the community could be very toxic and make me feel like i was not "gifted" enough
Competition math can be really discouraging for students who want to get into math but can’t “keep up” with everyone else. I never even thought that I could ponder problems for a day or more. Everything was kinda fast-paced. It’s funny that continuity can work perfectly fine only using rational numbers. Talk about lifting up your pencil! Oh and that room problem, I didn’t find that easy at all lol. Although, convex functions ARE quite interesting *cough* *cough* Legendre transform. Love your content! Can’t wait for more on diff eqs and probability
the room problem really isn't that hard. click read more if you want a hint (to find the solution I came up with) think about the room with the smallest number of people.
For the room problem: a "physical/intuitive" interpretation of the sum-of-squares is as a weighted count of the population, where each person is weighted by the number of their roommates.
Can’t help but mention that it can also be viewed as the variance of a distribution + the mean squared, where in this case the mean is constant, so it basically reflects the variance.
6:47 Regarding the concept of being gifted vs. not: I've always held to the belief that there *are* people who are better in a given field by their nature, but *anyone* can become skilled through effort. Anyone can become skilled, but some people have a head start. Regarding math specifically, yes. There are individuals who are naturally good at it; their brains are wired in a way that makes learning mathematics natural. These are the individuals we'd typically call "gifted." For others, learning math is harder. The ""not gifted"" individuals. It's very easy to see someone breeze through things that you struggle with and come to the conclusion that you're just not gifted enough to progress, but that's categorically untrue. Just because it takes more effort for you to learn something doesn't mean you can't. If you're willing to put your head down and push through, you can learn any skill you want. Whether you're gifted or not determines where you start, not where you end up.
I'm not sure it's true that _anyone_ can become skilled at math just by putting in more effort. Frankly, I don't think that's true of _anything._ But I agree that there are most likely many people in the world who could become skilled at math, but are never given a chance or a reason to put in enough work.
I bet whoever came up with that sum of squares solution had some connection to Cambridge. They used to grade their exams by taking the sum of squares of the scores to each question, to encourage people to prioritise answering fewer questions more completely rather than doing a lot of questions partially. As you say, problem solving is a lot about your experience....
I've always enjoyed math and physics. The one thing I have learned over time is that numbers, math and all that follows they are nothing more than a product of the mind as they are all conceptual ideas. At the end of the day, math is nothing more than a model and it is what you make of it! I enjoy hearing all different kinds of perspectives and approaches. 3Blue1Brown is one of my favorites along with Mathologer and a few others... I love their presentations and visualization methods. It's almost like comparing or listening to a monotone college professor who'd soon enough put you to sleep on trying to explain the phenomenons of the Mandelbrot Set as opposed to writing a simple computer program to graph it visually being able to see and interact with the infinite patterns and levels of details that a simple equation produces. Grant and others like him allow mathematics to be fun and engaging instead of being monotonous, drab and dreary... Keep up the great work!
5:00 Yes yes yes yes, I am obsessed with Valadmir Arnold's writings, I have been collecting his works in English but most of his writings are in Russian. Currently learning a lot from his book on Catastrophe Theory. I feel VI Arnold changed my life, his essays on mathematical pedagogy are revolutionary and I feel you are the one who will make his dream come true of unifying maths of physics which were divorced by the Evil Bourbaki's.
Lol don’t let arnold brainwash you with his nonsensical opinions the bourbaki’s did a great job and the existence of Pure mathematics is super convenient and useful
As for the continuity example. I think the most discrete answer would be a function is continuous at a given input when both of its limits exist and are equal or converge to the same value. Take sin(t) for example, no matter what input you use within this function both the left and right hand limits exist and are equal. Now, take the tangent function and examine its left and right hand limits at tan(PI/2) or tan(45) and other multiples of them. Both limits exist, yet they are not equal. The limit approach towards the right tends to +infinity and the limit approach towards the left tends to -infinity. Here we can see that the sine and cosine functions are both continuous where the tangent, cotangent, secant and cosecant are not continuous for all of their domains. The challenging thing here is that we are typically taught about continuity with respect to functions and graphs before we are ever introduced to the concepts of limits. Yet I think the best proof and representation of continuity lies within the properties of a functions limits across all of its domain.
This is full of valuable suggestions. Just "the convexity of x squared" at 21:20 led me to an answer to a question that's bothered me for years, namely: "Why do we nearly always use & teach Standard Deviation, rather than MAD or AAD?" Not bad for a 5 second toss off remark!
I would say that as long as you've done a reasonable amount of preparation then the fault pretty much always lies with the textbook. One of many examples of this is I once supported ten different students through the exact same maths course. 7 had textbook A - Topic X was impenetrable, only ever to be understood by the best and the brightest. Topic Y was completely straightforward and I was never asked about it. 3 had textbook B - Topic X was completely straightforward and I was never asked about it. Topic Y was impenetrable, only ever to be understood by the best and the brightest. I taught Topic X to those with textbook A, and Topic Y to those with textbook B, made those topics seem completely straightforward and everyone thought I was a genius.
