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: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-m2MIpDrF7Es.html
3Blue1Brown could you do simplectic geometry in classical mechanics ? Maybe a topic from Arnold's or mardsens book on classical mechanics. Those are graduate text to mathematics, but they cover physics
aa That's not the point of the channel. Anyone who has done a maths degree has seen the proofs. This channel is to show maths to people who haven't done a maths degree
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.
@@jsutinbibber9508 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).
Im sorry to hear you didnt go to a good one. In europe its very cheap and right now im studying at University of Twente in systems and control, it is very good. The teachers are competent and caring, our facilities are good. The campus promotes sports and social associations that help you develop skills outside of the core classes. I might be lucky but it is not a scam.
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.
I think one of my favorite proofs I have found of Euler's formula was in a book I picked on relativity, where it starts with only the equations ds²=dx²+dy² and rθ=s, and ends up showing rθ=x+iy, without even defining the number π.
We need a series for Essence of Group Theory. Students in high school are taught this subject in a purely numerical and definitional way, nothing to give them intuition or any representation of what they learn about at all. Everything is just symbolic algebra thing without sense, created to torture students. We need light like this in a dark educational system.
I love your channel. I usually end up watching your videos right when they come out and I tend to just go with the flow bc I hv no idea what's going on lol. But then I watch them a few more times, maybe not to completion, or with huge gaps of time in-between viewings, and I suddenly feel like switch go off in my head. And then once that happens and everything falls into place, it just feels beautifully awesome and I can extend my gratitude enough for making such abstract concepts relatively easy to digest and really comprehend. Because at the end of the day, I could read a textbook, but this gives so much more depth which is extremely helpful especially when starting out.
It's difficult to appreciate just how amazing this video is until you've actually done a course on group theory. When I watched this a couple of years ago, I got the main gist of the idea and I was impressed. However, even though the idea in itself is beautiful, I appreciated how succinctly you summarized the actual rules of group theory so much more. Also, in my education, I didn't really look at group theory as symmetric in nature, so when you showed the different numbers as actions that preserve symmetry it blew my mind. Great video, probably one of the best on this channel (and that's saying something!) :)
group theory has bought us the solution of various problems which remained unsolved for hundreds of years...i didn't know it was that easy.Thank you very much 3Blue1Brown.
The extent with which you can wrap my brain around some of these identities and concepts is SO WORTH a commercial or two. THANK YOU for feeding my visual learning preference. I am teaching myself Data Science and Machine Learning along with AnimateCC (Flash :) I have been trying to recreate the animation in my head as I learn - fascinating.
When you started to explain that additive and multiplicative actions are kinda the same I suddenly noticed that it looks similar to, how you explained, higher dimensional objects act upon lower dimensions, or how they look in this dimension. How the stretching of a line is actually just the rotating action in the next dimension
Thanks for the video. The analogy is really helpful. I have just one question: In 20:00, you explain that the additive action maps to the multiplicative action, and that the action of vertical sliding maps to rotation as a result. How do you come up with the numbers for the rotation in radians? What is a more intuitive way to understand this scaling apart from using Euler's formula? (which is what we are trying to prove so it does not make sense to use the formula before proving it).
Me 3! At 19:40, he said "it happens to be 0.693 radians", which is the moment that made me really want to now *why*! The explanation at 21:06 kinda touches on it, because it hints that the derivative of different exponentials would mean a different rate of change around the unit circle, meaning that Euler's formula only works for e. That said, he still didn't elaborate on it enough I think. Edit: If you do the derivative with 2^x instead of e^x, you get i(ln(2)2^(i*t)) which might be the explanation we were looking for! Because the factor in the derivative is converted into a factor in the velocity around the circle. This explains the angle we get if we use 2^x instead of e^x, as well as e^x is special.
Excellent. My only quibble is that in discussing multiplication on the number line, you only referred to the "multiplicative group of POSITIVE numbers." And while it's pretty obvious what multiplication by negative numbers should do to the line, you didn't spell it out. And yet, when talking about i*i = -1, you referred to -1 as the "unique action" without having earlier made clear that negative numbers are also a valid part of the multiplicative group.
