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Change of basis ｜ Chapter 13, Essence of linear algebra | If I have a vector sitting here in 2D space, we have a standard way to describe it with coordinates. In this case, the vector has coordinates 3-2, which means going from its tail to its tip, involves moving 3 units to the right and 2 units up. Now, the more linear algebra-oriented way to describe coordinates is to thi... |
From Newton’s method to Newton’s fractal (which Newton knew nothing about) | You've seen the title, so you know this is leading to a certain fractal. And actually it's an infinite family of fractals. And yeah, it'll be one of those mind-bogglingly intricate shapes that has infinite detail no matter how far you zoom in. But this is not really a video about generating some pretty picture for us ... |
Q&A #2 + Net Neutrality Nuance | Hey everyone, no math here, I just want to post two quick announcements for you. First, the number of you who have opted to subscribe to this channel has once again rolled over a power of two, which is just mind-boggling to me. I'm still touched that there are so many of you who enjoy math like this, just for fun, app... |
Fractals are typically not self-similar | Who doesn't like fractals? They're a beautiful blend of simplicity and complexity, often including these infinitely repeating patterns. Programmers in particular tend to be especially fond of them, because it takes a shockingly small amount of code to produce images that are way more intricate than any human hand ever... |
How to lie using visual proofs | Today I'd like to share with you three fake proofs in increasing order of subtlety, and then discuss what each one of them has to tell us about math. The first proof is for a formula for the surface area of a sphere, and the way that it starts is to subdivide that sphere into vertical slices, the way you might chop up... |
Dot products and duality ｜ Chapter 9, Essence of linear algebra | Traditionally, dot products are something that's introduced really early on in a linear algebra course, typically right at the start. So it might seem strange that I've pushed them back this far in the series. I did this because there's a standard way to introduce the topic, which requires nothing more than a basic un... |
But what is a convolution？ | Suppose I give you two different lists of numbers, or maybe two different functions, and I ask you to think of all the ways you might combine those two lists to get a new list of numbers, or combine the two functions to get a new function. Maybe one simple way that comes to mind is to simply add them together term by ... |
Solving the heat equation ｜ DE | We last left off studying the heat equation in the one dimensional case of a rod. The question is how the temperature distribution along such a rod will tend to change over time. And this gave us a nice first example for a partial differential equation. It told us that the rate at which the temperature at a given poin... |
Higher order derivatives ｜ Chapter 10, Essence of calculus | In the next chapter about Taylor series, I make frequent reference to higher order derivatives. And if you're already comfortable with second derivatives, third derivatives, and so on, great, feel free to just skip ahead to the main event now. You won't hurt my feelings. But somehow, I've managed not to bring up highe... |
Tattoos on Math | Hey folks, just a short kind of out of the ordinary video for you today. A friend of mine, Cam recently got a math tattoo. It's not something I'd recommend, but he told his team at work that if they reached a certain stretch goal, it's something that he do. And well, the incentive worked. Cam's initials are CSC, which... |
How colliding blocks act like a beam of light...to compute pi | You know that feeling you get when you have two mirrors facing each other, and it gives the illusion of there being an infinite tunnel of rooms. Or if they're at an angle with each other, it makes you feel like you're a part of a strange, kaleidoscopic world with many copies of yourself all separated by angled pieces ... |
Essence of linear algebra preview | Hey everyone, so I'm pretty excited about the next sequence of videos that I'm doing. There'll be about linear algebra, which as a lot of you know, is one of those subjects that's required knowledge for just about any technical discipline. But it's also, I've noticed, generally poorly understood by students taking it ... |
Q&A with Grant Sanderson (3blue1brown) | What is a grobner basis? If that is your intent for what this Q and A episode is going to be, as far as technicality and deep explanation is concerned, you're going to be grossly disappointed. Same goes to whoever asked about what the Fourier transform has to do with quantum computing. I can say at a high level it's b... |
Lockdown math announcement | As many of you know, with the coronavirus outbreak still very much underway, there's a huge number of students who are left to learn remotely from home, whether that means doing distance classes over video conference, or trying to find resources like Khan Academy and Brilliant to learn online. So one thing that I want... |
