3Blue1Brown - 2017-03-03
Intuition for e^(pi i) = -1, and an intro to group theory. Home page: https://www.3blue1brown.com/ Brought to you by you: http://3b1b.co/epii-thanks And by the Emerald Cloud Lab: - Application software engineer: http://3b1b.co/ecl-app-se - Infrastructure engineer: http://3b1b.co/ecl-infra-se - Lab focused engineer: http://3b1b.co/ecl-lab-se - Scientific computing engineer: http://3b1b.co/ecl-sci-comp There's a slight mistake at 13:33, where the angle should be arctan(1/2) = 26.565 degrees, not 30 degrees. Arg! If anyone asks, I was just...er...rounding to the nearest 10's. For those looking to read more into group theory, I'm a fan of Keith Conrad's expository papers: http://www.math.uconn.edu/~kconrad/blurbs/ ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted about new videos, subscribe, and click the bell to receive notifications (if you're into that). If you are new to this channel and want to see more, a good place to start is this playlist: http://3b1b.co/recommended Various social media stuffs: Website: https://www.3blue1brown.com Twitter: https://twitter.com/3Blue1Brown Patreon: https://patreon.com/3blue1brown Facebook: https://www.facebook.com/3blue1brown Reddit: https://www.reddit.com/r/3Blue1Brown
This is just so, so great. Approachable but rigorous, and in its way, kind of thrilling. -John
Hi , I enjoy Art Assignment , Crash Course Theatre and Crash Course History .
@O. Kyklop you should probably take a high school geometry course
@Canol Onar
I hate this kind of Fortune Cookie way to communicate .
What exactly bothers you?
@Bruce Frizzell so basically everything Hank.
@Hrishik _ And John .
By the way: You have Feynman levels of explanation skills!
When the robot uprising comes, we will spare valuable humans like you! :)
I'm not cheking but just let me believe that you end every single one of your comments with "When the robot uprising comes, X"
Lel
Final year math undergad.
NOBODY (not even any top level book) has explained isomorphism/homomorphism to me the way you did.
Much,much love and respect from India.
which uni?
Same!!!
I've seen both Essence of series multiple times, watched space filling curves and towers of hanoi, but this video, this one right here, is the most beautiful thing on this channel. Maybe I care too much about math, but the transition near the end from 2^x to 5^x to e^x, building up to that e^i*pi chokes me up every time. Pretty sure I had tears the first time I watched it. This video transitions from "why would this ever make sense" into "what else could it be" over the course of 20 minutes. Truly a masterpiece...
+Tim H. Wow, thanks so much! That's kind of you on so many levels, I'm glad you enjoyed.
His videos have the same effect on me!
Ohh i thought i was the only weird guy who cried over mathematics videos 😂
This might be the best channel on YouTube.
this is the best channel on youtube
How do compare this channel to others? Is there a some kind of general category of all channels that you can say it the best?
@PiErDzoncy7 shut up moron..u don"t have any idea about quality content..
@imran tahir exactly 😂
It is as good as a lot of other channels ie kurzgesagt
This Channel is better than University. You are the man 3Blue1Brown
@Sophisticated Coherence but he does not teach everything else
@jsutin bibber Yeah, but he gives a basis for understanding that I can cultivate on my own time. Where uni can often leave you with nothing to build off of. once you have an inclination or visual aid it is much easier to comprehend the concepts and build on them to abstract spaces.
@jsutin bibber What stage are you at in your education? Because these videos become absolutely indispensable at some point and fourth (not gonna get ya to the finish line - bad analogy - but gives some ideas about stamina (for example).
@Sophisticated Coherence only bad universities/students operate like that
@Brandon Walker Well, I took classes at a public university and I'm not taking them at an esteemed private university and I can tell you I have experienced this in both. This may not be true for all, but it's true for many. I pay the money for the credibility, the learning I am doing on my own.
So the tales were true: there really are friendly, constructive and informative comment sections on youtube.
This is such a nice crowd of subscribers.
I am the most grateful human being after discovering this. Thank you so much. I was almost about to quit my master's thesis in maths (just because we never visualize anything, just learning stuff with the speed of light but no-one ever explains us how to imagine things and that really makes me nervous), but you're a part of the people/things that helped me get my thrill back. This is pure gold, thank you so much Sir, from the bottom of my heart.
Prvpat gledam makedonec da komentira na vakvo video lol
No offense but I find it really surpising a master's student would say this. Don't get me wrong, I'm up late watching this for the hell of it because I like the teaching elements, but I'm just shocked someone in a graduate program needs to be taught to imagine things.
@Sean Kelly its not easy to stay at the speed of university and at the same time not to lose the initial genuine passion many has at the start, cause pretending to learn even if not at 100% of your own capacity, and you know it when it happens, becouse you need to be fast and have loads of other concepts to study, then you lose some bit quantity of affinity for what you study and going on and on you could lose all the interest you had at the start. Its really not easy.. And sorry cause i'm really bad at english, i know
You must be pretty stupid to only here about group theory while doing your masters. Didnt know a math education degree was so easy
@NASA JET PROPULSION LABORATORY Why so hateful dude :)
So all e^(i*pi) = -1 is saying is that "rotation by 180 degrees is the same as a reflection". Neat!