I actually found that quite genius of him. The point was that positive real numbers are pure stretchings while negative numbers also include rotations, which are a complex thing (and notice that the real exponential function only puts out positve numbers, so the real exponential function only gives you a relation between the additive real numbers and the multiplicative positve numbers)
uhm.... it's genius of him because frankly multiplication by real negative number is just stretching and flipping through 0, it's not really a complex thing. In 1D, there is no need for rotation to involve. It happens that in complex, "flipping" through y-axis is 180 degree rotation.
Exactly. If he had talked about multiplying by negative numbers in 1D, it could have looked like flipping. But that would have created a misconception when extending to the complex plane because multiplying by negative 1 is not flipping in the complex plane but just rotation.
It's 'cause only the positive reals are in the range of the exponential map e^x (where x is real), but all complex numbers other than zero are in the range of the exponential map e^x (where x is complex).
After reading a basic algebra book at least one month, I was blocked by the content of the group theory and could only drink ... , but this video made me start enjoying the world of group theory. Thanks for the excellent presentation!
I appreciate the thought, effort and the new ideas that this generates. I just hope to see other comments that ask questions and answer instead of just appreciation. No shade on the channel, i was just hoping to see ask questions and get answers here, no way that can happen on youtube, hopefully, youtube implements a 'forum' type of section apart from comments where people can ask and answer just questions regarding the video. Not realistic i know, but i wish for it anyhow. Questions always popup when i see 3b1b videos, there are some leaps that are taken that are perhaps very common to some, but require a lot of effort for me. it took a lot of time for me to understand the idea that, the multiplicative transformation happens along ONLY the unit circle is cause the sum of how the eulers formula expands, cos(theta) + isin(theta). A more intuitive explanation for this would have helped me a ton. Thanks you once again!!
0:48 "so here, two years later" *whince* This much time already? D: After checking, I'm relieved, that for me at least, it's only a little bit more than one year. But still, time passes! :/
Odd how your vocalization has slowed slightly over two years. Is the aging process exponential or linear? Will your voice be twice as slow in another 2 years or slowed to a barely audible rumble? ;-)) Wish we had vids this back in the Sixties. Nice!
I left a couple tech universities 15 years ago because I couldn't understand calculus, then in a while started my programming career, and always thought I wouldn't ever understand such high materials as why multiplying by vector must work like that. Now I finally do.
At 17:36 , according to preceding explanation, -1 should mean sliding to the left, 2 sliding to the right which should make 1 locates to the place where original 0 does.
Essence of Abstract Algebra incoming?! I've spent the last 10 years of my life thinking groups were just "what would happen if we deleted some axioms." I never got this kind of intuition for groups in my college class. Thank you as always!
Thanks! This is one of my new favorite videos. I took an abstract algebra course and learned all about homomorphisms but it never clicked until now to define exponentiation as the function that preserves the homomorphism between additional of exponents and multiplication of exponentiated values.
I have a feeling this is going to get a second wave of views in the wake of Alan Becker's "Animation Vs Math," given it has Euler's number right in the thumbnail, being the "main antagonist" of the video.
Thanks for sharing! It is like a kind of encapsulation. You can use this formula not really understanding why it works, as like as you can drive a car knowing nothing about of the Laws of thermodynamic. Your wonderful explanation allows to look under the hood of the Universe.
Three years later, but here's how I understood it. I don't think there's necessarily a direct relation between the operations themselves, meaning a way to deduce the operation that must be performed below by seeing the operation performed above, but what happens is: On the top, you slide by whatever the power is. So if the power is 3, you slide by 3. If it's -1, you slide by -1. On the bottom, you squish by 2 to the respective power. So if it's -1, you stretch by 2^(-1)=0.5. If it's 3, you stretch by 2^3=8. This is because exponentiation is repeated multiplication, so if you raise 2 to the third power, it's the same as stretching by 2, then again by 2, then by 2 once more.
Isn't it trying to say that both additive and multiplicative actions have similar result for the same expression? Sliding 1 unit left and then 2 unit right lands on the number 1. Squish by 0.5 and then stretch by 4 lands on number 1 also right?
@@cliffordwilliam3714 If you start the arrow on 0, squishing and stretching by any amount shouldn't move the arrow away from 0. I don't understand at all what he's trying to get at in that part of the video.