But what is a Fourier series？ From heat flow to drawing with circles ｜ DE4 | Here, we look at the math behind an animation like this one, what's known as a complex Fourier series. Each little vector is rotating at some constant integer frequency, and when you add them together, tip to tail, the final tip draws out some shape over time. By tweaking the initial size and angle of each vector, we ... |
Why is pi here？ And why is it squared？ A geometric answer to the Basel proble | I'm gonna guess that you have never had the experience of your heart rate increasing in excitement while you were imagining an infinitely large lake with lighthouses around it. Well, if you feel anything like I do about math, that is gonna change by the end of this video. Take 1 plus a fourth plus 1 9 plus 1 16th and ... |
Gradient descent, how neural networks learn ｜ Chapter 2, Deep learning | Last video, I laid out the structure of a neural network. I'll give a quick recap here just so that it's fresh in our minds and then I have two main goals for this video. The first is to introduce the idea of gradient descent, which underlies not only how neural networks learn, but how a lot of other machine learning ... |
The Brachistochrone, with Steven Strogatz | For this video, I'm doing something a little different. I got the chance to sit down with Stephen Strogatz and record a conversation. For those of you who don't know, Steve is a mathematician at Cornell. He's the author of several popular math books and a frequent contributor to, among other things, Radio Lab and The ... |
e^(iπ) in 3.14 minutes, using dynamics ｜ DE5 | One way to think about the function e to the t is to ask what properties does it have? Probably the most important one, and from some points of view, the defining property, is that it is its own derivative. Together with the added condition that inputting 0 returns 1, it's actually the only function with this property... |
Derivative formulas through geometry ｜ Chapter 3, Essence of calculus | Now that we've seen what a derivative means and what it has to do with rates of change, our next step is to learn how to actually compute these guys. As in if I give you some kind of function with an explicit formula, you'd want to be able to find what the formula for its derivative is. Maybe it's obvious, but I think... |
But why is a sphere's surface area four times its shadow？ | Some of you may have seen in school that the surface area of a sphere is 4 pi r squared. A suspiciously suggestive formula given that it's a clean multiple of the more popular pi r squared, the area of a circle with the same radius. But have you ever wondered why this is true? And I don't just mean proving the 4 pi r ... |
Backpropagation calculus ｜ Chapter 4, Deep learning | The hard assumption here is that you've watched Part 3, giving an intuitive walkthrough of the back propagation algorithm. Here we get a little bit more formal and dive into the relevant calculus. It's normal for this to be at least a little confusing so the mantra to regularly pause and ponder certainly applies as mu... |
Why “probability of 0” does not mean “impossible” ｜ Probabilities of probabilities, part 2 | Imagine you have a weighted coin, so the probability of flipping heads might not be 50-50 exactly, it could be 20% or maybe 90% or 0% or 31.41592%. The point is that you just don't know, but imagine that you flip this coin 10 different times and 7 of those times it comes up heads. Do you think that the underlying weig... |
Circle Division Solution | In my last video, I posed the following question. If you take N points on a circle, then connect every pair of them with a line, how many sections do these lines cut the circle into? What was strange is that when N is less than 6, N when N is 10 for some reason, the answer is always a power of 2. But for all other val... |
Researchers thought this was a bug (Borwein integrals) | Sometimes it feels like the universe is just messing with you. I have up on screen here a sequence of computations, and don't worry in a moment we're going to unpack and visualize what each one is really saying. What I want you to notice is how the sequence follows a very predictable, if random, seeming pattern, and h... |
But how does bitcoin actually work？ | What does it mean to have a bitcoin? Many people have heard of bitcoin that it's a fully digital currency with no government to issue it and that no banks need to manage accounts and verify transactions, and also that no one really knows who invented it. And yet many people don't know the answer to this question, at l... |
We ran a contest for math explainers, here are the results (SoME2) | In the last month or two, there's been a measurable increase in the attention to a wide variety of smaller math channels on YouTube. My friend James and I ran a second iteration of a contest that we did last year, the Summer of Math Exposition, which invites people to put up lessons about math online. Could be a video... |
The medical test paradox, and redesigning Bayes' rule | Some of you may have heard this paradoxical fact about medical tests. It's very commonly used to introduce the topic of Bayes rule in probability. The paradox is that you could take a test which is highly accurate in the sense that it gives correct results to a large majority of the people thinking it. And yet, under ... |