The Hubbard text is rather good for all the topics covered. I think the book is especially great for those who have studied already Linear Algebra, Real Analysis, and Multivariable Calculus and are in need of a more in depth review. We used that book for Math 23a & b with Paul Bamberg at Harvard. When I took the class, we used that very same 4th edition.
2:25 woah! Grant, in your interview with Steven Strogatz, he mentioned teaching at a math program at Hampshire College (HCSSiM). I just realized today that turns out this quote from Erdos was actually in 1985 when he visited HCSSiM and was giving a talk. "yellow pig camp" :)
Recently read Mathematical Methods of Classical Mechanics by Vladimir Arnold and it's probably my favourite book. Super cool topics and ingeneous presentation, just a beautiful textbook.
You mentioned that it would be great if there were emojis in the margins of textbooks to help with understanding. In the Park City Math Institute (PCMI) problem sets used in their math teacher seminars, there are those sorts of comments (sadly, no emojis) in the margins. Sometimes they’re there for humor, but other comments are informative. I really like them!
Besides everything good about Grant Sanderson, he's also got the best diction/ voice on the internet.. or anywhere! I can listen to him for hours on end.
I've just realized the reason why the channel is called 3b1b - Grant's right eye has a mix of brown and blue. It's amazing how this detail only struck me today!
Judging over gifted thing can sure demotivate you, but practising and learning perquisite can sure bring surprising change to learn math. Also, I guess visualisation really helps to understand what it meant instead mere wandering in head, though it will not always the case as context , reality and drawing are critical to relate all the time as so many known and unknown parameters we have to dealt with.
I took some Chinese back in high school, and spent a couple weeks in Beijing. Sadly I've forgotten most. This was a greeting a translator handed to me, likewise for the outro.
A different way to think about the room problem: for each person, count how many people they share a room with (including themselves) and add all of those numbers up. It's easy to see that the result is the sum of squares of the people in each room. But the number increases for the person moving rooms, it increases by 1 in the people of the room they're moving into and decreases by 1 for the people in the room they're moving out of. By the fact that there are more people in the former, the result is a net increase in the total.
I find it interesting that your channel reached 1M subscribers just after Steve Mould's and ahead of Matt Parker's (which hasn't happened yet, he's still around 900K). But all in all it's a great win for STEM.
His main TH-cam is at 3.85M right now, this video celebrates the Chinese channel (on BiliBili) reaching 1M! The rest of the translated channels have between a few hundred and a tens of thousands subs each.
I don't believe in the philosophy that frustration is apart of your interest. The reason is that I have found plenty of scenarios where reading a book has been less or not at all frustrating compared to jumping blindly into a project. I enjoyed the not at all frustrating approach and I learned more from that.
You can easily add one plus one is two. But real mechanical trace for that is a different story. There are so many ways but there is no proof that there is only one way to solve a problem. Though the idea is the same result. Because source and destination are same for every problem. Visualisation is the different forms of how human brains work. Almost 90% of mathematical solutions lie on filling the gaps. For example a random numbers sequence to an orderly is filling the gaps to regular.
Dear Grant, The one-third of Linear Algebra of the Hubbard and Hubbard book is a great resource of LA but is rather short. And the longer, more elaborate textbooks of Linear Algebra that I could find were unintuitive(Like Axler, Strang, etc). Books like Hoffman Kunze and Halmos seem to be at a higher level than what I currently am at. Are there other detailed textbooks of Linear Algebra that teach the subject primarily as the study of *linear transformations* rather than the study of mathematical objects consisting of a bunch of numbers arranged in an array and weird operations on those objects like multiplication whose rules kind of seem arbitrary, pointless and unmotivated.
I just took a college course called "abstract linear algebra" and we used the first 4 chapters of a book called "linear algebra done wrong." Rather than just starting with the algorthims on vectors and matricies, the book starts from the most abstract definition of a vector space, then looks at transformations between vector spaces, then shows exactly why matricies characterize linear transformations. The book then uses that as a jumping off point to learn the algorthims and stuff like row reduction, rank, and eigenvalues. Based on your description, it sounds like it's what you are looking for
I'd recommend Axler but it sounds like you already tried him. He's all about emphasizing linear transformations over matrices. What did you find unintuitive about his book?
@@martinepstein9826 I like that Axler's book tells us abstract definitions rather than focusing on computations. But my problem with the book is that it feels less motivated. Throughout the book, all sorts of operations and mathematical objects with their different properties are introduced. However, we are never told once why mathematicians invented that concept (I am not so concerned with real life applications but I really want to know why mathematicians defined things in one way rather than the other. Surely, all of the definitions in the book were later formalizations of intuitive ideas of mathematicians. Those ideas that motivated them to define those objects with those properties in *that specific way and not any other* is what I want to know so that I can appreciate what the definition actually stands for). And Axler's book while being abstract, is not intuitive or motivated (atleast to me).
