e to the pi i, a nontraditional take (old version)
HTML-код
- Опубликовано: 8 сен 2024
- Wait! There's an improved version: • Euler's formula with i...
Also, for the calculus-savvy, you'll prefer this one: • e^(iπ) in 3.14 minutes...
Home page: www.3blue1brow...
The enigmatic equation e^{pi i} = -1 is usually explained using Taylor's formula during a calculus class. This video offers a different perspective, which involves thinking about numbers as actions, and about e^x as something which turns one action into another.
For more information on viewing exponential functions in this new light, check out this article: goo.gl/LPyc4R
Music: "Wyoming 307" by Time For Three, www.tf3.com/
This video is not exactly my proudest work, I have since made an improved version of this video, giving some of the surrounding context and presenting things at a more reasonable pace. Check it out here: ruclips.net/video/mvmuCPvRoWQ/видео.html
(Edit 2 years later): And now yet another! If you're comfortable with calculus, I think this 3.14-minute explanation is the best way to understand it: ruclips.net/video/v0YEaeIClKY/видео.html
Hi,Can you tell me which microphone you switched on to? as your voice in latest videos in very pleasant.
I liked the music of this one better though
It is difficult to understand because π is not a number in that identity. There is no valid parameter to allow us to identify π with a number.
IT means, that the value for π emerging from this function is just generated by the function itself to satisfy the identity. With the function you're intruding beforehand the value of π you will obtain. It has nothing to do with the real π function of perimeter/ diameter.
Why have you deleted the"Crash course on complex derivatives "video????!?WWHHHHYYYYYYYY????????????
3Blue1Brown you're amazing
Before the video:
I came here to understand Euler's Rule
After the video:
Scrolling through comments to see whether anyone else was also lost
It seems everyone was
But I watched this video over 5 time throughout the years
Only now when I came with the question why in multiplying complex numbers we rotate by the angle and multiply by the magnitude that I truly understood it.
I'm fucking lost
you're welcome :)
I think that was the point of this video...
at least the animations are good
So m I...
wow, feels like i'm entering a time mahine
First 3 minutes: okay I can keep up with this
Last 3 minutes: wat
Derek Leung but you're an Aperture scientist ... How did ApLabz hire you if you don't know college grade maths ?
Mihai Lazar tbf he made the comment 2 years ago
The amount of knowledge passed down at each instant during this video, is pretty well described by e^(timeline)
Thank you for leaving this old video up. It's an inspiration to those of us wanting to do our own videos: it tells us your amazing style didn't happen overnight, but rather took some experimentation to get just right.
samlawhorn yea it felt like someone else’s videos, then i saw it was really old
He talked about this in his 3B 1B podcast, he though this video was bad in terms of explaining things. So you are right it really took him some experiments.
Wow he still has the same beautiful voice 9 years later🥺
At about 3 minutes....actually a bit before, you begin to speak very rapidly about complicated concepts. You speak slowly in the beginning and explain the easy stuff in great detail, but then ramble quickly through the material that gives me pause. I enjoy your video as you have a nice speaking voice and good visuals, but I wish you had moved quickly through what was simple and then slowed down for the more complicated issues.
+Mark G If you click on the 'gear' symbol in the bottom right hand corner of the video, you will be able to adjust the speed of any video to be slower or faster at different points at your preference. You can always achieve this more easily by using typing the < symbol to slow down or > symbol to speed up. This can introduce distortions on videos with backing music though so I understand your frustration. I just thought I'd share as I often use these in other videos to rapidly parse through their content.
+Mark G I always have this problem with teachers in general. They start to explain the simple things very clearly and then as things get more complex, they increase their pace.
+Apricots I tend to think this is because people that really love a subject tend to get ahead of them self. You start slow, because it is boring, than you get excited because you get to the interesting stuff. And you forget that not everyone loves a subject as much as you do :D
*****
That's what I REALLY(!) hate. I got my self a couple of books about math, and some are clearly much better worded and written compared to others. I am sure, they are all "correct", but I don't need someone to explain the most simple steps. I need a book that isn't leaving out steps in the complex matters. And that happens sadly way to often, so I end up looking for other sources where they actually show what they did :/
+Mark G
That's actually the better way to do it and here is why:
Easy stuff is the basis from which you go forth. Clearing that part slowly enables you to recall information that is attached to it so it's ready to use. The complex stuff afterwards takes different amounts of time to understand for different people. As it's a video you can pause and rewind as you like but having it go slowly would impede this phase of understanding because you'd be slowed down by the video when trying to double and triple up on what he said if it's too slow for you. If it's too fast, pause longer. If it's too slow, you're stuck.