Your Brilliant!!! I've had several Professors attempt to explain this concept.. I watch your video for 15mins and BAM!! Keep creating AMAZING CONTENT your truly game changing force in mathematical literacy!
A positive statement propels hope toward a better future, it builds up your faith and that of others, and it promotes change. Jan Dargatz, Publishing professional
Bro, it's super nice that you remove ads from the videos for month one, but as far as I'm concerned, when I watch a video of your with the intent of actually understanding it, I have to pause it myself so often that honestly I didn't even notice the ad breaks 🤣
Does this mean the vertical axis of the additive group undergoes modulus arithmetic when its transformed? Like with e as the base, a vertical transformation of 2pi maps to a 360 degree turn, or the same as doing nothing? Do you "lose" information then when this is done?
Wow thats a pretty good catch. Indeed if you look back at the group of rotations of the square, this group is the same as the integers modulo 4, and the circle group (group of all rotations) is just the Real numbers modulo 2*Pi. The loss of information you describe comes from the fact that exp(2*Pi*i*k)=1 for all integer k, so multiple points are mapped to 1 (adn thus multiple points are mapped to any point). In general, "loss of information" like this are so important they are described through the fundamental theorem on homomorphisms. the most important theorem of grouptheory
Everything is great until 16:58 and onward. (The following references the graphic at 16:58): What are the inputs and outputs exactly?? I was under the impression that the variables x and y were the inputs, each side of the equation represented a function with two arguments, and the equal sign signified that the two functions, while different in operation, represented the same mapping of inputs to outputs (i.e. the function 2^(x + y) where the two inputs are added and applied as the exponent to the base 2 with the power being the output of the function will map those same arguments x and y for any specific inputs to the same output as the function (2^x)(2^y) where each parameter is applied as the exponent to the base and the results of each of those operations are multiplied together with the product being the second function's output. I have more questions that extend past that timestamp but I assume they stem from whatever my misconception at this point is. Does someone understand what I'm missing?
What he is trying to demonstrate graphically is that he has a function... Let’s say F(n) = 2^n. The input is n, and the output is 2^n. Now, consider what happens if we have two inputs x & y. Does it matter if we add x + y first, and put the result through the function F? Or do we get a different result if we put each of x and y through F separately and then apply the multiplying operation on the outputs of this function? F(x+y) = 2^(x+y) = (2^x) x (2^y) = F(x) x F(y). He’s basically building a graphical way to describe how doing the adding action on the inputs in the input space is the same as doing the multiplying action on the outputs in the output space. I hope this helps.
Ruslan Goncharov When you map the exponentials in the complex plane it corresponds to rotations around the unit circle. So 2^x can be written as e^xln(2). Taking the rate of change (i.e derivative) of the rotation, you get the original function times ln(2). So ln(2)2^x. Where ln(2) is approx 0.693. Same with say ln(5), as 1.609.
+Marcel.M But, why take the derivative? Shouldn't it be that if I were to have, say, x = 1, mapping it by 2^x would just directly rotate it by 2 radians...? On which part did I misunderstand?
I find it easier to understand why groups are useful for everything with the abstract definition: a group is a set, plus an operation with some particular properties. Change the properties of the operation, and you get things that are not quite groups, like a semigroup or a monoid. This is then easier to apply to different fields of study. I find this framework easier to understand than 5:55 now. Numbers being able to form groups is also simpler to understand: they are analogous to actions, but not necessarily literally actions. I didn't get how important the phrase "group operation" (7:33) is when I watched this in the past.
Your thoughts sound not far off from transformations in graphical math, such as moving and rotating a point in 3D space. Have you examined quaternions? They follow a similar line of thinking to this 2D complex plane, except that they're 4D numbers: a real number line and 3 imaginary number lines, all orthogonal to each other. I wrote an extensive essay for myself on them a couple years ago and their big brother, the dual quaternion (the dual operator was interesting).
John Cox they don't sound far off because they are not. Crystallography is the direct application of group theory in the graphical math sense. It can be used to describe the symmetries of the atomic positions in crystalline materials.
I don't get why symmetry is a condition for defining a group, after all, for a purely asymmetric object (especially for such an object actually) you can still say that : some transformation plus another transformation is equivalent to doing a third transformation only. Why bother with the symmetry ?