This video came up in my feed at EXACTLY the right moment. Just a couple of hours ago, I finally decided that the vexation of a pop-math book I was trying to learn from was too much for me to deal with, and it's time for me to move on, so as not stymie my progress. (I've always struggled with math since my algebra days, but aced logic, and have managed to teach myself enough of the relevant math for my physics to not fall too far behind.) I'm not going to say the book was 'bad'. Just that it's way of presenting the information--some stuff about a family of 6 boys, 6 girls, a mom and dad, and trying to see how many dance pairs the siblings can make--was not helpful at all. But at least I learned what FACTORIAL is!
"...because Pi has legs." I was not expecting this; I always assumed it was related to the circular nature of the 3blue1brown logo. At any rate, the Pi creatures are a great lure to catch the eye of casual TH-cam viewers who might not realize they have an interest in math. "Come for the cute critters; stay for the learning."
My recommendation: "Elements" from Euklid. First learn the basics and learn to think mathematically, then the rest is easy. It's like jumping in a pool. First, the water is just cold. After a while, it makes really fun.
Hello Grant. I am immensely benefiting from your videos. Can you please make a video where you could maybe show a roadman for self learners of mathematics? Your favorite books?
I think I have a problem. I dig all the axioms and definitions when I start reading about a branch of mathematics but I don't really like the how all this is usefull part that comes after.
07:49 There's absolutely no innate gift factor in any area, taking Von Neumann as an example: he was the child of a banker and he was instructed on all sort of things by governess so that between the age of 6 and 8 he knew multiple language, knew how to divide two eight-digit numbers in his head, knew to speak in ancient greek and from 8 onwards he learned about differential and integral calculus, by twelve he understood the Borel's Théorie des Fonctions. He wasn't born like that, he was meticulously crafted by his father, the study at home, the stimulating environment, he didn't went to public school, he had several figures available at all times to satisfy his curiosities, curiosities that all children will have if grown up in the right environment, he was nothing special, everybody at his place would've become a stunning polymath. So everybody can feed properly his brain and become a polymath, it's just that public schools and society in general is build on top of a hyper sectoral criteria within which everybody got to do one stuff and do it good, you don't have time for being curios in a average family, Von Neumann's family wasn't an average one so that Neumann itself became a non average mind, I know that is hard even to start something in certain age automatically comparing yourself with people like this but all you have to do is to compare yourself of today with yourself of yesterday and break any boundaries, don't let things weight you down never and ever, remember that the brain is the same for everybody is just like a muscle but stunning complicated, when people analysed Einstein brain noticed that the difference wasn't in the neurons count or divergence itself but in the glia, he had a gigantic macro convoy of glia, the stuff that carry axons around the brain and microglia that weld neurons when new connections are always proposed at the brain, just like a bodybuilder have giant strange muscle. It's not too late, just give all you can and you can't fail, believe in yourself and if no one ever believed in you just believe it on behalf of all of them and obviously you need to work extremely hard. P.S. thank for all the great content
There are thousands of kids who grew in similar environment as von Neumann but there is only one von neumann. You speak that how brain is like muscle and can be trained but I think you are not aware of the fact that every body does not have equal athletic talent. Athletic talent is genetic and innate and now research has even identified some specific genes associated with better athletic performance . People are genetically different and some people just have better muscle potential than others and this is similarly true for intelligence too. Some people are just naturally more intelligent than others because of genetics.
Honestly, I feel sad that I have learnt math in a way that I can write proofs, solve problems but not yet learnt the way that I can recite proofs in a poetic way (like your videos), in a simple- sleak, clear way., like a story. I am inspiredby your videos, 2 days from now, I have to give a seminar on Fourier series. And I will try my best to make it motivating, that will make audience interested in its maths. I was going to lecture like dumping the theory infront of them, but then I saw your video and I clearly see what's smelly, unattractive and whats has a nice fragrance, makes us wonder. Your videos have opened a closed gate that only allowed theory to pass, now I am interested in simple exxplanation. I know this leap will be difficult for me, to be patient, read and write in my own words, but I want to promise that I beleive its going to be rewarding, I would love it. I wont give up. Thankyou so much.
The problem at 19:00 really looks like some sort of inverse of entropy since entropy is a time dependent phenomina I think this is a very interesting problem.
Grant giving a student friendly perspective on Mathematics, is probably on of the best things on TH-cam. What I am missing in the book recommendations list is a discrete mathematics book, is there any good textbook about the subject for autodidacts?
1) John H. Hubbard ( Vector Calculus, Linear Algebra, and Differential forms ) 2) Steven H. Strongatz ( Non-Linear Dynamics and Chaos ) 3) Martin Aigner Gûnter M Ziegler ( Proofs from The Book ) 4) J. Michael Steele ( The Cauchy - Schwarz Master Class )
Was not expecting that at the start 😂
台湾！
To any of you wondering, the outro (下次一定) is a common Chinese phrase/meme to say "maybe next time" or "take a rain check".
