You never know this can just be simulation by that " hovering superintelligence " and it is so powerful that it can create complete accurate virtual characters.
Hey 3blue1brown, I recently saw a math problem that I think you could give a neat explanation to. It goes like this: “Pick x,y uniformly form (0,1). What is the probability that x/y rounds to an even number?” I’d love to get a good intuitive argument for why pi shows up in the answer (I trust the math but it comes out of nowhere).
pi? When I work it out I get (1 - ln(2) / 2) as the answer, which also lines up with the numerical experiments I did. Are you have the right puzzle/answer pairing? It was a lovely puzzle, thanks for sharing, I may very well make a quick video on it at some point.
Okay, yes, it looks like the answer (5 - pi) / 4. That's very pretty. It comes from first looking at the unit square representing pairs (x, y), then slicing it into regions that look like (n/2) * y < x < ((n + 2)/2) * y for odd n. The presence of pi stems from Leibnitz formula, that pi/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - ....
You seem too young, I had imagined much mature mid-aged person given the deep voice and domain knowledge. My mind still feels a young man lip-synced to the voice.
His voice isn't deep, but it's definitely rich and slow. The kind of voice that sounds like it should have taken much more experience than 27 years can give.
@@subaruyagami2327 Audio engineer here: shrillness is usually due to the frequency spectrum of the voice, not the pitch or fundamental frequency. A sine wave at 1kHz is not shrill (think of a high pitched whistle), but a 90-10 pulse-width waveform at 1kHz is, because there's a build-up of harmonics at frequencies the human ear is very sensitive to (1-10kHz).
"no matter what you do, your kids are gonna do things just like you" remember. be curious, i guess.. that is like a seed to grow a whole forest. don't teach the content, teach the pattern that unlocks all these different kinds of contents and applications. everyone who is good at a subject, is passionate about something. be passionate and your kids will see how to be just that. it's a metaskill contentwise
Applied math is far superior to theoretical math for introducing people with less experience to new concepts, because learning works by forming connections to things you already understand. However, if you've studied enough math, then you can start doing theoretical math and using your prior math knowledge as the thing you already understand. The only thing to avoid is becoming so engrossed in the theoretical that you can no longer relate fascinating new ideas back to reality in some way. I believe this applies to essentially all things, not just math. Music theory vs. playing with instruments. Computer science vs. writing code. Color theory vs. pottery. All practices have some theory and some pragmatism, and you can either focus on one side of things, or dive just far enough into both to find a happy middle ground where you can have the best of both worlds.
dandymcgee - This is actually how I got to know this channel! I was studying audio editing, and I had to learn a lot about mathematical concepts that can be applied on signals, like the Fourier Transformation for an example.
@@Schattengewaechs99 I came from an audio background too, so it's cool to see someone else did. Out of interest, how come you needed to know about the FFT for audio editing? Is it for a course you're taking or was it out of curiosity? I write realtime audio software (VST plugins), and would happily help you out. Learning about the z-Transform is what really made things click for me regarding the DFT, and I have some great accessible resources to share if you're interested.
damn, this does sound like a trap i fallen into, i was way too into math of music, how sound is actually made, making patches in softsynth was my favorite pastime. and now i barely make any music at all. in all honesty i subbed to this channel because it was talking a lot about FFT, something i sort of "worship", as visualization of sound helped me so much to understand and "see" it. eventually i realized it was counter intuitive, and in a lot of ways was a crutch, relying too much on eyes than ears.. and forgetting that music was about creativity... not math
@50 u gotta go do some more basic applied math first, if you're trying to do applied math at the same level as your pure math it's not going to be good for learning. i'm studying maths and physics and i've found that just doing physics problems alongside the standard 1st-2nd year math made me good at applied. i've had good success taking a pure and applied perspective on any applied math problems, it took me a few years to really develop it (2-3), but it has been so useful as a mindset. Ex. GF(2) is both interesting from a pure and applied perspectives. Pure: smallest finite field of {0,1}. Applied: has addition equivalent to XOR and multiplication equivalent to AND. it's super useful in information theory from the lil bit i learnt.
First time I've ever felt compelled to comment on anything: thank you for addressing the anxiety/depression/adhd question. It took me off guard to see it come up on your channel (which I subscribe to partially as a treatment for all those conditions). I can't imagine it being answered better. Your calm, modest brilliance is what the world needs right now. Thank you!
Claude Shannon, a man who loved the pairing of the curiously useless to the absolutely crucial. One of the only people to formulate, substantially develop, and write the seminal paper on one of the most important theories in history. Simply unbelievable.
