e^(iπ) in 3.14 minutes, using dynamics | DE5
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- Опубліковано 7 чер 2024
- Euler's formula about e to the i pi, explained with velocities to positions.
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Not familiar with the calculus referenced in this video? Try taking a look at this one:
• What's so special abou...
Another perspective on this formula, from Mathologer:
• e to the pi i for dummies
Another perspective from this channel:
• Euler's formula with i...
And yet another from the blog Better Explained:
betterexplained.com/articles/...
I'm not sure where the perspective shown in this video originates. I do know you can find it in Tristan Needham's excellent book "Visual Complex Analysis", but if anyone has a sense of the first occurrence of this intuition do feel free to share. It's simple and natural enough, though, that it's probably a view which has been independently thought up many times over.
Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld
Vietnamese: @ngvutuan2811
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These animations are largely made using manim, a scrappy open source python library: github.com/3b1b/manim
If you want to check it out, I feel compelled to warn you that it's not the most well-documented tool, and it has many other quirks you might expect in a library someone wrote with only their own use in mind.
Music by Vincent Rubinetti.
Download the music on Bandcamp:
vincerubinetti.bandcamp.com/a...
Stream the music on Spotify:
open.spotify.com/album/1dVyjw...
If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then "add subtitles/cc". I really appreciate those who do this, as it helps make the lessons accessible to more people.
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3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with UA-cam, if you want to stay posted on new videos, subscribe: 3b1b.co/subscribe
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Complex exponents are very important for differential equations, so I wanted to be sure to have a quick reference for anyone uncomfortable with the idea. Plus, as an added benefit, this gives an exercise in what it feels like to reason about a differential equation using a phase space, even if none of those words are technically used.
As some of you may know, Euler's formula is already covered on this channel, but from a very different perspective whose main motive was to give an excuse to introduce group theory. Hope you enjoy both!
Please make a series on tensors and imaginary numbers because these topics are the least understood among students and misjudged due to their weird names. Also they are really very interesting, Intuitive and fundamental to the universe.
I am not feeling well😗😗😌😌
Loved the Euler's formula video. This one too.
I'm about to start a course in computing this august. ( I took the advice from your numberphile podcast ) Just want to thank you for these videos the animations are honestly so beautiful to watch. I'm looking forward to the next videos!
@Wolfgang Kleinschmit did you see his Essence of Calculus series?
Last I checked, 3.14=/=4.08
This is by far the most intuitive explanation of this identity I've ever seen!
CozmicK G I would check out the explanation with the differential equation y’’ + y = 0
@@johnlesslie1963 where is this at? i can’t find it
KRUH
I would say that there is simpler proof that also shows the real engineering operation behind that function.
We should know that the value of e is related to having $1 dollar in the bank and with 100% interest continuously compounded, then at the end of the year one gets $e = $ 2.7182
This compounding of interest in a numeral $ dollar operation, makes the money grow. If we had to represent the compounding of the value of $ money in the bank, we could use the length of a broomstick, where $1 dollar would be a unit length along with the broomstick, while the continuous compounding at the end of the year would be shown by a length of 2.7182 hence $2.718
But compounding on a broomstick can take place along the length, AND ALSO IN QUADRATURE WITH THE LENGTH OF THE BROOMSTICK, where this would result in a rotation or an orientation of the straight long broomstick. So if we invest in rotating the unit length of the broomstick acting as a rotating vector, rather than elongating it along its length, then e^iA would be the final orientation of the unit broomstick after it terminates it compounding "rotational banking interest" rather than the conventional linear compounding which just changed its length!
All we have to do is to draw the vector e^iA where "i" indicates the quadrature direction of the compounding with respect to the length of the unit broomstick length 1, and A is the final compounded angle in radians reached where the unit length of the broomstick will remain a unit length.
Hence taking the components of the final rotational compounding e^iA on the two basic axes which we used as a reference, we have e^ iA= cos A +isin A.
I would say that the students would appreciate all this OPERATION which describes the real engineering actions behind the " compounding of interest" when the interest is linear or when it is (i. interest angular displacement ) the rotational interest on a unit length of a broomstick, or it could even be a $1 dollar paper sheet, wrapped up like a thin cigarette to represent the unit vector, and the wrapped up thin $ sheet will act as the rotating vector accumulating the rotating angular interest which finally compounds to e^iA.