Basically never lol
下次亿定
Yes, great pun, wasn’t expecting that!
Like the mexican "Ahorita!" ?
'next time for sure'
i will now write all my texts and articles with the following notation:
:) signifies the end of a proof
:/ signifies a question
:( signifies a limitation in the experiment
edit: :o signifies a theorem or lemma
god dang it I edited and lost the heart
I really like your notation - will be suggesting it to my students for their notes. Thanks.
I like to draw a smiley on the inside of the end of proof square
@@nealcarpenter3093 thank you! are you a professor? It's cool to imagine this idea influencing a class somewhere in the world
@@yurigouveawagner9432 Hey, good or humorous (if there's a diff.) ideas spread. I teach math, grades 6-8, though that does include some Calculus.
This guy also speaks Chinese, Grant has everything covered.
I'm absolutely blown away by this man!
his tones are off though, but i never spoke mandarin in a colloquial setting so maybe there's something about when it's ok to change tones in certain contexts.
@@donsanderson Mandarin is what people commonly mean by the language “Chinese”.
@@donsanderson Really where did you find this information? I really want to hear him speak chinese
This is the advantage of a powerful memory.
12:00 this is why Grant is among the best maths educators out there. He talks about the emotion in maths, about stressing what's important vs the detail. It explains why the first time I properly intuited linear independence was after looking at his Linear algebra series. Some people will argue you can't visualise all maths but I think there's a happy medium between slavishly following symbols on a page and characterising what they represent more fully. This is why Grant is a bit of a hero for me.
Agree.
Well just to tell as a fact , here in India , textbooks do have these kind of emojis , more like characters as whole in textbooks interacting with children and trying to tell them the concepts like a story or something through the textbook .
I remember here in my state board textbook (local educational board's published textbook ) , there was a Rubik's cube interacting with us asking us questions or giving us a recap of the previous chapter at the start of a new chapter and also giving us some tips , they put him into corners and it was interesting.
I would recommend Jiří Matoušek's "Thirty-three miniatures".
It's about 33 beautiful applications of linear algebra.
Also, all textbooks by him that I checked out are gold.
Sounds amazing, I’ll check it out!
+inf
Matousek's books are incredible!
Thank you, sir!
I come from China, and I learned quite a lot from your teachings. They are very insightful!
Thank you very much!
感谢！
is china fun?
@@Benderhino Why don't you come to China and get the answer by yourself?
@@与我无关-z4k nah man busy with poverty stuff, this is embarrassing I wish everyone was dead, anyways, have a nice day.
When talking about natural talent, I think about crafting. When I have just started teaching someone to knit or crochet, at some point, after they've gotten just good enough to reliably make stitches, they'll watch me do the same craft and say something like "gosh you're so fast and I'm so slow, I wish I was fast". And when it's about crafting, the response to this is actually really intuitive. I say "well, I've been doing this about two decades longer than you have, so I should be faster; if I wasn't, there would be something going rather wrong".
But, bio-communism isn't real.
Dude, you're a legend. Thank you for the hard work you put into your videos to make things clearer. Seriously, thank you!!
Being a practicing engineer, I'm really interested in theory of nonlinear dynamics and mathematics behind. The Nonlinear Dynamics Book will be a real help. Thanks very much, Grant! ❤
I have been following and looking up to this guy for years, and never would have expected him to speak Chinese so fluently.
I am really enjoying these videos on this second channel. Great to hear conversations and interviews that give more depth behind the characters in mathematics.
On the nature vs nurture aspect. I really do feel that there is an in-built ability to engage with mathematics in some people. I know that I saw friends with whom I performed on level with in undergrad really outpace me in graduate courses. It really showed me there was a difference in ability. There is a spectrum of human ability and Tao and Von Neumann are really at the peak of this. If there are people at the top and the bottom, there are going to be people at every point in between.
Of course, this is really hard to quantify, and I don’t think there is value in approaching education this way. I think most topics taught in school should be approachable to everyone, but after a point, you see real differentiation between students.
What’s hard to tell is if this is actually innate, or based on instincts learned so young that it might as well be innate.
@@GrantSanderson yeah, it really is hard to characterize and identify. Though it is undeniable with people like Gauss, where they comprehended arithmetic at 3 years old. But that may be apocryphal too.
that question about not being gifted enough was hard to hear. coming from a competition math background, i often felt that the community could be very toxic and make me feel like i was not "gifted" enough
@@markv785 this is just my opinion, but SATs are a terrible mode of evaluation, especially for college admissions.
Competition math can be really discouraging for students who want to get into math but can’t “keep up” with everyone else. I never even thought that I could ponder problems for a day or more. Everything was kinda fast-paced. It’s funny that continuity can work perfectly fine only using rational numbers. Talk about lifting up your pencil! Oh and that room problem, I didn’t find that easy at all lol. Although, convex functions ARE quite interesting *cough* *cough* Legendre transform. Love your content! Can’t wait for more on diff eqs and probability
the room problem really isn't that hard. click read more if you want a hint (to find the solution I came up with)
think about the room with the smallest number of people.