Watched this video pretending to myself I understood a thing.
Daniel Astillero same here
+Evi1M4chine How about next time, you edit your post until it is free of logical, grammatical, and spelling errors?
Same
In Mathematics it is not that simple to understand things. I myself have training in mathematics for I am a Statistician and we use complex numbers. But I had to see this video 5 times, take notes, stop it from time to time, memorize a couple of things, until I understood the whole idea. At the end the effort made me feel satisfied. This video is just fantastic.
Me too
What an experience it is to watch this video years later after learning advanced math in college. I remember seeing this years ago and not understanding a thing, now I understood almost everything. You have given me a fantastic new view of mathematics. Learning this stuff in Calculus, Linear Algebra, and Differential Equations was great, but this video really brings it together with an amazing new perspective.
It's so amazing when you get a completely fresh perspective on things you've already learnt the conventional way
Both your pinned comment and the top comments are retrospective and think of this video in lesser terms as if this should meet todays standards. I cannot disagree more, these videos were unique for their time, your skill in translating topics has improved and as a result so has general knowledge.
For people who only have themselves as a test, mistakes are the most memorable teachers.
I...I think I'll stick with the calculus proof.
Faster, please.
+balanemate LOL
Yeah kinda slow, wasn't it... I felt like I understood everything in the first 30s :)
+G Yogaraja Then you should never use a macroscope.
I have found that "faster, please" usually means "harder, please."
Try the One Minute Poem...
This would work so much better if you had actual numbers on there. You're rotating.. ok great.. hard to see what is actually happening when there are no numbers!
+muesk3 Thats the point, everything is relative. You can make those numbers anything you want, as long as they are scaled proportionally.
Its just easier to visualize. This wasn't a very intuitive way to show it, in my opinion. I still don't grasp the rotating.
+muesk3 yeh, I'd definitely agree to that! I'd also welcome some numbers. I know it's a general explanation, but explanations live by examples :)
I am a programmer, so it made complete sense to me.
Putting in numbers breaks the whole damn example... you do not understand if you need numbers to make sense of it, the point is that the exact numbers do not matter, only the distance and relationships.
We are doing relation based calculations with an infinite set of numbers. We are in a different realm of math than that of the real number line.
Let me give you an example. Lets say you have a graphed line. This line is a wave that extends indefinitely in both directions, but you have to measure the average of that squiggly line.
Trying to calculate this with actual numbers will be hopeless, and completely impossible to do with perfect accuracy, so you need to use calculus to define properties of this line. This is what he is explaining here. The rotation of the point is referencing the effect an aspect of the formula will have on the result.
I hope this does not confuse you more.
+Richard Smith the operation itself is confusing. I'm merely saying that if given some example using normal numbers, I'd understand more of the rotating on the plane. it was done on the line segment, but not on the plane.
alright folks, I totally agree 3b1b's level in 2020 is just unmatchable, even by 2015 3b1b himself. but just (pause and ponder) consider his progress. I mean.
also, it's not that this is a better proof for e^(pi*i) or something, but the approach is really unconventional and creative. and since this is such an isolated equation, it's a perfect place to start your channel.
+ notice how back then, he already hat his oustanding and brilliant phrasing, like when he said "the life's ambition of e^x is to transform adders into mulitpliers" - that's just talent. this doesn't have to be your (or Grants) favorite way to think about e^(pi*i), but it's a different approach so it's inherently worth considering.
I've just begun to visit his old videos after being a longtime fan, and I really liked this! I certainly see where some people may have gotten lost, but I am hugely fond of the graphic representations as soon as the 2d plane appeared here!
@@aepokkvulpex I totally agree, it's so satisfying to watch.
0:31
This video proves that
pi is "What even are numbers?"
I really think you should revisit these earlier videos, especially this one and the one you link to at the end of it, with your newer, clearer style. Your later videos have animations that are even more beautiful and with narration that is easier to follow.
A good point. Somewhere on my list of videos to make is a better explanation of this e^(pi i).
After seeing the first chapters of your algebra series, most concepts on this video are clearer!
Mathologer has already done this one very well I think.
+
Yes, Mathologer's explanation is beautiful.
I think I understood my hideous calculus 3 teacher better than I understood this video.
Oh, it's all clear to me now. How could I not have seen this before?
Because you are an egg
@@elijahr_1998 bully -_-
Maybe take your Oakleys off and you'll see better.
You are great at making things sound simple without making them actually understandable
OMG. All engineering students and professor and engineers should watch this to get the real sense of e. This episode should have 1 billion views.