Who is this? The perfect math teacher? Indeed.
So in this view, adding 1 to both sides to 'beautify' the equation as e^(i*pi) + 1 = 0 actually obscures its true meaning?
All e^(i*pi) = -1 is saying is "rotation by 180 degrees is the same as a reflection".
And if we indulge the tau-ists, then all e^(i*tau) = 1 is saying is "rotation by 360 degrees is the same as doing nothing", which is pretty neat. Euler's formula almost seems like a tautology in this sense, in that this is the only natural way to define e^(i*x).
e^(i*tau) = 1 is the same as multiplying by 1. As for e^(i*pi) = -1, it is the same as e^(1/2*i*tau) = --1 or rotation of half turn. So, yes pi obscures its true meaning.
You lost me when you brought up exponentials in, I can't see the relation between the top and bottom at 17:21
"Those who know do, those who understand teach." And your are one of the best examples of the second one.
20:08 As soon as you say "what makes the number e special" it all clicks in this big eureka moment. Incredibly beautiful and succinct explanation.
12:50 GOOD LORD IT ACTUALLY MAKES SENSE NOW
If you were to start from square one and do a complete course covering math from calculus I, it would become the new standard reference of the modern world, replacing all those crappy books.
Haskell has prepared me well for this “weird and strange thing of functions that take actions and return other actions”. :D
You gave me that "aha! oooooh...." moment that makes life worth living.
Thanks!
18:00 I Don't seem understand to how Grant(3b1b) is relating the two groups. I would highly appreciate it if someone helped me to understand the correlation. thank you
Mohith S N Raju all it means is that given the two actions, the result is the same for these two groups (so in this case -1 and 2 give the same result (but different paths to get to them, which is of course to be expected))
I just want to say, Grant, that I really love this revisiting of the Euler formula. I didn't quite understand the whole idea you were trying to get across in the first video, about stretching, moving, and rotating being related to certain operations. But the group explanation has actually really helped me to understand it.
This made my day, I finally went through this while understanding everything (even the beautiful, beautiful calculus)
I've been waiting for Lie algebra the entire video, got disappointed :(
So sorry, at first I was thinking of talking a little about general exponential functions in a lie algebra sense, but ultimately cut it out as being too much. Maybe in a future video, though, don't worry :)
It's an amazing video anyway, thanks for your work! Looking forward to the future videos.
17:24 i don't get that... it looks like x+y = 2^x*2^y
P W I think this part is just trying to tell us that put in a x+y first and calculate 2^x+y can mean the same as doing 2^x*2^y directly on the graph though it seems that there’s no direct connection.
We will never say this enough : your animation is gorgeous
Archibald Belanus it is.
17:15 and computer science :D
great video!
This was your all time best clip! Thank you for helping me understand.
Wow.... this is the best introduction for group theory that i have ever seen.
Thanks a lot.
awesome !!!! when you uploaded the video about e^pi i two years ago, I couldn't understand it completely and this video helped me alot in gaining insight and complete my understanding of what "taking adders to multipliers" is-the phrase you used in that video. I am always thankful to you for making such amazing videos. I wish I could meet you and learn math directly from you.....keep up the great work of making people enthusiastic in math !! waiting for your essence of calculus series ........
I just love how youtube sent me this video to recommended just to see that this 2 year anniversary happened 2 years ago
Your videos are so inspiring and very captivating. The animations and the explanations are great. I have learned a lot from your channel. Keep up the great work.
EmeraldCloudLab looks amazing
This is, hands down, one of my favorite video's on Youtube
By seeing these theories finally makes sense, helps a lot, thanks.
Is there any program which I can play with the same way.
(I am not a programmer so somewhat user friendly one would be desirable
and if it would be free and open source that heavenly perfect)
and some mathmatician said that "The greatest shortcoming of the human race is the inability to understand the exponential function."😅
20:28 mind blown. Love that feeling.
Thank you for the many subtitles .
sadly I can't like your video more than one time.
Lol
I desperately need the love button for this video!
share all of his videos on all your social networks and to your friends on messages
I'll help you in your quest for likes on this video
Because clicking on the like button twice in this group of actions equals not clicking on it at all 😂
A little bit confused
19:43 why 2^i gives rotation 0.693
20:01 why 5^i gives rotation 1.609
Natural log of the number.
because 2^0 * In2 , 5^0 * In5 ?
@ganondorfchampin Hi, thanks for the explanation, so intuitively, the transformation from addition to multiplication, does the adding action of sliding vertically in the complex plane same as saying how "fast" you are sliding relative to time?
Jerick Tan The radius is always 1 because you start from the point 1, as x^0 = 1. Changing the base doesn’t change the curve, it changes how quickly you move along it.