The agon is all.
Please do not stop making those videos 🙏🙏
You are changing my life 🙂
For the room problem: a "physical/intuitive" interpretation of the sum-of-squares is as a weighted count of the population, where each person is weighted by the number of their roommates.
Can’t help but mention that it can also be viewed as the variance of a distribution + the mean squared, where in this case the mean is constant, so it basically reflects the variance.
I like to think that it is because of the fact that (x + a)^n > x^n + a^n, you can always choose x^n with n > 1 and you'll get the same proof
6:47 Regarding the concept of being gifted vs. not:
I've always held to the belief that there *are* people who are better in a given field by their nature, but *anyone* can become skilled through effort. Anyone can become skilled, but some people have a head start.
Regarding math specifically, yes. There are individuals who are naturally good at it; their brains are wired in a way that makes learning mathematics natural. These are the individuals we'd typically call "gifted." For others, learning math is harder. The ""not gifted"" individuals. It's very easy to see someone breeze through things that you struggle with and come to the conclusion that you're just not gifted enough to progress, but that's categorically untrue. Just because it takes more effort for you to learn something doesn't mean you can't. If you're willing to put your head down and push through, you can learn any skill you want. Whether you're gifted or not determines where you start, not where you end up.
Agreed, I think the same.
I'm not sure it's true that _anyone_ can become skilled at math just by putting in more effort. Frankly, I don't think that's true of _anything._ But I agree that there are most likely many people in the world who could become skilled at math, but are never given a chance or a reason to put in enough work.
Given a chance: ramanujan. Or has the chance
Richard Feynman said hard works he didn't want honors.
Tho yes genes is a factor. Eg cofactor factor constanta etc iirc l'hospital rule
I bet whoever came up with that sum of squares solution had some connection to Cambridge. They used to grade their exams by taking the sum of squares of the scores to each question, to encourage people to prioritise answering fewer questions more completely rather than doing a lot of questions partially.
As you say, problem solving is a lot about your experience....
You are an extremely talented communicator, for real.
I've always enjoyed math and physics. The one thing I have learned over time is that numbers, math and all that follows they are nothing more than a product of the mind as they are all conceptual ideas. At the end of the day, math is nothing more than a model and it is what you make of it! I enjoy hearing all different kinds of perspectives and approaches. 3Blue1Brown is one of my favorites along with Mathologer and a few others... I love their presentations and visualization methods. It's almost like comparing or listening to a monotone college professor who'd soon enough put you to sleep on trying to explain the phenomenons of the Mandelbrot Set as opposed to writing a simple computer program to graph it visually being able to see and interact with the infinite patterns and levels of details that a simple equation produces. Grant and others like him allow mathematics to be fun and engaging instead of being monotonous, drab and dreary... Keep up the great work!
5:00 Yes yes yes yes, I am obsessed with Valadmir Arnold's writings, I have been collecting his works in English but most of his writings are in Russian. Currently learning a lot from his book on Catastrophe Theory. I feel VI Arnold changed my life, his essays on mathematical pedagogy are revolutionary and I feel you are the one who will make his dream come true of unifying maths of physics which were divorced by the Evil Bourbaki's.
Lol don’t let arnold brainwash you with his nonsensical opinions the bourbaki’s did a great job and the existence of Pure mathematics is super convenient and useful
As for the continuity example. I think the most discrete answer would be a function is continuous at a given input when both of its limits exist and are equal or converge to the same value. Take sin(t) for example, no matter what input you use within this function both the left and right hand limits exist and are equal. Now, take the tangent function and examine its left and right hand limits at tan(PI/2) or tan(45) and other multiples of them. Both limits exist, yet they are not equal. The limit approach towards the right tends to +infinity and the limit approach towards the left tends to -infinity. Here we can see that the sine and cosine functions are both continuous where the tangent, cotangent, secant and cosecant are not continuous for all of their domains. The challenging thing here is that we are typically taught about continuity with respect to functions and graphs before we are ever introduced to the concepts of limits. Yet I think the best proof and representation of continuity lies within the properties of a functions limits across all of its domain.
This is full of valuable suggestions.
Just "the convexity of x squared" at 21:20 led me to an answer to a question that's bothered me for years, namely: "Why do we nearly always use & teach Standard Deviation, rather than MAD or AAD?"
Not bad for a 5 second toss off remark!
It seems to me that you can add up other functions and it will still work, for example adding 2^x where x is the number of people for each room
@@columbus8myhw i think we just need to use a strictly convex function
Dude, your eye IS the 3b1b logo! Never noticed it before!
Mind = blown
Hubbard’s book is fantastic! It was the textbook for my first proof-based class, and it made me switch to the math major
I would say that as long as you've done a reasonable amount of preparation then the fault pretty much always lies with the textbook. One of many examples of this is I once supported ten different students through the exact same maths course.