I think this was made harder than it really is
+shaba supermayn
It might only seem that way if you are already familiar with how it 'really' is.
Someone new to complex numbers might get a useful insight from the video, though.
Still, nothing can really beat reading and working the exercises in a proper textbook. Complex numbers are especially magical and well worth the effort, in my opinion.
lol what i meant was this video makes it harder than it is there are other videos on youtube that explain it more simply and more to the point so this video makes it seem harder than it is
+shaba supermayn I couldn't agree any more - this is the worst explanation I have ever seen!
its impossible to follow those rotating shrinking and sliding graphs. there is no sense of relation to anything or absolutes of anything.
clear as mud
hahah exactly
Lol
Says more about you than it does about the vid.
@@sloaiza81 that's the point
You mean muskeg
Lol
Why am I interested in this stuff I am not comprehending... I am more confused than before I watched this.
Follow your curiosity
Dw I am 3rd year maths and don’t get it
Same, I'm in grade 11 but I'm watching this
@@notsoclearsky I'm 12 years old.
@@mariomario-ih6mn go learn algebra first.
These older videos are way more artistic in animation but they're 2fast, 2 quick 4 anyone to understand
It's a good thing he started talking slower as time went on and those moments where they stop by and have the pi figures talk is a great time to let the facts sink in and not rush though the entire video at break-neck speed
I am a student studying in school and honestly I didn't understand 70% of it but appreciate the effort done!
I think this is really an explanation for mathematicians more than for lay people; to really follow it, one needs to be comfortable with reframing things in an abstract way, defining functions by functional equations, and choosing things based on naturality. To a mathematician, this makes perfect sense: view the real numbers as their actions on the geometric line. What could be more reasonable? It is a familiar thing to do, also, with many examples in mathematics of great success with this abstract (dare I say relative (a la Grothendieck)) point of view. And then all you have to do is choose whatever is most natural, which most mathematicians will 'naturally' do, and everything falls out beautifully. But to many it seems like hand waving, even though it's not; it's definition waving, which I have no problem with at all!
WOW redefining what numbers are intrinsically. This is what I've always been curious about but never actually been able to ask about because when I ask my teachers "how can I think about multiplication fundamentally" they look at me like I'm stupid because I don't know what it is.
This is it man, thank you so much!!!!!
Connor King If you’re interesting in this, then you should really consider studying group theory.
The thing is, the notion of numbers you have is probably nothing close to the most generalization notion of what a number is. The best way to think of a number is as an iterator, it describes how much you . Adders are just shifting by a certain amount, or shifting a certain number of times, multiplication is just stretching by a certain amount, or stretching a certain number of times. This notion extends to complex numbers as well, it's just only easy to define with natural numbers, the rest of it is just an extension of that notion to geometric space.
same here. I always wondered about this, especially how complex numbers work, and how in class it involved a graph and i was SOO lost.
Putting it in this frame work makes perfect sense and it comes from a fundamental perspective change on how you visualize numbers...it's like you can probably get more and more different number types by just going up in dimensions...that's also probably how genius's like Hawking and Einstein see the world, as geometries rather than numbers.
This video doesn't have the same explanatory power as your others.
+
Why do you think that?
And most of the non-subject related animations are rather distracting than helping.
take notes and watch it again
It's his very first video as well. It's nice to see how he has evolved.
I watched this a week ago, and I only understood until the addition and multiplication point. I came back a week later, and now I understand everything.
Holy shit
now i know why that equal to 'cos(180)+sin(180)' in electricity
I love your videos, but I feel like they're more geared to people with masters in math, rather than people who are essentially lay men. Like you have to already know what it means before you learn the definition.
Like they're good visualizations for these advanced mathematical concepts, but us laymen are not really the intended audience.
Andre Tsang i understand it and i dont take the advanced classes
Ethan Marsingill Nice.
ikr
Mathematical Theories!
See my latest post, and I assure you you will understand the math/geometry
This explanation is a little more intuitive for new learners with e^(i tau) = 1.
I GET IT NOW THANKS
Tau = 2pi, ie (e^(pi*I))^2 which is just the square of e^(pi*I).
What are y'all talking about
@@dee-mv1os Look up "The Tau Manifesto". It is an idea that we'd be better off if we had defined tau as the circle constant, instead of pi. Tau is would be the ratio of a circle's circumference to radius, rather than using pi as the ratio of circumference to diameter. Since 2*pi appears so often in mathematical formulas, this would be replaced with tau, which can simplify a lot. Particularly radians, since when you state radians in terms of tau, the fraction in front of tau directly tells you what fraction of the circle it is. This might seem like it would make the area of a circle more complicated as 1/2*tau*R^2, rather than pi*R^2, but the leading constant of 1/2 tells you something, as it makes it immediately obvious that the area of the circle is the integral of its circumference relative to radius.