I wager there are a few smiles after readers read all this! One would agree that the philosophy of compounding the $1 dollar paper sheet in a linear manner of its growth in monetary value, or its broomstick length representation, and the rotational angular manner, is exactly the same! where all that engineering actions and reactions are described so accurately in the meaning of the symbols in that expressions. We all need to learn what they mean in engineering actions and reactions and not only the rules of differentiation of the product function used in the video as ...... (the first function)multiplied by (the differential of the second function) plus (the second function) multiplied by (the differential of the first function). I always wonder how many people see the engineering/physical activity in that operation!!!!!!!!!!!!
Ok
"Multiplying by i has the effect of rotating numbers 90°" - Both lost and mind blown.
Think of multiplying by negative one as turning around to face backwards. If you do that twice you will be facing forwards again. That is (-1)*(-1)=1. Now if we ask what number times itself equals negative one, it will be a rotation that when done twice has you facing backwards, either a quarter turn counterclockwise i or a quarter turn clockwise -i. (the complex plane doesn't actually distinguish between clockwise and counterclockwise because that would require it physically be facing in some direction relative to us, and it is an abstract mathematical object, but we aren't abstract and generally draw i above the real number line and -i below, and numbers on the real number line getting larger as you move right)
@@paulfoss5385
So, for real numbers, |R, (-1)^2 = 1, and for complex numbers, |C, (i)^4 = 1. I think it's really cool that that exponent is off by a factor of 2 for |R and |C. I'm trying to think how this generalizes to quaternions (|Q) and maybe octonions (|O) but it's not obvious. |Q also holds that i^2=-1, same as |C, so it's still (i)^4=1. Octonions, well... lol.
I think he has a video about group theory that explains more about this rotation.
theres a really good series from another channel about imaginary numbers that goes into depth about it from the absolute basics, Id recommend giving it a watch
@@sdoman7215 WelchLabs is probably the channel you're thinking of.
I have known that e^i*pi = -1 for many years.
This video was the moment I understood that fact. Thank you.
That's surprising, considering how I was exposed to it was through the MacLaurin series. That answers it immaculately too. As I've matured more in mathematics, still quite infantile, I realized that 3B1B's videos are very visual. Though this video is more approachable, I'm often leaning towards traditional textbook like material with the leap of intuition than his videos that are ambitious visualizations.
im having the same experience
i've watched other 3blue1brown videos about this but i never really got it until now for some reason. holy crap it's so intuitive now, i don't know what i didn't get it before
His channel is pretty magical! But so is yours, so.
Xidnaf I loved your videos , I hope your doing well
That warm fuzzy feeling you get when one of your favorite youtubers watches the videos of another :))
Please make a new video
Oh hey Xidnaf, didn't expect you here. We miss you!
Oh my god that makes so much sense.
Why was this never explained to me like this?
because it's wrong...
@@umair5602 Expand?
@@umair5602 go on
The great thing about math is that you don’t have to take anyone’s word for whether something is right or wrong. You can think through if this explanation makes sense for yourself (spoiler alert, it checks out). To engage with it that way will force you to think critically about what it means to take a derivative of a complex-valued function, which in turn builds good intuitions for complex analysis.
If e^πi=-1 then
π=ln(-1) / i
Then 2πr= 2r ln (-1) / i
= ln (-1^2r) / i
Which equals ln (1) / i
Which equals 0
Which means that if Euler's identity is true, the circumference of a circle is always 0.
*uses τ*
Ah, i see you are a man of culture aswell
I too was very pleased to see that.
You mean why not 2*pi?
isn't pi cultural lmaoo
*Uses both π and τ*
What a fantastic way of avoiding an argument!
Pi is better for pretty much every application
I was comfortable with this identiy before, but I had a great “AHA!” moment. This is probably the most concise and surprising way to look at it I’ve ever seen. Thanks a bunch
"I'll spare that detail for the next video"
Almost 2 years passed, he's finally done it 😭😭😭
Bro, which one is that next video you are
referring to?