7 had textbook A - Topic X was impenetrable, only ever to be understood by the best and the brightest. Topic Y was completely straightforward and I was never asked about it.
3 had textbook B - Topic X was completely straightforward and I was never asked about it. Topic Y was impenetrable, only ever to be understood by the best and the brightest.
I taught Topic X to those with textbook A, and Topic Y to those with textbook B, made those topics seem completely straightforward and everyone thought I was a genius.
The Hubbard text is rather good for all the topics covered. I think the book is especially great for those who have studied already Linear Algebra, Real Analysis, and Multivariable Calculus and are in need of a more in depth review.
We used that book for Math 23a & b with Paul Bamberg at Harvard. When I took the class, we used that very same 4th edition.
22:13
Being a majorly doubtful visual learner of mathematics, I've been dying to know this. Thank you so much Grant .
Heard the lil intro theme phrase and couldn't help but to think on 3Blew1Blown
Fastest 25 mins of my life. Love listening to you Grant, always a pleasure.
2:09 actually during my third year in university I had a course which was dedicated only to proofs from this book.
Thank you prof. Yuval roichman!
我从很早的时候就开始看你的影片，这么长时间一直都很喜欢你的影片。你给了我很多对于数学新的想法。我也很喜欢manim。
2:25 woah! Grant, in your interview with Steven Strogatz, he mentioned teaching at a math program at Hampshire College (HCSSiM). I just realized today that turns out this quote from Erdos was actually in 1985 when he visited HCSSiM and was giving a talk. "yellow pig camp"
:)
It's amazing that the first couple minutes of this are the reason I now know where the name "3Blue1Brown" comes from. Genuinely so damn cool.
I was moving stuff around and now my computer screen sits closer to me. Finally understand why your channel is called 3 Blue 1 Brown! :D
It's always amazing the first time you notice!
Great video! Really hope you keep posting on this channel. (The podcast is also fantastic!)
Recently read Mathematical Methods of Classical Mechanics by Vladimir Arnold and it's probably my favourite book. Super cool topics and ingeneous presentation, just a beautiful textbook.
You mentioned that it would be great if there were emojis in the margins of textbooks to help with understanding. In the Park City Math Institute (PCMI) problem sets used in their math teacher seminars, there are those sorts of comments (sadly, no emojis) in the margins. Sometimes they’re there for humor, but other comments are informative. I really like them!
is it weird that i almost cried at ( 6:47 - Not being "gifted" enough ) ?
Why you no doctor??
Besides everything good about Grant Sanderson, he's also got the best diction/ voice on the internet.. or anywhere! I can listen to him for hours on end.
I've just realized the reason why the channel is called 3b1b - Grant's right eye has a mix of brown and blue. It's amazing how this detail only struck me today!
I only just now noticed how one of your eyes is 3/4 blue and 1/4 brown. Guess that's where the name comes from xD
And the logo design too, it’s an iris!
Imagine Next time grant come up with
"Namaste, To kese hen aap log"
I hope someone from India will sponsor you for making some series.
@@quirtt HM orz 🛐
😂
Wow, what a beautiful narrative voice! So glad you didn’t dumb this one down by putting music over this fascinating Information 😇👍
That intro woke me up!
Besides about everything good about Grant Sanderson, the best diction on the internet.. or anywhere! I can listen to him for hours on end.
Hey! I thought of the sum of squares solution almost instantly. That feels pretty neat
when he spoke chinese in the intro i SCREAMED he did so well i love my bilingual king
@@donsanderson yeah!! his pronunciation was so good and he even added some tones which is super impressive
18:31 Grant tries to remember a mathematical puzzle, his platonic brainworks appears to screen.
Got some books for Christmas this year tbh -- pretty excited thanks for the recommendations
Judging over gifted thing can sure demotivate you, but practising and learning perquisite can sure bring surprising change to learn math. Also, I guess visualisation really helps to understand what it meant instead mere wandering in head, though it will not always the case as context , reality and drawing are critical to relate all the time as so many known and unknown parameters we have to dealt with.
Great video. Agreed on visualization as a gateway to help. I have some stuff on symmetry and computation. Have a great day and keep up the great work.
the intro!, does grant know Chinese or did he just learn the greeting?
I took some Chinese back in high school, and spent a couple weeks in Beijing. Sadly I've forgotten most. This was a greeting a translator handed to me, likewise for the outro.
@@GrantSanderson you are so gangster that I just assume you knew mandarin.
@@le0nz ahahhahahahaha samee
A different way to think about the room problem: for each person, count how many people they share a room with (including themselves) and add all of those numbers up. It's easy to see that the result is the sum of squares of the people in each room. But the number increases for the person moving rooms, it increases by 1 in the people of the room they're moving into and decreases by 1 for the people in the room they're moving out of. By the fact that there are more people in the former, the result is a net increase in the total.
probability series when? 下次一定! 下次一定!