In order to get a group this way, you need this functions to be invertible. And invertible functions are in a way symmetries on the set you are looking at.
But an asymmetric object can be rotated 90° to the right and then 90° to the left and still be asymmetric. The symmetry argument apply to member of the set and not the transformations on them, right ?
@@LesSpinsSo in the case of the asymmetric object that you're rotating, there is still an underlying symmetry in your rotations, namely the symmetries of space itself.
Awesome video! I adore the anthropomorphic Pis, but e is still my favorite 😂 This took me back to my senior year in high school. Still love the theory, but actually proving group properties for operations on complex numbers was such a hassle.
10:03 Schizophrenia is not split personality. It's not even related. It's more of a hallucinatory disease, mixed with psychosis. (A person with schizophrenia imagine things, and cannot tell what's real and what's not.) Conversations with people who aren't there, or general paranoia are some manifestaions.
Well, confusing what is real and what is not and being stuck in a rather complex state of mind seems perfectly fitting for this video. Now, let's ask that group of people who aren't there whether they are interested in switching places, and whether they care about keeping the same neighbors.
You should’ve drawn the arrow is the exponent differently, so it’s clear that the point at 0 in the additive group results is the point at 1 in the multiplicative group (f(x)=2^x as opposed to f(x)=0)
I used to be a "I don't like math" kind of person and now I hope that one day I can at least understand some of your videos lol. I'm in my mid 20s and I haven't even studied calculus yet and I'm currently learning some very basic math on Khan Academy... Wish me luck!!!! ~_~
This is amazing. I wish I'd studied more mathematics before approaching crystal structure. We used these concepts, but I didn't understand them in any rigorous sense. Good stuff.
Thanks for explaining this visually! There's still thing I don't understand though: We've established that: x + Re = horizontal transformation x * Re = horizontal stretch x + Im = vertical transformation so logically, x * Im = vertical stretch but in fact x * Im = rotation Why is this? Thanks for your help!
x * Re = stretch in all direction with 0+0*i fixed, or, horizontal and vertical stretch with 0+0*i fixed, or, x-direction stretch x * Im = no stretch... no more than rotation
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 :)
Another fun side-effect not covered by this video is that Euler's formula gives us a neat and concise way of turning polar complex numbers into rectangular complex numbers. The polar coordinate (x, θ), that is the angle θ and the magnitude x can be written as xe^(iθ). This notation does two neat things: 1) it encodes the polar notation directly, and 2) when calculated it becomes x (cos θ + i sin θ). So in a sense, it's a neat way of getting both polar and rectangular complex numbers in one notation for free.
At first it seems like a problem that n^q*i can equal n^p*i for unequal a and p, but even the definition you originally learn for exponents leads to i^(4+a)=i^a so one should just accept exponents aren’t unique when it comes to complex numbers
I took group theory last quarter and ring theory this quarter... ... and we never got a chance to explore the content from this angle. The closest we got was groups acting on sets and partitioning them into orbits or creating permutations of strings of numbers (which, I guess, is the same) but I don't think the creative uses for group theory were explored adequately. A similar thing is happening with rings too. Idk why but math at this level stops focusing on the creative side and starts focusing on getting everyone to do quantum mechanics.
So in the complex plane, when you do a^b it turns the sliding action of b into a multiplicative action of b with a rate proportional to lna? So when you do e^i it turns the sliding action of i to a multiplicative action of i (which is rotation) with the rate of ln e, or 1. Makes sense
I think your visualization at 17:52 is off. The final solution to 2^-1 * 2^2 should be 2, but the visualization shows the result as being at 1/2. Otherwise, great video - clear explanations with good visualizations that help build my intuition with group theory. I've watched this one twice.
It's funny because a studied group theory in the context of inorganic chemistry where it finds some applications, but I naively didn't know that it was the same thing as group theory in math. I really don't know why I thought that lol
A very good video - the only thing that seriously spoils it for me it constant, grinding muzak in the background. I know this is sort of trendy, but I really wish people would just leave it out. A worthwhile video like this does not need to distract the viewer from what is being said - leave that to the conspiracy theorists.
Isn't weird how one increase in the complex number line moves the circle exactly one radian? It's strange how perfectly each one relates to each other perfectly. I wonder if there's a underlying reason....