@@aarjith2580 The latest one. He hadn't touched the differential equation for 2 years.
@@chizhang2765 oh so the one about probability that he's yet to make is still a pretty long way away huh
stop getting so emotional
@@Fire_Axus stop being a dumbass
The idea that multiplying a number by i rotates it 90 degrees into an imaginary axis is probably the most important thing I've ever gone my whole life without knowing.
Wow, how did anyone understand maths before 3b1b
@@ASLUHLUHCE They teach you polar coordinate multiplication literally in 11th Grade or so
After four years of studying electrical engineering in college, no one ever explained it that way until a professor casually mentioned it in a elective I took in my last semester :/ It would have helped so much to learn that earlier
"We're sorry, the number you have dialed is imaginary. Please rotate your phone 90 degrees and try again."
@@FranziskavonKarma They just say that cis(a)*cis(b) = cis(a+b). They don't go beyond that
Kudos! This is pedagogy brought to the level of art. I hope your work, together with that of other excellent UA-cam educators, will pave the way for a new generation of teaching.
The only 4min video that actually led me to understand the dinamic behind Euler's identity. Well explained! Congratulations!
the e^iπ part comes right at 3.14 but also the e^i𝜏 part comes right at 3:14
nice
*i* wonder whats e^iφ
@@squibble311 if you are wondering about anything else,
Generally e^ix=cos(x)+isin(x)
There's a really easy explanation for this, it has to do with trig
@hiqwertyhi you've forgotten to explain that since a minute is only 60 seconds... 3.14 minutes is in fact about 3:08... And then tell readers to look closely in the bottom right corner of the video at that time... :-B
wow, actually amazing
3:14 is not π minutes into the video.
gotta click fast when 3blue1brown posts
3B1B: *uploads *
Me: I have the fastest hand in the west.
QUICKLY!
Why?
Translation: Look how smart I am everybody! I watch 3B1B!
What's the rush? This guy takes time to make quality videos. It's not like there's one every day.
No wonder the Euler’s formula in Alan Becker’s “Animation Vs. Math” is given sentience, it’s all about position and movement. It can do so much.
I can’t even stress how important your work is for me. You explain everything in such a beautiful and simple way; you were BORN to teach. Congratulations on all of your videos, sir.
THAT WAS SUCH A GREAT EXPLANATION
I always struggled with that one, despite the fact you've made quite a few videos about it
this was so good though, it makes so much sense now
I gotta watch the other ones you've made to see if I can understand them better now
I did not understand why the yellow line (velocity ) was moving faster then the blow line (position ) even though he said the dervative is just equal to to the position at that time,.
Can u explain
@@hassanomer7777 it's not yellow it's green
@@hassanomer7777 It doesn't move faster. If the right side of green line moves at 2n, while left side moves at n, this means its length increases by n. The "speed boost" comes from blue line's end
This is the simplest and most intuitive expression for this I have ever seen. Thank you. I'm a grad student in robotics so ODE's and PDE's are my life blood. So, this has always bothered me that the intuition for this concept was so difficult. With such a simple explanation I wonder why I have never seen this before.
2:32 Good thing you initially drew more than 4 arrows to illustrate this point XD
r/angryupvote
I really like the buildup in this video. You start at derivatives and everything just blends into each other until you're expanding it further. From the effect of positive factors in the exponent, introducing the negative factor and eventually the imaginary factor. Definitely hard to understand if you don't know about the underlying concepts, but once you do, this almost feels like the most perfect way to explain e to the i pi. Great video
YeS! Last time i watched this i didn't know about e, now we have been learning it in school and i understood the introduction way better. 🙂
It's amazing how helpful Grant's use of color is. This was particularly pronounced on his superb Fourier series videos. The old analog chalkboards of my youth did not have this added dimension.
Even with your previous videos about e^ipi I couldn't fully grasp WHY it behaves so. Why it has this undeniable link to rotations in the complex plane. As SOON as you brought derivatives into it, and e^x's nature of being its own derivative, it makes absolute perfect sense. Dude, you and blackpenredpen are the math teachers I needed so desperately when I was precalc in high school.
I think this way of teaching "visually" should be introduced in every school in the world... it's amazing how you manage to find the right animation for whatever concept, bravo!