The more general uncertainty principle, regarding Fourier transforms | You've probably heard of the Heisenberg uncertainty principle from quantum mechanics. That the more you know about a particle's position, the less certain you can be of its momentum and vice versa. Michael here is for you to come away from this video feeling like this is utterly reasonable. It'll take some time, but I... |
Taylor series ｜ Chapter 11, Essence of calculus | When I first learned about Taylor series, I definitely didn't appreciate just how important they are. But time and time again, they come up in math and physics and many fields of engineering because they're one of the most powerful tools that math has to offer for approximating functions. I think one of the first time... |
A quick trick for computing eigenvalues ｜ Chapter 15, Essence of linear algebra | This is a video for anyone who already knows what eigenvalues and eigenvectors are, and who might enjoy a quick way to compute them, in the case of 2x2 matrices. If you're unfamiliar with eigenvalues, go ahead and take a look at this video here, which is actually meant to introduce them. You can skip ahead if all you ... |
How (and why) to raise e to the power of a matrix ｜ DE6 | Let me pull out an all differential equations textbook that I learned from in college. And let's turn to this funny little exercise in here that asks the reader to compute e to the power a t, where a, we're told, is going to be a matrix. And the insinuation seems to be that the results will also be a matrix. It then o... |
Hamming codes part 2, the elegance of it all | I'm assuming that everybody here is coming from part one. We were talking about hamming codes, a way to create a block of data where most of the bits carry a meaningful message, while a few others act as a kind of redundancy. In such a way that if any bit gets flipped, either a message a bit or a redundancy bit, anyth... |
Matrix multiplication as composition ｜ Chapter 4, Essence of linear algebra | Hey everyone, where we last left off, I showed what linear transformations look like and how to represent them using matrices. This is worth a quick recap because it's just really important, but of course if this feels like more than just a recap, go back and watch the full video. Basically speaking, linear transforma... |
How pi was almost 6.283185 | I'm sure that you're already familiar with the whole pie versus tau-to-bate. A lot of people say that the fundamental circle constant we hold up should be the ratio of a circle circumference to its radius, which is around 6.28, not the ratio to its diameter, the more familiar 3.14. These days we often call that larger... |
But what is a neural network？ ｜ Chapter 1, Deep learning | This is a 3. It's slobily written and rendered at an extremely low resolution of 28x28 pixels, but your brain has no trouble recognizing it as a 3. And I want you to take a moment to appreciate how crazy it is that brains can do this so effortlessly. I mean this, this and this are also recognizable as 3s, even though ... |
Music And Measure Theory | I have two seemingly unrelated challenges for you. The first relates to music, and the second gives a foundational result in measure theory, which is the formal underpinning for how mathematicians define integration and probability. The second challenge, which I'll get to about halfway through the video, has to do wit... |
Exponential growth and epidemics | The phrase exponential growth is familiar to most people. And yet, human intuition has a hard time really recognizing what it means sometimes. We can anchor on a sequence of small, seeming numbers and then become surprised when suddenly those numbers look big, even if the overall trend follows an exponential perfectly... |
Why 5⧸3 is a fundamental constant for turbulence | The arrow around you is in constant and chaotic motion, replete with nearly impossible to predict swirls, ranging from large to minuscule. What you're looking at right now is a cross section of the flow in a typical room, made visible using a home demo, involving a laser, a glass rod, and a fog machine. Predicting the... |
Using topology to solve a counting riddle ｜ The Borsuk-Ulam theorem and stolen necklaces | You know that feeling you get when things that seem completely unrelated turn out to have a key connection? In math especially, there's a certain tingly sensation I get whenever one of those connections starts to fall into place. This is what I have in store for you today. It takes some time to set up, I have to intro... |
2021 Summer of Math Exposition results | This summer, James Schloss and I ran a contest that's meant to encourage more people to make online math explainers. And here, I want to share with you some of my favorites. I wrote up a much fuller blog post about the event and the process for selecting winners, but let's open here with some of the key points. First ... |
Eigenvectors and eigenvalues ｜ Chapter 14, Essence of linear algebra | Icon Vectors and Icon Values is one of those topics that a lot of students find particularly unintuitive. Things like why are we doing this and what does this actually mean are too often left just floating away in an unanswered sea of computations. And as I've put out the videos of the series, a lot of you have commen... |