My opinion: if it ain't broke, don't fix it. Just write "let tau = 2*pi" at the top of your work, if you prefer to work in terms of tau.
I've struggled with math my entire life, it's one of the only disciplines that hasn't just come naturally to me. Your videos make way more sense to me than anything that my math teachers could have come up with growing up.
I cannot believe this came out over 5 years ago. I could have swore it was later. I'm now in college hoping to finish with a degree in math.
Adding simple number line labels to the graph animations would drastically improve understandability (particularly in the case of rotations). This is truly a fantastic video- a great tool for a new way of thinking about basic algebra and complex numbers. Thank you!
You seem to have a very good grasp of math itself, but an extremely poor grasp of how people learn. When introducing a new concept, especially when people first have to unlearn / let go of their previously learned paradigma, information *REALLY* isn't absorbed as efficiënt as possible when you ramble through your newly introduced paradigma in a few short minutes.
+Smonjirez It's a bit like: "Okay people today we are going to introduce a new language. Let's say we replace every letter in a word by the sum of the letters before it. So: 'hello' would become [h(8)], [h(8) + e(5) = m(13)], [h(8) + e(5) + l(12) = y(25)], [h(8) + e(5) + l(12) + l(12) = k(37 - 26 = 11)], [h(8) + e(5) + l(12) + l(12) + o(14) = y(51 - 26 = 25)], so our final word would be 'hmyky'. Now it's obvious to see what the works of Shakespear would become if you apply this operation to every word."
+Smonjirez I like the effort you put into this comment
That's why the learner must be active. Replay, rewatch multiple times.
That's a really good constructive criticism
"move the line, stretch the line!" All I got from it
I love the aesthetics of your video, but I feel like your explanation was too brief and too rushed to understand, and I ended up being more confused than I already was by the end of your video. You're trying to introduce too many concepts which go against our traditional education; I suggest creating a series of videos instead which build on each other that introduce these novel approaches beforehand.
Having said all that, I thoroughly enjoyed the presentation and think you're onto something here.xx
You're better than my calculus teacher..
Wish you taught me first so I didn't fail my calculus.
WOW, those animations are sick! This viedo is one of the bet Vodds i'v ever ssen! (content+optical)! Love it.
[Edited--see comment below that corrects something I said here originally]
I don't think you succeed in explaining why e raised to pi i equals -1. Your visualization of adding and multiplication are fine; that's a good way of looking at it. Even in the complex plane I think your definition is fine--multiplication rotates and stretches. That's fine. So, how pi is related to -1 is even fairly easy to see based on what you're saying here. You're rotating half way around the circle.
But you do MAJOR hand-waving when you try to explain the relationship between e and -1. Your explanation completely falls apart with this "relation" between adding and multiplying. How do we get from e^(x+y) = e^x * e^y to 1 + 1/2 * x + . . .?!?! What are all those fractions? What are all those powers of x? You didn't explain anything about that! And then, you also didn't explain how e relates to the unit circle at all.
The adding and multiplying are all well and good, but the ACTUAL difficulty of that equation--wtf is e doing there--you completely glossed over. Because the real question you have to answer is, why does e, of all numbers, rotate without stretching?!?!?!
I know already that the reason usually given is that the Taylor series for e is what you get when you add together the Taylor series for cos and sin, which will then give you a circle. But THAT is the question that I don't think you even touched on--what is the relationship between e and pi?!
+greg55666 All excellent points. I tried to show how the adders-to-multipliers view will gives you the series with factorials in the short article (link in the description), and I tried to shed a little more light on where the pure rotations come from in the follow-on video.
Numbers like 2 and 3, raised to x.i behave almost exactly like e^x.i, in that they all trace a unit circle, *not* any kind of spiral.
The base e is special, because only base e turns to -1 when x is _exactly_ pi.
When x = pi for base 2, the rotation is a bit short of reaching -1, and x = pi for base 3, the circle has gone past -1.
*Any* real base, raised to x.i must trace the unit circle. The base only determines the rate of rotation on the unit circle.
+Paul Gross Oh, you're totally right. 2^i*pi==e^i*pi*ln(2).
The relation between e and pi, then, is DEEP and I still don't understand it. (Why, I mean, not whether.)
greg55666 Don't worry, I made exactly the same mistake when I first tried to get my head around the complex exponential.
The full story is this:
Any (real) base to the power of _i_ traces the unit circle on the complex plane when we then multiply _i_ by a real number, say _x_. The _rate_ is the base times the natural log of the base.