Being frustrated by my math textbooks is how I found this channel in the first place. 😛
I find it interesting that your channel reached 1M subscribers just after Steve Mould's and ahead of Matt Parker's (which hasn't happened yet, he's still around 900K). But all in all it's a great win for STEM.
3blue1brown hit 1 million subs in 2018, I'm pretty sure that's well before Steve Mould
His main TH-cam is at 3.85M right now, this video celebrates the Chinese channel (on BiliBili) reaching 1M! The rest of the translated channels have between a few hundred and a tens of thousands subs each.
The question with the room made me think about the Herfindahl-Hirschmann that measures market concentration.
Ahhhhh! Didn’t expect this at all lol. Love your contents!
Incredible
Love your work Grant ❤❤❤
I don't believe in the philosophy that frustration is apart of your interest. The reason is that I have found plenty of scenarios where reading a book has been less or not at all frustrating compared to jumping blindly into a project. I enjoyed the not at all frustrating approach and I learned more from that.
your right eye really is 3blue1brown
wait what? that's crazy!
好的 3蓝色的1棕色的 .... I almost thought entire video will be in Chinese lol
waiting for your next videos, just the old good 3b1b
the intro got me shook
You can easily add one plus one is two. But real mechanical trace for that is a different story. There are so many ways but there is no proof that there is only one way to solve a problem. Though the idea is the same result. Because source and destination are same for every problem. Visualisation is the different forms of how human brains work. Almost 90% of mathematical solutions lie on filling the gaps. For example a random numbers sequence to an orderly is filling the gaps to regular.
Dear Grant,
The one-third of Linear Algebra of the Hubbard and Hubbard book is a great resource of LA but is rather short. And the longer, more elaborate textbooks of Linear Algebra that I could find were unintuitive(Like Axler, Strang, etc). Books like Hoffman Kunze and Halmos seem to be at a higher level than what I currently am at.
Are there other detailed textbooks of Linear Algebra that teach the subject primarily as the study of *linear transformations* rather than the study of mathematical objects consisting of a bunch of numbers arranged in an array and weird operations on those objects like multiplication whose rules kind of seem arbitrary, pointless and unmotivated.
I just took a college course called "abstract linear algebra" and we used the first 4 chapters of a book called "linear algebra done wrong." Rather than just starting with the algorthims on vectors and matricies, the book starts from the most abstract definition of a vector space, then looks at transformations between vector spaces, then shows exactly why matricies characterize linear transformations. The book then uses that as a jumping off point to learn the algorthims and stuff like row reduction, rank, and eigenvalues. Based on your description, it sounds like it's what you are looking for
I'd recommend Axler but it sounds like you already tried him. He's all about emphasizing linear transformations over matrices. What did you find unintuitive about his book?
@@simply42 Oh that sounds good, I would surely like to try that.
@@martinepstein9826 I like that Axler's book tells us abstract definitions rather than focusing on computations. But my problem with the book is that it feels less motivated. Throughout the book, all sorts of operations and mathematical objects with their different properties are introduced. However, we are never told once why mathematicians invented that concept (I am not so concerned with real life applications but I really want to know why mathematicians defined things in one way rather than the other. Surely, all of the definitions in the book were later formalizations of intuitive ideas of mathematicians. Those ideas that motivated them to define those objects with those properties in *that specific way and not any other* is what I want to know so that I can appreciate what the definition actually stands for). And Axler's book while being abstract, is not intuitive or motivated (atleast to me).
Axler's book is not abstract lol
I love your Chanel 💕💕
You are the best!!
This video came up in my feed at EXACTLY the right moment.
Just a couple of hours ago, I finally decided that the vexation of a pop-math book I was trying to learn from was too much for me to deal with, and it's time for me to move on, so as not stymie my progress.
(I've always struggled with math since my algebra days, but aced logic, and have managed to teach myself enough of the relevant math for my physics to not fall too far behind.)
I'm not going to say the book was 'bad'. Just that it's way of presenting the information--some stuff about a family of 6 boys, 6 girls, a mom and dad, and trying to see how many dance pairs the siblings can make--was not helpful at all.
But at least I learned what FACTORIAL is!
You got me scared af at the beginning..lol
"...because Pi has legs." I was not expecting this; I always assumed it was related to the circular nature of the 3blue1brown logo. At any rate, the Pi creatures are a great lure to catch the eye of casual TH-cam viewers who might not realize they have an interest in math. "Come for the cute critters; stay for the learning."
So, you created Manim!... you are the best!
My recommendation: "Elements" from Euklid.
First learn the basics and learn to think mathematically, then the rest is easy.
It's like jumping in a pool. First, the water is just cold. After a while, it makes really fun.
Do you mean Euclid?
Thank you soooo much, I love your videos!
Hello Grant. I am immensely benefiting from your videos. Can you please make a video where you could maybe show a roadman for self learners of mathematics? Your favorite books?
"it's a curve you can draw without lifting your pencil"
-- Euler, as reported by Ian Stewart.