What does area have to do with slope？ ｜ Chapter 9, Essence of calculus | Here, I want to discuss one common type of problem where integration comes up. Finding the average of a continuous variable. This is a perfectly useful thing to know in its own right, but what's really neat is that it can give us a completely different perspective for why integrals and derivatives are inverses of each... |
Why do colliding blocks compute pi？ | Last video I left you with a puzzle. The setup involves two sliding blocks in a perfectly idealized world where there's no friction and all collisions are perfectly elastic, meaning no energy is lost. One block is sent towards another smaller one, which starts off stationary and there's a wall behind it, so that the s... |
Euler's Formula and Graph Duality | In my video on the Circle Division problem, I referenced Euler's characteristic formula, and here, I would like to share a particularly nice proof of this fact. It's very different from the inductive proof, typically given, but I'm not trying to argue that this is somehow better or easier to understand than other proo... |
The other way to visualize derivatives ｜ Chapter 12, Essence of calculus | Picture yourself as an early calculus student about to begin your first course. The months ahead of you hold within them a lot of hard work, some neat examples, some not-so-need examples, beautiful connections to physics, not-so-beautiful piles of formulas to memorize, plenty of moments of getting stuck and banging yo... |
Vectors ｜ Chapter 1, Essence of linear algebra | The fundamental root of it all building block for linear algebra is the vector. So it's worth making sure that we're all on the same page about what exactly a vector is. You see, probably speaking, there are three distinct but related ideas about vectors, which I'll call the physics student perspective, the computer s... |
Oh, wait, actually the best Wordle opener is not “crane”… | Last week I put up this video about solving the game Wordal, or at least trying to solve it, using information theory. And I wanted to add a quick, uh, what should we call this? An addendum, a confession. Basically, I just want to explain a place where it by made a mistake. It turns out there was a very slight bug in ... |
The Wallis product for pi, proved geometrically | Alright, I think you're gonna like this. I want to show you a beautiful result that reveals a surprising connection between a simple series of fractions and the geometry of circles. But unlike some other results like this that you may have seen before, this one involves multiplying things instead of adding them up. No... |
The impossible chessboard puzzle | You walk alone into a room and you find a chess board. Each of the 64 squares has a coin sitting on top of it. And taking a step back, this is going to be one of those classic prisoner puzzles, where a strangely math obsessed warden offers you and a fellow inmate a chance for freedom, but only if the two of you solve ... |
Thinking outside the 10-dimensional box | Math is sometimes a real tease. It seduces us with the beauty of reasoning geometrically in two and three dimensions where there's this really nice back and forth between pairs or triplets of numbers and spatial stuff that our visual cortex is good at processing. For example, if you think about a circle with radius 1 ... |
Visualizing the chain rule and product rule ｜ Chapter 4, Essence of calculus | In the last videos, I talked about the derivatives of simple functions. And the goal was to have a clear picture or intuition to hold on your mind that actually explains where these formulas come from. But of course, most of the functions you deal with in modeling the world involve somehow mixing or combining or tweak... |
What's so special about Euler's number e？ ｜ Chapter 5, Essence of calculus | I've introduced a few derivative formulas, but a really important one that I left out was exponentials. So here, I want to talk about the derivatives of functions like 2 to the x, 7 to the x, and also to show why e to the x is arguably the most important of the exponentials. First of all, to get an intuition, let's ju... |
Some light quantum mechanics (with minutephysics) | You guys know Henry from Minifysics, right? Well, he and I just made a video on a certain quantum mechanical topic. Bales in equalities. It's a really mind-warping topic that not enough people know about, and even though it's a quantum thing, it's based on some surprisingly simple math, and you should definitely check... |
Winding numbers and domain coloring | There's two things here. The main topic and the meta topic. So the main topic is going to be this really neat algorithm for solving two-dimensional equations, things that have two unknown real numbers, or also those involving a single unknown, which is a complex number. So for example, if you wanted to find the comple... |
Linear combinations, span, and basis vectors ｜ Chapter 2, Essence of linear algebra | In the last video, along with the ideas of vector addition and scalar multiplication, I described vector coordinates, where there's this back and forth between, for example, pairs of numbers and two-dimensional vectors. Now, I imagine the vector coordinates were already familiar to a lot of you, but there's another ki... |
Beyond the Mandelbrot set, an intro to holomorphic dynamics | Today, I'd like to tell you about a piece of math known as holomorphic dynamics. This is the field which studies things like the mandalbrut set, and in fact, one of my main goals today is to show you how this iconic shape, the poster child of math, pops up in a more general way than the initial definition might sugges... |