So if you choose _e_ as your base, the natural log of the base is therefore unity, and the rate of rotation is just _x_.
Additionally, choosing _e_ as the base means that the rotations are multiples of 2.pi, so you are back where you started when _x_ = 2.pi, 4.pi, 6.pi etc.
And of course, you rotate through -1 when _x_ = pi, 3.pi, 5.pi etc...
The key to complex numbers, and the complex exponential, is to think of multiplication as rotation.
The complex exponential is just another, very useful, way to do complex multiplication.
+Paul Gross Well yes I know all that (except for the mistake of the bases). The question is WHY is e the number that fits perfectly.
please make a vid to illustrate the sliding stretching and rotating as ways of viewing adding multiplying numbers.
Made no sense. I'll stick to the calculus of it.
cryptexify You should study some elementary group theory, which is exactly what this is.
I agree. If this guy were my math teacher, I would have dropped his class right after the first session.
If one were to imagine addition as sliding a finger across a smartphone, and multiplication as zooming in, then for complex numbers the smartphone would probably display a grid, and the direction in which fingers move during the zoom in gesture on the screen would cause rotation. If, however, the bad 'e to the power of x' came and changed our settings, then moving a finger horizontally would do what zoom in does, and moving a finger vertically would cause the grid to rotate. I wonder what the 'zoom in' gesture would cause with such settings?
as always Beautiful video sir, your videos always shows the Beauty of Mathematics
sir are you a PhD in Mathemarics??, if yes then what is your thesis subject?, please reply me sir please
do you know how much Beautiful your video on Taylor's Series is? really believe me sir when I first saw it my eyes came to tears
I love your videos sir because your videos always contain 'Intuition' and 'Beauty' and that is Mathematics all about..
How does e^x turn adders into multipliers?
hand waiving. this is what he needs to go way more in depth on in order for people (including myself) to understand this perspective
I think what he's doing is to reverse the usual idea of multiplying numbers to the same base, and seeing that you can simply take the base, then raise it to the sum of the indices, to get the result (x^a)(x^b) = x^(a+b). This indeed 'Takes adders to multipliers' if you read it 'x^(a+b) = (x^a)(x^b)' But you already have to know this before you can follow his explanation - or I did, anyhow.
Input i for x in e^x: e^i = a point on the unit circle. Which point? 1 radian on the unit circle. Basically, if you were to wrap the distance between 0 and i on the complex plane along the unit circle, starting at the point 1 and going counter clockwise, you would get to the point e^i. This is why e^(pi*i) is one half along the unit circle because pi many radians is exactly half.
e^x doesn’t turn adders into multipliers. Rather with the inclusion of imaginary numbers [and complex number], classical Cartesian arithmetic doesn’t really do the intuition to what’s going on here justice. Instead he brings up a new concept where instead of simply counting up a number line or following a (x,y) system, he introduces a new way of looking at real numbers. 1 you can just slide to get to one, 2+3 you can slide two then slide three to get to five.
Now the reason why multipliers are important is because we’re not using a regular Cartesian plane. Rather, we’re using a complex plane where the y-axis is comprised of imaginary numbers. Now say we have i then we multiply it by itself and get i^2 this is simply -1 which we can illustrate by rotating the graph in accordance to where it should be (-1,0) on the complex plane.
TL:DR watch his newer rendition of it. It uses the same words, same concept, goes slower, and is a lot more visually clear.
How would 0 work as a multiplier in your graphical depiction? You'd have to bring the point at 1 to zero, but keeping everything evenly spaced would mean your entire number plane collapses to a single point at zero, which seems to imply that you can't MULTIPLY by zero either.
Yup, you're right! It collapses everything to the point 0, but there's nothing wrong with that. If anything, it lines up nicely with the fact that there is no value (real or complex) such that e^z = 0.
e is the base of the natural log therefore it doesn't revolve around logs but logs revolve around it they wouldn't work without NL logs would not work. e=1+ 1 over 1 + 1 over 1'2 + 1 over 1'2'3 etc, meaning e= 2.7182818284590452353602874713527
Beautiful conceptualization, explanation, and graphics, not to mention a great choice in sound tracks.
I will join in the critique of the pace of the narration in the middle of the video. I think it would have been better to pace that slower and allow for points to pause and reconsider. I have enjoyed several of these videos and think this applies to several others as well.
great job. I hope you continue with these.
+Thomas R. Jackson Thanks for the kind words, and for the feedback. As I look back, I cannot help but wholeheartedly agree with the pacing complaint.