12:12 The book, _Cryptological Mathematics_ by Robert Edward Lewand, has a few characters in it that react, and "interact" with the information.
7:45 You don't understand mathematics, you get used to it. And apparently, so does van Neuman, because that's his quote.
did anyone else just realize the 3blue1brown comes from his right eye? thats insane!
What's crazy is that i recently picked up each of the books Grant mentions except for the Cauchy schwarz master class
I think I have a problem. I dig all the axioms and definitions when I start reading about a branch of mathematics but I don't really like the how all this is usefull part that comes after.
19:30 The problem of rooms is very closely related to the Raising Operator on integer partitions
Hey Grant, can you in your next podcast interviews interview a famous math lecturer on youtube like Gilbert Strang or Tom Leighton?
@@markv785 He is one of my professors for mathematics at my university!
Yes , bring prof. Strang
This dude is amazing
07:49 There's absolutely no innate gift factor in any area, taking Von Neumann as an example: he was the child of a banker and he was instructed on all sort of things by governess so that between the age of 6 and 8 he knew multiple language, knew how to divide two eight-digit numbers in his head, knew to speak in ancient greek and from 8 onwards he learned about differential and integral calculus, by twelve he understood the Borel's Théorie des Fonctions. He wasn't born like that, he was meticulously crafted by his father, the study at home, the stimulating environment, he didn't went to public school, he had several figures available at all times to satisfy his curiosities, curiosities that all children will have if grown up in the right environment, he was nothing special, everybody at his place would've become a stunning polymath. So everybody can feed properly his brain and become a polymath, it's just that public schools and society in general is build on top of a hyper sectoral criteria within which everybody got to do one stuff and do it good, you don't have time for being curios in a average family, Von Neumann's family wasn't an average one so that Neumann itself became a non average mind, I know that is hard even to start something in certain age automatically comparing yourself with people like this but all you have to do is to compare yourself of today with yourself of yesterday and break any boundaries, don't let things weight you down never and ever, remember that the brain is the same for everybody is just like a muscle but stunning complicated, when people analysed Einstein brain noticed that the difference wasn't in the neurons count or divergence itself but in the glia, he had a gigantic macro convoy of glia, the stuff that carry axons around the brain and microglia that weld neurons when new connections are always proposed at the brain, just like a bodybuilder have giant strange muscle. It's not too late, just give all you can and you can't fail, believe in yourself and if no one ever believed in you just believe it on behalf of all of them and obviously you need to work extremely hard. P.S. thank for all the great content
There are thousands of kids who grew in similar environment as von Neumann but there is only one von neumann. You speak that how brain is like muscle and can be trained but I think you are not aware of the fact that every body does not have equal athletic talent. Athletic talent is genetic and innate and now research has even identified some specific genes associated with better athletic performance . People are genetically different and some people just have better muscle potential than others and this is similarly true for intelligence too. Some people are just naturally more intelligent than others because of genetics.
Freshman year we used Hubbard and most of my friends made fun of me for liking the book. At last, someone else likes it.
Alot of people like that book
Geometric/Clifford algebra when? :)
you know, his chinese is actually really good, he has the tones correct and everything
Thank you for this video.
Honestly, I feel sad that I have learnt math in a way that I can write proofs, solve problems but not yet learnt the way that I can recite proofs in a poetic way (like your videos), in a simple- sleak, clear way., like a story. I am inspiredby your videos, 2 days from now, I have to give a seminar on Fourier series. And I will try my best to make it motivating, that will make audience interested in its maths. I was going to lecture like dumping the theory infront of them, but then I saw your video and I clearly see what's smelly, unattractive and whats has a nice fragrance, makes us wonder. Your videos have opened a closed gate that only allowed theory to pass, now I am interested in simple exxplanation. I know this leap will be difficult for me, to be patient, read and write in my own words, but I want to promise that I beleive its going to be rewarding, I would love it. I wont give up. Thankyou so much.
The problem at 19:00 really looks like some sort of inverse of entropy since entropy is a time dependent phenomina I think this is a very interesting problem.
John b Hubbard.
Non linear dynamics and kaos .
Proofs from the book .
The causy stawd master class.
3:30 Thought provoking, indeed: why not mainstreamise masterclasses in mathematics?
We definitely should!
Nonlinear dynamics was phenomenal
I need a version of this video but every time he says "The Book" the volume goes way up
Grant giving a student friendly perspective on Mathematics, is probably on of the best things on TH-cam. What I am missing in the book recommendations list is a discrete mathematics book, is there any good textbook about the subject for autodidacts?
Calculus is brain of Maths and Alzebra its heart
Wishing to have a Calculus book from james stewart in my bday today
1) John H. Hubbard ( Vector Calculus, Linear Algebra, and Differential forms )
2) Steven H. Strongatz ( Non-Linear Dynamics and Chaos )
3) Martin Aigner Gûnter M Ziegler ( Proofs from The Book )
4) J. Michael Steele ( The Cauchy - Schwarz Master Class )