I am into video games, and I'm a self-taught programmer, as well as a (mainly) self-taught mathematician. And I absolutely adore your videos and your explanations! You... are... amazing! You've helped me a lot! You're intuitive and you should keep this going, because MANY people need just that in order to compensate for a bad/insufficient education system. The work and the results you've achieved while popularizing math is just amazing.
Great comment. But, he's not referring to any (imminent) child of HIS, but rather the child of the questioner. Fun fact to make you feel better about that confusion: His own mother watching this video got faked raw by this very same misunderstanding and believed she had just learned of her first grandchild's conception. (And no, I'm not kidding)
I've gotta say, I love your voice. Its very relaxing and there has been times that when I can't sleep I just put Essence of Calculus on autoplay and listen to your voice.
Hey 3blue1brown! As a current Mathematics Major, I absolutely love watching your video! You have s really deep intuition in terms of math or anything that is physics related! Dont stop making videos! >=
I’m a 17 year old high school student and I’m going to uni next year. I’ve got a profound passion for computer science and cryptography, however I also really love maths and most of the famous cryptographers were obviously mathematicians so I was in a bit of a dilemma which study to choose. But thanks to your little talk on “mathematician + x” I’ve now definitely decided to go study technical computer science and learn all the left over maths by myself if I have to. Thanks!
Why not double-major? You may be able to study both at the same time. That's what I did and even though it's a lot of work, it is worth it for someone with your interests. Many computer science people don't have the strongest math skills, and few math people have strong computer science or programming. Having both becomes very useful, especially if you want to study cryptography which spans both worlds.
I double majored in both, and while I don't use the pure mathematics from undergrad (I'm currently a software engineer), the problem solving skills and techniques for abstracting problems have been incredibly useful.
I avoid math majors because teachers in my country never seems to have interest in teaching math. From middle to high school, so college must be that way too. All they taught is to "Remember the Magic Algorithm of Math Formulas", i've had enough of that shit ever since i finally learn from internet that it's the wrong way to learn math.
Good choice. You don’t need math courses to learn math. I even believe self-study math is way better. Math lecture is like definition-theorem-proof. You don’t have time to think it first yourself. But self-study is very different. Sure listening to lecture is easier, but it’s worth to spend the extra time to tackle it yourself. Also, undergraduate math course often goes extremely slow.
Haha what a coincidence I just learned Stokes theorem in calc 3. It's funny what my professor told me, apparently Stokes was a professor at Cambridge and he made a problem for PhD students which was basically proving Greene's theorem in 3d
I love studying engineering and hearing " Shannon " in my control systems lectures about sampling, and then in this video 🤔 it really gives perspective
Math is like a violin. There are great violin makers and great violin players; but rarely if ever has the greatest creator of violins been the greatest writer of songs and the greatest performer. For a violin to have value, there must be music, musicians and---of course---a violin maker. The one time I watched a man (on RU-vid) play a piece he had written on a violin he had made, I felt like I was witnessing something a little different than ever before. Not necessarily great, but different. Even the most pure mathematician needs the symbols created first by people trying to reconstruct the world they heard, saw and touched.
@@jajefan123456789 Since the hallmark of math is precision of language (as much as possible), I changed the 2nd 'song' to piece. The 1st use of song just did not resonate as 'piece.' Thank you for the correction.
I just wanted to thank you for everything! You really work hard to explain things to us. Personally you’ve changed the way I’ve perceived mathematics and for that I’m grateful!
I marvel at your resolve to communicate mathematics in an interesting and enjoyable way. I have been a mathematician for over 30 years now, and I always take something new away from watching your videos. Thank you!
It is incredible how many subs you have gained. Good indicator of how many people prefer intuitive math videos rather than boring vids that don't explain why those topics are interesting or even where do they come from.
Totally agree with your point on "how to turn someone on to maths". I used to not particularly enjoy maths until I felt like I really needed them as a software developer. I finally got around to learning about vector geometry and linear algebra one day and it opened up a whole new world of possibilities for me as a programmer and I felt like it sharpened my "logical mind" if that makes sense. From then, I was hooked.
THAT'S SOOOOO TRUE!!! WHEN YOU KNOW HOW POWERFUL IT IS, HOW IT CAN TURN SOMETHING SEEMINGLY SO COMPLEX INTO SOMETHING MORE APPROACHABLE, MORE UNDERSTANDABLE, YOU CAN'T HELP FALLING FOR IT!!! WELL SAID!!
Education for me is very social. Although I have a genuine interest in science, a huge part of the appeal for me is learning stuff and then sharing it, meeting likeminded people and talking about puzzles, problems, theories etc. Just so much fun.