+3Blue1Brown It's alright, I understood it at your pace. Blame the Bell Curve!
What software did you use to make the astonishing animations of the planes in this video? I would like to mess around with it and try to better understand mathematical concepts such as Fermat's Last Theorem.
I would like to know what he used here. Smooth astonishing animations.
+Luiz Meier +Bobby Sanchez He's using Python for his animations
Be honest your here because of Alan Becker.
Same
I see there are some comments that this is not the clearest explanation, and that pace is fast, and I understand why people say that. Sure, it has a faster pace, BUT, if you came here with some background (perhaps you watched the newer video on the topic), I think you can see the beauty of this explanation. It also elaborates on what Grant said just before the end of the newer video, and I'm really glad this video is still here and that I stumbled upon it.
No its still needlessly fast. Its been years since i first saw it and even so, I just barely kept up on this watch even though I'm fairly comfortable with it.
da fuck did just happen?
Many key details to the explanation were left out. This is a common problem with some instructors, I think they do it because they only care about showing off their own knowledge and not teaching others what they need to know to understand it.
Gordon Chin he is not a good teacher at all
So why don't you stop being another dull commenter who is showing off their "math skills" by just saying something is wrong. Point out what's missing and add it through the comments, smartass.
+Gordon Chin I also found this video disappointing. I intend to understand this properly, its very intriguing, but maybe because it won't be easy if I then had to explain it I could communicate better than this guy. If you could intuitively see the connections his presentation implies you can at this pace then you wouldn't have needed the video at at all.
+Gordon Chin
There is a link in the description leading to a short paper which goes through everything in detail. If you're interested in understanding, you'll have to put the time and work in.
I actually learn plenty from these videos. I don't agree that its just showing off. He intrigues, makes it look fascinating. It's a good thing that you actually have to do some brain exercise to understand everything.
i am more confused now that i was at the start of the video. :/
0:00 A legend was born.
That background music really suits ... My situation 😁😁
It's kind of adorable when math people think they're explaining things simply for people who don't understand math.
I've spent the entire day watching your videos
This was one of the most amazing videos about math. Very well produced and superb content. Now I intuitively undestand the character of i as a rotation, instead of some magic rule.
I began to wonder is there operators to rotate the number plane in the other 2 axles. I don't know does it make sense, as it would just warp the number space of imaginary/real numbers.
+RealationGames I'm afraid I don't understand the question, what do you mean by "other 2 axles"?
3Blue1Brown
Well, if you look directly at a piece of paper, you can rotate it in 3 dimensions.
That would kind of require a "super imaginary numbers" that occupy the space within the depth of the screen.
Real numbers = X axle(horizontal)
Imaginary = Y axle(vertical)
"Super imaginary" = Z axle(depth)
I bet that this doesn't make sense for some reason.
+RealationGames the number "i" is a mathematical object that was introduced because mathematicians wanted to find all the zeroes of a polynomial. So, using complex numbers, you have a set of numbers that allows you to do all the algebraic operations you want. There is no need to extend them in 3 dimensions. In the beginning of maths people only used natural numbers (1,2,3...) then they wanted to know what 3-5 was and so they extended the numbers to the integers (...,-2,-1,0,1,2,...). They asked themselves what 4/5 was and they invented fractions and so on. So we had real numbers that were perfect except for square roots of negative numbers and, using the same process as before, we invented complex numbers. Now we know all the numbers to solve an algebraic equation and as i said before, there is no need to extend them. If you want to do operations on n-dimensional spaces you must use linear algebra (the complex plane in fact is just a 2D space). I hope i helped:)
+RealationGames you can observe that a complex number like 2+3i is like a vector that starts in (0,0) and ends in (2,3). You can define all the rules of addition and moltiplication to match the ones of complex numbers. For example (0,1)x(0,1)=(-1,0) just like ixi=-1. So if you want to see what happens in 3D you can just use 3D vectors like (1,3,8) and use the tools of linear algebra to see what happens:)
Francesco Ragghianti
Okay, but my view is that the "i" has been there all along until we discovered it, and I want to ponder what more is lurking in the big picture that we have not yet discovered.
I don't think that math is 100% ready yet as we know it.
As you pointed out, my views including this are 99.9-100% just silly and invalid because I don't understand the whole scenario, but that's how new discoveries are always found. By someone who doesn't know the preset rules and boundaries.
There is one other continuous function that satisfies f(x+y)=f(x)f(y) for all complex x and y. It is the identically zero function f(x) = 0. This function is equal to g(x) = 0^x only if we restrict the domain to positive real numbers; elsewhere, g is undefined. Assuming the axiom of choice, there are also uncountably many discontinuous functions satisfying that functional equation, but none of them can be constructed. (A proof of the last statement relies on the existence of a Hamel basis of R treated as a vector space over Q, which relies on the axiom of choice.)