Snell's law proof using springs | So in my video with Steve Strogat's about the Braquista Cron, we referenced this thing called Snell's Law. It's the principle and physics that tells you how light bends as it travels from one medium into another where its speed changes. Our conversation did talk about this in detail, but it was a little bit too much d... |
Linear transformations and matrices ｜ Chapter 3, Essence of linear algebra | Hey everyone, if I had to choose just one topic that makes all of the others in linear algebra start to click and which too often goes unlearn the first time a student takes a linear algebra, it would be this one. The idea of a linear transformation and its relation to matrices. For this video, I'm just going to focus... |
Why slicing a cone gives an ellipse | Suppose you love math, and you had to choose just one proof to show someone to explain why it is that math is beautiful. Something that can be appreciated by anyone from a wide range of backgrounds while still capturing the spirit of progress and cleverness in math. What would you choose? Well, after I put out a video... |
Binary, Hanoi and Sierpinski, part 1 | Today, I want to share with you a neat way to solve the towers of Hanoi puzzle just by counting in a different number system. And surprisingly, this stuff relates to finding a curve that fills Serpinsky's triangle. I learned about this from a former CS lecturer of mine, his name's Keith Schwartz, and I've got to say, ... |
Cramer's rule, explained geometrically ｜ Chapter 12, Essence of linear algebra | In a previous video, I've talked about linear systems of equations, and I sort of brushed aside the discussion of actually computing solutions to these systems. And while it's true that the number crunching is typically something we leave to the computers, digging into some of these computational methods is a good lit... |
But what is the Fourier Transform？ A visual introduction | This right here is what we're going to build to this video, a certain animated approach to thinking about a super important idea from Matt, the 48 transform. For anyone unfamiliar with what that is, my number one goal here is just for the video to be an introduction to that topic. But even for those of you who are alr... |
Fractal charm： Space filling curves | I'm going to add a little bit of the color. I'm going to add a little bit of the color. I'm going to add a little bit of the color. I'm going to add a little bit of the color. |
Quaternions and 3d rotation, explained interactively | In a moment, I'll point you to a separate website hosting a short sequence of what we're calling Explorable Videos. It was done in collaboration with Ben Eater, who some of you may know is that Guy who runs the excellent computer engineering channel. And if you don't know who he is, viewers of this channel would certa... |
A Curious Pattern Indeed | A two points on a circle and draw a line straight through. The space which was encircled is divided into two. To these points out a third one, which gives us two more courts, the space through which these lines run has been fissured into four. Continue with the fourth point and three more lines drawn straight, now the... |
Why do prime numbers make these spirals？ ｜ Dirichlet’s theorem, pi approximations, and more | The full title of this video might be something like, how pretty but pointless patterns and polar plots of primes prompt pretty important pondering on properties of those primes. I first saw this pattern that I'm about to show you in a question on the math stack exchange. It was asked by a user under the name Dwimeark... |
Divergence and curl： The language of Maxwell's equations, fluid flow, and more | Today, you and I are going to get into divergence and curl. To make sure we're all on the same page, let's begin by talking about vector fields. Essentially, a vector field is what you get if you associate each point in space with a vector, some magnitude and direction. Maybe those vectors represent the velocities of ... |
Cross products in the light of linear transformations ｜ Chapter 11, Essence of linear algebra | Hey folks, where we left off, I was talking about how to compute a three-dimensional cross product between two vectors, v cross w. It's this funny thing where you write a matrix whose second column has the coordinates of v whose third column has the coordinates of w. But the entries of that first column weirdly are th... |
e to the pi i, a nontraditional take (old version) | E to the pi i equals negative 1 is one of the most famous equations in math, but it's also one of the most confusing. Those watching this video likely fall into one of three categories. 1. You know what each term means, but the statement as a whole seems nonsensical. 2. You're lucky enough to see what this means in so... |
The essence of calculus | Hey everyone, Grant here. This is the first video in a series on the Essence of Calculus, and I'll be publishing the following videos once per day for the next 10 days. The goal here, as the name suggests, is to really get the heart of the subject out in one binge watchable set. But with the topic that's as broad as C... |