8:34 Here's a Mental health PSA/personal story. There are three decisions in my life I regret not making sooner. I wish I had broken up with an abusive ex earlier, I wish I had dropped out of college earlier, and I wish I had addressed my mental health earlier. I'm now in a happy 4-year relationship, planning financially to go back to finish my degree in a year or two, and seeing a psychiatrist and therapist regularly. It is not weak to seek help. On the contrary, it is one of the bravest things you can do. Things will not stay bad forever. It may take a long time to realize how much progress you're making, but it will happen.
TLDR: necessity is the mother of invention Being a 'Mathematician + X' has certain unique benefits which make it my preference. Being an X is important for fundamental and initial research in any area. For example, nowadays it's normal to talk about n-dimensions but the person who initially thought about representing "dimension" through a line and locating specific places on it ( points) must have started out with just one dimension in mind. The question now becomes why did they feel the need to think about the world in such a manner. This is where the X comes in. Being that X is maybe what necessitates thinking about these problems in a new way. Maybe the person being talked about was involved in trade and needed a way of measurement. Although I agree not every example will abide by this POV but a surprising number of them will.
Congratulations on 2^21! Each of your videos are incredibly inspirational to me. I would love to see some videos about the relationship of math and physics. This relationship is incredibly fascinating to me but I don't feel like I have a solid enough understanding of physics (or math for that matter!) to make any meaningful connections between the two.
I think he should make more videos where he shows himself. I loved the numberphile video featuring him about the hyper-darts problem. I highly recommend it!
@@thedoublehelix5661 Well, maybe you're right but I really like how Brady interacts with his guest mathematicians and how he makes some appreciations that would have been averted otherwise like when he asked about what happens with the center or the perimeter.
Wow it's so amazing to finaly see the face behind the voice. Your explantions are very coharentic, with a great graphic demo and a calm tone. I send you from Israel my BIG FAT LIKE!
7:27 This is so real. I switched to the math major because I thought it would be the easiest major to switch to that would lead me to a job without going to grad school. Growing up, I thought math was boring and easy, but I ended up falling in love with it and will be pursuing a PhD in the fall
As an engineering student, I won't be taking any more further studies into math past basic discrete mathematics and some basic levels of vector calculus. Your channel is there to keep me interested in the subject and act as a way to show me the true beauty behind the world of mathematics on an intuitive level. Thank you so much for your work.
Or you could try writing a software that'll detect the camera shake, and then program a mechanical stabilizer to cancel it out for you. Works every time. Try it!
Great video, interesting to hear your thoughts on random subjects. Your channel is awesome, you are sharing fantastic and important content with the world. Kudos sir!
@@moartems5076 nearly all physics in simple engines is frame-based iterative summing of the velocity and position, then some comparisons to do collision. Even computing collision time is a closed-form quadratic equation in most cases. I've seen some suggestion that partial derivates are needed to handle things like cloth simulation reasonably, because energy conservation is really important there, but otherwise the systems tend to be too dynamical for inaccuracies to matter much. What has your experience been?
@@moartems5076 Eh. I see the point you're trying to make, but the actual math knowledge required is primary school level. You're probably not really doing numerical analysis here until you're doing something like computing error bounds from the exact solution, eg. to figure out physics update rate lower bounds.
Euler was always the guy that amazed me. He had a hand in everything and always seemed to tackle hard important problems. Euler Lagrange equations, e^i*π+1=0, etc. Love Grant and his philosophical approach to mathematics. The meaning and the psychology around mathematics.
@Suki Desu People who don't like math should not be viewing this video. I believe that if you don't like a video on RU-vid, you should just move on. There's no need to be negative.
Love your insight regarding the mathematician +X. Ever since I've been in high school, my passion has been psychology. Still is, that is my major with a math minor. Recently, however, mathematics has become a second passion because of its beauty and my excellence at it. It has elicited confusion making me question whether I want to go into mathematics or psychology, I've even thought about changing my major to pure math. But I've realized I shouldn't just leave psychology and what I love because I really dont have to to continue pursuing mathematics. There is actually a field called mathematical psychology that I could concentrate toward that does modeling in cognition, psychophysics, and human factors where I could combine my passions and go deeper with my knowledge. So I thought that was a very insightful point. Beautiful nature walk by the way.
What an interesting perspective, on why individual authors help people to get a foothold, by explaining it 'wrong' first. This is why experts in a field have so much difficulty simplifying an introductory lecture....