Keshu Nazneen Not how fast you are sliding, how fast you are rotating around. This is important because it’s periodic, rotating more than a rotation is like going negative.
it's like seeing the light of minds from the past and tge present fusionning together
made so much sens i'm a bit poetic :p
So watching this again two years later, it makes a bit more sense... except for one thing. I actually came back to this video because I've been searching for an explanation for this particular part that I can't make sense of:
(Starting at timestamp 18:32) "We already know that when you plug in a real number to 2^x, you get out a real number, a positive real number, in fact. So this exponential function takes any purely horizontal slide and turns it into some pure stretching or squishing action. So wouldn't you agree that it would be reasonable for this new dimension of additive actions, slides up and down, to map directly to this new direction of multiplicative actions, pure rotations?"
No, I can't say that's reasonable to me. The hangup is with the "pure" qualification of the rotation. A pure imaginary number as an additive action is a pure vertical slide... but a pure imaginary number as a multiplicative action is a slide AND a stretch, for every point on the imaginary axis except for i itself. So I don't see how it follows that this stretch should be dropped when discussing a transformation from additive to multiplicative actions in regards to i.
when you multiply something by a non-zero imaginary number, it should cause rotation.
19:36 - why is that so? <curious>
I'm currently in grad school studying robotics. One of the cruxes of engineering is taking maths on faith. We don't necessarily have the time to make sense of these things. Your videos have helped me understand Linear Algebra and Diff Eq so much better beyond it's applications. I really appreciate what you do. e makes more and more sense every day.
The crazy thing is even in math programs (at least at my undergraduate institution), a lot of what we had to do was take math on faith. We were basically given the definition of a group as though they were handed to us on stone tablets like Mosses on Mount Sinai. Never asking "why the hell are we using this as our definition of a group?" or "what even IS a group and what is the broader concept it's supposed to represent?" It would be like teaching a six year old: multiplication is the process of repeating the addition property an indicated number of times. Sure it's correct, but you aren't really learning multiplication at a deeper level and it becomes harder to later generalize to multiplying by zero, fractions, or God forbid, negative numbers. Visuals of rotating a 4x5 grid to show that 4x5 = 5x4 are a much better way of understanding the fact that multiplication is commutative than simply being told it's an axiom and just something you have to accept.
These 3Blue 1 Brown videos are a great example of the rotating the grid type of visuals that give one a much deeper understanding of what these concepts actually represent as opposed to some vague abstract construct with which to work.
17:00 was my mind blown moment. Amazing video.
These videos are amazing. I wish I had these videos when I was growing up. If like me you, love collecting the most important clips from these videos either to revisit or to share with other students, visit renoted . com an innovative solution to learning from videos for the new-age learners
Whenever I get demotivated with my studies, I watch your videos to remind myself the beauty of mathematics to motivate me to learn more of it. I know a lot of people say thank you, but through all my struggles, you make me want to continue to learn! So, Mr. Brown, thank you!!!!!
It's amazing how we can get to Euler's identity from many ways. I love it.
Please make more videos on group theory! Group theory in general is such a good topic to give intuition about because no one know what it is unless you study math. Also, I really need a good way to picture quotient groups!
Don't forget p-adic-ness.
oh yaa for sure
I concur, please make more videos on Group Theory! If there is any way you can connect it to chemistry/physics then it would be that much better, though I understand this isn't exactly the point of your channel. I've just taken two bioinorganic courses (biochem/chem bio major) and the prof spent some time on the application of groups & group theory in Molecular Orbital Theory. It would be amazing if we also had your stylized/visualized interpretation of molecular symmetry as well!
Or if you're a chemist
I agree: please 3Blue1Brown, more on group theory! And category theory.
15:50
Say... 2 to the "1 and a 1 to the 3".
This made me intrested in group theory, especially since you had cryptography mentioned. Gotta see if it is possible to study in my school or at University.
3Blue1Brown - 2017-05-03
For those who want to learn more about where the number e comes from, and why that constant 0.6931... showed up for 2^s, there's a video out it in the "Essence of calculus" series: https://youtu.be/m2MIpDrF7Es
Rajendra Misir - 2018-09-26
A clever way to introduce group theory. Fantastic work! I enjoyed watching your videos. According to Mathologer, that identity was conceived by Roger Cotes.
Overwatch Philosophy - 2019-05-14
Can you do a video on the wave equation?
Arnon Marcus - 2019-06-16
I have a question/observation: At 12:23 you define multiplication as scaling + rotation, but that doesn't really follow with what you showed with the number line. Following with it would be to scale each line (axis) by its respective component - that's just a non-uniform scale of the plane, NOT a rotation(!). To rotate the plane, you'd have to apply the same 2 scalings to each basis vector (like with 2D rotations), as that's changing the orientation of each axis/line itself.
Redefining multiplication to be a different kind of operation on each axis is non-trivial - can you explain why it's chosen to be redefined this way?
Arnon Marcus - 2019-06-16
Another question: At 17:20 sliding the exponent to -1 should be squeezing the number line, and sliding it to +2 should be stretching it - why is the animation showing it the other way around?
italianpro 2005 - 2019-09-05
so can we Say that (1+e) × (1)^i pi is right? ?