Summer of Math Exposition 2 Invitation | I've got that wanderlust gonna work this scene. We will get back to this buddy at the end of the video. But before that, I want to tell you about the Summer of Math Exposition number two. So last year we did the Summer of Math Exposition number one, which was essentially an open invitation for people to make explanato... |
Hilbert's Curve： Is infinite math useful？ | Let's talk about space-filling curves. They are incredibly fun to animate, and they also give a chance to address a certain philosophical question. Math often deals with infinite quantities, sometimes so intimately that the very substance of a result only actually makes sense in an infinite world. So the question is, ... |
Visualizing the Riemann zeta function and analytic continuation | The remanzata function. This is one of those objects in modern math that a lot of you might have heard of, but which can be really difficult to understand. Don't worry, I'll explain that animation you just saw in a few minutes. A lot of people know about this function because there's a $1 million prize out for anyone ... |
blue1brown channel trailer | Hey there, welcome to 3BlueOneBrown. So I make videos that animate math. For a lot of them, I just try to find something kind of interesting or thought-provoking, not necessarily in the typical progression of subjects that a student would see in school. To see what I mean there, check out some of the recommended playl... |
Alice, Bob, and the average area of a cube's shadow | In a moment, I'm going to tell you about a certain really nice puzzle involving the shadow of a cube. But before we get to that, I should say that the point of this video is not exactly the puzzle per se. It's about two distinct problem solving styles that are reflected in two different ways that we can tackle this pr... |
How secure is 256 bit security？ | In the main video on cryptocurrencies, I made two references to situations where in order to break a given piece of security, you would have to guess a specific string of 256 bits. One of these was in the context of digital signatures and the other in the context of a cryptographic hash function. For example, if you w... |
How to count to 1000 on two hands | nd nd nd nd nd nd nd nd nd nd nd nd nd nd nd nd nd |
The most unexpected answer to a counting puzzle | Sometimes math and physics conspire in ways that just feel too good to be true. Let's play a strange sort of mathematical croquet. We're going to have two sliding blocks and a wall. The first block starts by coming in at some velocity from the right, while the second one starts out stationary. Being overly idealistic ... |
All possible pythagorean triples, visualized | When you first learned about the Pythagorean theorem, that the sum of the squares of the two shorter sides on a right triangle always equals the square of its hypotenuse, I'm guessing that you came to be pretty familiar with a few examples, like the 3, 4, 5 triangle, or the 5, 12, 13 triangle. And I think it's easy to... |
Simulating an epidemic | I want to share with you a few simulations that model how an epidemic spreads. There have recently been a few wonderful interactive articles in this vein, including one in the Washington Post by Harry Stevens, and then another by Kevin Simler over at Melting Asphalt. They are great, you can play with them, they're ver... |
What does it feel like to invent math？ | Take 1 plus 2 plus 4 plus 8 and continue on and on adding the next power of 2 up to infinity. This one is crazy, but there's a sense in which this infinite sound equals negative 1. If you like me, this feels stranger, obviously false, when you first see it. But I promise you, by the end of this video, you and I will m... |
Implicit differentiation, what's going on here？ ｜ Chapter 6, Essence of calculus | Let me share with you something that I found particularly weird when I was a student first learning calculus. Let's say that you have a circle with radius 5 centered at the origin of the x-wide plane. This is something defined with the equation x squared plus y squared equals 5 squared. That is, all of the points on t... |
Integration and the fundamental theorem of calculus ｜ Chapter 8, Essence of calculus | This guy, Growth Indique, is somewhat of a mathematical idol to me. And I just love this quote, don't you? Too often in math, we dive into showing that a certain fact is true with a long series of formulas before stepping back and making sure that it feels reasonable and preferably obvious, at least at an intuitive le... |
But what is a partial differential equation？ ｜ DE2 | After seeing how we think about ordinary differential equations in chapter 1, we turn now to an example of a partial differential equation, the heat equation. To set things up, imagine you have some object like a piece of metal, and you know how the heat is distributed across it at any one moment, that is, what's the ... |
Nonsquare matrices as transformations between dimensions ｜ Chapter 8, Essence of linear algebra | Hey everyone, I've got another quick footnote for you between chapters today. When I've talked about linear transformations so far, I've only really talked about transformations from 2D vectors to other 2D vectors, represented with 2 by 2 matrices, or from 3D vectors to other 3D vectors, represented with 3 by 3 matric... |