Applied Math. Two-year-olds or thereabouts have been watching others open and close doors and drawers by pulling and pushing on knobs and handles. For days and weeks and months they struggle to learn the secrets and perfect their processes of opening and closing doors and drawers by pulling and pushing and turning every which way on knobs and handles and changing their trials and techniques as they observe the fruits of their labor. Adults may see their efforts as play or mischief, but their brain is learning a form of applied math that only a small number of physics or engineering graduate students may study and understand. Suppose that your task is to move some object or articulating mechanism in some way using your finger(s) and/or hand(s). In a few seconds you analyze the object or mechanism and then apply your finger(s) and/or hand(s) to it in some way(s) and expect to see resulting movement(s). You evaluate the result(s) as feedback and may adjust your finger(s) and/or hand(s) input(s) to change the observed result(s). Our brains are quantifying what may be referred to as mechanism mobility associated with specific spatial directions. Suppose you want to open a closed door using a rotatable knob. Your brain believes that your hand must turn the knob left or right to release an internal latch and then your hand must pull or push on the knob in a direction perpendicular to the door's surface. Your brain believes that these may be the best directions because experience has shown that they generally offer higher mobility. Your brain is also aware of other directions of lower to zero mobility, which it believes would take much higher levels of effort to open the door, if at all. So it avoids them or may eventually try some as a last-ditch effort if all else has failed. My future applied math challenge will be to describe a numerical methodology that will provide quantitative/qualitative assessment(s) of mechanism mobility, similar to what our brains are doing with very little effort on our part.
I don't know if anybody ever told you that: your voice is absolutely great; it's a pleasure to listen to your videos. Which is to mean that your videos are phantastic anyway, but the sound of your voice just makes them even more enjoyable.
Since Terence Tao needed less than two hours to find three independent proofs of an identity about neutrino oscillations given by Stephen Parke, Xining Zhang, and Peter Denton, will they now rename the tau neutrino as Tao neutrino?
I am just studying information theory for my EE degree and when you named Shannon as one of your favorite mathematicians I was so happy. I also love information theory, it is a real shame not many people know about this topic.
Disagree! Most people are not able to grasp advanced math. They are able to learn arithmetic but many start to fail when algebra is coming up in class already. I know that i cant. Its not the teaching method.
@@Steuben1978 I don't think so, because whenever people asked questions in class the teacher would shame that person. I was able to learn algebra with a better teacher outside of school, and Kahn academy has amazing lessons on there website. I agree that people form put in effort to understand concepts well enough, but the school systems also promote plug and play mathematics, they don't answer why we do it a certain way, they just give formulas and you put in number and answer them
Being very good at arithmetic is not a prerequisite for maths. They are two different subjects. In fact algebra is far easier than arithmetic if it is taught correctly. Teachers are not mathematicians and generally work as teachers because they are not experts in any subject (not all of them, but most of them). That is what is wrong with schools. If you selected the best teachers, then maths teaching at schools would work. But that is not how it is done.
@@kevinmccallister7647 That does not refute my point! People differ greatly in their intellectual ability. Yes, a bad teacher or a dysfunctional family can make things worse, but for the many people who lack the necessary spatial reasoning or mathmatical ability or have bad pattern recognition....even a good teacher will not be able to make them grasp advanced math.
I love your videos. Graphics, style, how you explain things. I love the intuition behind your videos. You don't just explain things, you convey a feeling. And I love your voice! It's overwhelming to see the face to that voice! Excuse me for my bad englisch please! Greetings from Germany ❤
I *Grant* you this, in all the years I watched your video's, I pictured you somewhat differently, and being around *40* years. And I must add, you have the ideal voiceover personality for attracting people's attention.
Good choices on favorite mathematicians. Personally I’m a fan of David Huffman, who was behind a lot of modern compression theory (he’s the H in LZH) and also was passionate about paper folding as mathematical art.
"The Information" by Gleick is a masterpiece. I reread it every couple of years. Exactly how humans gained mastery (drastically decreasing entropy) by "learning" 🙂 to control information. Add a book on the evolution of RNA and metabolism (in the ocean) and the big picture comes into focus.
Perhaps a question to be answered in a future Q&A: What is the average angle between the hour and minute hand on a clock? I came up with this problem randomly while driving and immediately began working on it when I got home, it is a lovely little problem with a very unintuitive result. (I think)
Interesting! Can you clarify what is meant by the angle? Like if its over 180 degrees do we just count the small angle, or should we imagine it is 0 degrees when they intersect and then it ranges up to 360 degrees?
I never found the usefulness of maths until I used to google things like “what’s the significance of matrix? or what’s calculus?” and other questions that teachers found baseless. But I found the answers in quora, math exchange, eddie woo’s yt channel and of course 3blue1brown. I pray for all the people, resources who were the reasons in my life to turn the hatred into Love ❤️ Also, the way Grant inspires us to actually enhance our thinking of math is absolutely amazing! He’s a blessing, especially for students like us in the subcontinent where they barely have any good resources to quench the thirst of learners