If you don't understand just look up taylor series expansion of sine and cosine. Then once your confused look up the required prior knowledge in order to understand taylor series and then understand that there's really shortcut to learning this.
Quite frankly if you are watching this video then 00:23-00:28 is all you need - yes 5 seconds.
+William Collyer It's not as intuitive though, I think this explanation gives a very natural explanation as opposed to a long derivation using taylor series.
I simply said all you need is 00:23-00:28 if you understand the concept of Taylor series then this will more than likely be enough to convince oneself. I don't think the derivation of sin and cos is long using Taylor series either.
+William Collyer But people watching this particular video are more than likely wanting an explanation /other/ than the well known equations. The whole point of this video is to explain this formula without directly using those equations.
The Taylor series explanation is garbage because it comes across as just a coincidence, witchcraft from algebra. It doesn't give any notion of what it actually means to raise something to an imaginary power. This video kinda goes into it by concluding raising to an imaginary power has to be a rotation rather than a stretch, but I still don't think it gives a great idea about what raising to an imaginary power means, while I think I managed to come up with one on my own.
You are talking too fast! There isn't any time to digest what you're saying. SLOW DOWN.
simple, just set the video speed to 0.75x
and some people are saying normal speed is too slow
Use the pause button, think about what is being said, watch again until clear. It's a shame people want to fully understand a six-minute video expressing a complicated and novel concept in no more than six minutes flat. It's well worth an extra time investment.
think outside the box, the answer is always A
Brane Games
Two trainz
As someone who started their journey on Fourier Series, and didn't want to take for granted the rotational power of complex exponential, this video opened up another door and pushes me to ask more questions about the intrinsic properties of e and the properties of the defined Taylor series. This video was helpful and I think it does a great job, but does require some mathematical optimism and open-minded careful thinking.
An excellent video. Thanks. It was not until AFTER college that I was exposed to the idea of using numbers as OPERATORS rather than simply for counting. That vastly expands the concepts of mathematics for me.
0:37 I’ve always wondered what numbers are
I love your videos and your unique approach to all things math, however I agree with some of the criticisms here. You have a tendency to blow through things pretty quick, and as the video went on you got faster and faster. Love the content, just wish it was a little more drawn out.
Can you make a video about the meaning of 3Blue1Bown?
@@twopie6911 It is not a disorder, just an artefact. He has sectoral heterochromia.
I already knew that you could pull out and seperate two adding exponents, but when you explained it you freaking blew my mind.
This video is like the final boss of 3b1b videos, and I'm proud to say I mostly understood it on this watch
BRUH THIS IS FUCKING LIT
xima dont u ever
Alright, can you repeat the 2nd to the last word for me, I didn't catch them.
I want a new brain from you, mine just exploded
The idea of understanding arithmatics here reminds me of linear algebra (linear space transformation/base vectors), I think it would be a brilliant way to re-understand the basic operations (and it reminds that real numbers could be seen as 1D vectors)
I used to always think how to how to visualize a^b where b is in decimals because a^b = a*a*a*...(b times) but that makes no sense for decimals. Thank you so much for redefining the definition of adder and multiplier!
Honestly, the last part is like an explosion. Throughout the video, the ending bombarded me a lot like a huge it twist in a movie.
I never imagined that someone could explain that equation. And yeah, this explanation pulls me stronger to subscribe to the channel and really appreciate math in dynamic and imaginative way. Kudos to you guys!
"What even are numbers"
Wow, what software do you use. It's gorgeous
The last minute was pure hand waving... I'll check out the follow up vid.
What are ADDERS AND MULTIPLIERS?
Just think on this, you have the WHOLE numeral line, then you add any number to every number on this line (...,-2, -1, 0, 1, 2, 3,...)
We'll add 3
This 3 will be added to every number in this line and their value will change (-1 +3, 0 +3, 1 +3, 2 +3,...)
So now, the position of every number in the line is not the one it must be (we added 3 to 0, so now it will be 3) hence, we have to "move" the line sideways for make things normal (THIS IS LINEAL CHANGE OF POSITION, here the number "3" is the adder)
The same thing occurs if we multiply every single number of the line, excepting that 0 will still in the same spot ( 0 × n = 0), but every number will change exponentially, so that this line have to "stretch" or "retract" in order of "keeping things normal"
Again, all of these is a LINEAL change, in a 1 dimension plane
Now Imagine a 2 dimensional change of position
Although I am a programmer I am really bad at math, but your videos make me see math is a description of the world
Be wary that description will never lead you to the whole thing.