Three-dimensional linear transformations ｜ Chapter 5, Essence of linear algebra | Hey folks, I've got a relatively quick video for you today, just sort of a footnote between chapters. In the last two videos, I talked about linear transformations and matrices, but I only showed the specific case of transformations that take two-dimensional vectors to other two-dimensional vectors. In general through... |
Pure Fourier series animation montage | 2 Cloves 20十 Macs Пока 800 °. ※1-2-1L ※1-2-1L ※1-2-1L ※1-2-1L ouch ouch ouch ouch ouch ouch ouch ouch ouch ouch ouch ouch ouch ouch ouch ouch ouch ouch ouch ouch ouch ouch ouch ouch ouch ouch ouch ouch ouch ouch ouch ouch ouch ouch ouch ouch ouch ouch ouch ouch ouch ouch ouch ouch ouch ouch ouch ouch ouch ouch ouch ou... |
Differential equations, a tourist's guide ｜ DE1 | Taking a quote from Stephen Stroghats, since Newton, mankind has come to realize that the laws of physics are always expressed in the language of differential equations. Of course, this language is spoken well beyond the boundaries of physics as well, and being able to speak it and read it adds a new color to how you ... |
Other math channels you'd enjoy | This is almost surreal to say, but the channel recently passed 1 million subscribers. And I know what you're thinking, just 48,576 more to go before the next big milestone. And indeed, I'll hold off proper celebrations until then, but you know, seeing that seventh digit really does make you reflect on what led to this... |
Solving Wordle using information theory | The game Wordal has gone pretty viral in the last month or two, and never one to overlook an opportunity for a math lesson. It occurs to me that this game makes for a very good central example in a lesson about information theory, and in particular a topic known as entropy. You see, like a lot of people I got kind of ... |
Euler's formula with introductory group theory | 2 years ago, almost to the day actually, I put up the first video on this channel about Euler's formula, e to the pi i equals negative 1. As an anniversary of sorts, I want to revisit that same idea. For one thing, I've always wanted to improve on the presentation, but I wouldn't rehash an old topic if there wasn't so... |
Binomial distributions ｜ Probabilities of probabilities, part 1 | You're buying a product online and you see three different sellers. They're all offering that same product at essentially the same price. One of them has a 100% positive rating but with only 10 reviews. Another has a 96% positive rating with 50 total reviews. And yet another has a 93% positive rating but with 200 tota... |
Who cares about topology？ (Inscribed rectangle problem) | I've got several fun things for you this video, an unsolved problem, a very elegant solution to a weaker version of the problem, and a little bit about what topology is and why people care. But before I jump into it, it's worth saying a few words on why I'm excited to share the solution. When I was a kid, since I love... |
The paradox of the derivative ｜ Chapter 2, Essence of calculus | The goal here is simple, explain what a derivative is. The thing is though, there's some subtlety to this topic, and a lot of potential for paradoxes if you're not careful. So kind of a secondary goal is that you have an appreciation for what those paradoxes are and how to avoid them. You see, it's common for people t... |
Limits, L'Hôpital's rule, and epsilon delta definitions ｜ Chapter 7, Essence of calculus | The last several videos have been about the idea of a derivative, and before moving on to integrals, I want to take some time to talk about limits. To be honest, the idea of a limit is not really anything new. If you know what the word approach means, you pretty much already know what a limit is. You could say that it... |
Abstract vector spaces ｜ Chapter 16, Essence of linear algebra | I'd like to revisit a deceptively simple question that I asked in the very first video of this series. What are vectors? Is a two-dimensional vector, for example, fundamentally an arrow on a flat plane that we can describe with coordinates for convenience, or is it fundamentally that pair of real numbers, which is jus... |
Cross products ｜ Chapter 10, Essence of linear algebra | Last video I talked about the Dot product, showing both the standard introduction to the topic, as well as a deeper view of how it relates to linear transformations. I'd like to do the same thing for cross products, which also have a standard introduction, along with a deeper understanding in the light of linear trans... |
Q&A with Grant (3blue1brown), windy walk edition | Who's your favorite mathematician? I always find favorite questions kind of silly, but I will tell you about two different mathematicians that have been on my mind lately. So a month or two ago, I watched this documentary about Claude Shannon called The Bit Player, and then that prompted me to read a little bit more a... |
Pi hiding in prime regularities | This is a video I've been excited to make for a while now. The story here braids together prime numbers, complex numbers, and pi in a very pleasing trio. Quite often in modern math, especially that which flirts with the Riemann Zeta function, these three seemingly unrelated objects show up in unison, and I want to giv... |
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