I'm wondering if a single person understood the entirety of this video.
Euler
Fanime Productions T.V. He’s dead 💀
@@anthonychang2298 r/woosh
Keith Thedford II bro I get the joke I was just making another joke... people are so quick with the whoosh trigger these days
@@anthonychang2298 r/woosh times infinity
I think the video is ok, but too fast. I only could recognize what I already knew.
3b1b should collab with DDOI. They both have really good voices😄
Okay ..do you trade or invest Bitcoin I can recommend Mr William charle he is a good broker and also expert in crypto currency he can help you get good profit from Bitcoin
He's available on what's app
*+1 8 1 8 7 2 2 9 1 4 5*
I did not know that I don't know this, but it makes sense that I didn't know of this
Sorry, this is cheating. The crucial part of introducing the "e" is done via series. But... to understand how they work you need calculus which in turn is *based* on definition of numbers, operations on them and properties of both numbers and operations. You're getting into circular dependencies of your definitions there. And no, saying "it's the most natural way of ..." simply doesn't cut it. It's just saying that the equation you are trying to prove is the most natural! In a way - yes, it is. But you were trying to prove it, right?
In calculus, rotation on the complex plane comes up pretty much by itself, as a consequence of other things you're doing with those numbers. And sure, it's a handy little trick to visualise what's going on for those who use/need visualising (not all people do). But it's no substitute for proper definitions, much less proofs.
Over the years, I've seen a number of similar attempts to redefine how introduction to calculus (and maths in general) in taught to school-level students. I'm not sure if it works or not in school (no idea really, it might help teachers perhaps?) but, as soon as these students are faced with further studies, they are up against the wall: what they've learnt doesn't work. So we have to run remedial crash courses (as part of the intro usually) to help them unlearn the latest fad they were fed at school and start all over again. So, if you like this trick or find it handy to help visualise something - ok, maybe it works for you but remember - this isn't the real thing.
+SayNOtoGreens Well I don't think this is meant to be a rigorous proof but more of an intuitive explanation. Though I don't think you need the series definition necessarily to make a proof. There are many ways to define e^x.
Logarithms/exponentials were developed to simplify calculations by defining the multiplication/addition equivalency and I think that is the most natural or fundamental property they possess. I think that was what he was referring to.
I literally cannot understand this at all. Sliders? so you slide a point.. or stretch the graphical distance between two.. and say that's addition and multiplication.. without any real explanation as to what you mean. what?! this is intuitive? Im so very confused. Ill give this another watch because this really irks me, but then its back to the old way
+glenthemann Of course it's intuitive. Multiply any function you want by any number and you'll stretch it. Add any number to it and you'll shift it along the x-axis. You shouldn't watch a video about complex numbers if your early high school math is shaky. It's called a transformation. Hit up wikipedia.
A legends was born
I always try to watch all videos on english, but this time I had to put the spanish subtitles and pause every second, and I still didn't understand, I'm going to come back again some day
The "osmium" of all RUclips.
Most content delivered in such short and lucid format.
Forget about e^πi‚ i don't see what multiplayer do ... 😭
It's been a while since I watched this video for the first time. But really, I wanna emphasize that all the concepts introduced in the video are better to keep in mind than the traditional ones. In fact, in math when we get stuck, the best thing to do is to extend our set of axioms or at least extend our visual approach about many concepts. I really disagree with anyone who watched the video and dislike it simply because concepts are in introduced in different ways. You have no idea how this is very helpful. Suppose we have learned those ideas in school and how they would change fundamentally our thinking
about many things that we have encountered. To sun up this video is a rare and an exceptional piece that introduce the effective way to think about basic operations since they are the basis that we build on.
couldn't agree more
Wow this channel is a gold mine!
I'm using my 10% remaining brain power before sleep to stay awake and watch this video, that I couldnt possibly comprehend on the best of sundays after a pot of coffee.
First of all, this video inspired me alot to consider the action meaning of numbers, my only comment is that there's a gap not covered which may have caused some confusion at least to me. It's the role played by the function. Adders and multipliers applied directly to the number line vs adders that is applied through a function here is e^x
Hold up, I ain't this clever, I'll stick with Pythagoras theorem
Yep...its the dawn of the amazing 3B1B
Like if you are here from the Riemann Zeta video
looks like he decided to gtfo
Reading through the comments, I realize that I have friends! I thought at first I am the only one who failed to follow this beautifully-visualized-yet-rapidly-narrated video
0:00 a legend was born