It is incredible how far you've grown as an educator. Your first take on e to pi*i as your first video on a channel was really confusing to a lot of viewers, me included. But now you are providing an easily conprehensive explanations together with enjoyable visuals on this and many other topics. Thank you for all the work you're putting in your channel
That is by far the best, and most intuitive explanation of Euler's formula I have ever seen. Thanks!
I'm not first
and I'm not last
but when 3b1b uploads
I click fast.
I Meaaaan you were last.
@@fgvcosmic6752 have some patience
Someone summoned my name?
Roses are red
@@WackyAmoebatrons omfg
It's Magical !
The way you explain.
And I am not even a newcomer on this channel, yet my mind is always blown away by how you brilliantly decompose mathematical concepts to their core. So elegant, so beautiful.
This is the way to do it, the proper way to use technology in order to share science and wisdom.
So thankful for the Internet, that is allowing us to connect with such quality content.
So thankful for you, creators of such quality content.
Thank you for what you do !
Peace.
Interestingly, it's "creator" (singular). Just one guy does this
@@piman7319 Sure, but 3B1B is not the only great content creator on the Internet ;-)
I actually just come here after watching "Animation vs Math" 😅
To anyone who wants to think about something deeper with this concept:
Think about the function y=(-1)^x. Now this function is discontinuous, it’s not a line at all. It just seems to alternate between y=1 and y=-1 on the xy plane. But what happens if you include the complex numbers, say adding them in a z axis to make a three-dimensional graph. What you’ll find is that the function (-1)^x, or to write it another way e^(i*pi*x), traces out a line in 3d space. A line that is in the shape of a helix, a coil going around the x-axis, and intersecting the xy plane at 1 and -1 at appropriate points.
Hilfigertout woah.
And not just any helix, sinusoidal at that!
This is what I call high quality commenting.
@@nuralimedeu "woah" - Neo.
Wouldnt it be two helixes mirrored since the function is ambiguous on complex numbers?
I love this! Your ability to show the patterns in mathematics in an elegant, but understandable way is truly profound. Thank you for all you do!
I am constantly amazed at your ability to so elegantly show the beauty in mathematics. This was simply incredible.
Can't wait for the full decomposition video! This was fantastic (many thanks from an electrical engineering student!)
U know he's awesome when ur math professor teaches u using his videos.
Holy shit I love this. So beautiful and intuitive. Already understood Euler’s formula and more or less why, but this really made me feel it in my gut
You deserve a Nobel Prize for this video, such an intuitive explanation.
Amazing video. I've never had it explained this way to me. I've always just understood E^jX is equivalent to the unit circle and accepted it as so.
I was afraid that by 3.14 minutes he meant 3 minutes and 14 seconds, which is, of course, inaccurate. I'm glad that didn't happen!
Grant's too smart for that ;P
Which would instead be 3 minutes and ~8 seconds which doesn't really add up either. 🤷♀️ Not that it matters ultimately. It's still a really good and elegant explanation.
@@Gameboygenius look at the timestamp 3:08 in the bottom right hand corner
4 min 8 sec ≈ 4.13 min 🤔
@@conrada5 I missed that. So ok, fine I guess, but I feel that the explanation was not entirely finished at that point.
Explained my confusion about Euler s formula so elegantly! Thank you!
Which one?
Such an intuitive explanation of a formula that has always seemed so abstract to me. I will now always have this concrete image in my head when I think of it. Thank you :)
This is so beautifully intuitive! I was first introduced to Euler's formula years ago in calculus 2, but algebraically derived using Taylor series. This geometric explanation is so wonderful (as are all of your geometric explanations in your videos)! :) I've been a fan for years and I love how you give these simple clear pictures for explaining interesting mathematical concepts and equations.
Love this! So insightful.
Watching this video was like listening to a jazz piece for me. Everything was harmonious and made sense, but I would not have guessed the whole in advance.
This is a GLORIOUS way to think about e^t.
After a year or so of using it, I can now officially say I fully understand Euler's Formula. Bless up. 🙌
One of the most beautiful mathematical relations explained in the most beautiful, insightful, and intuitive way. KUDOS !!!
I wish I had this for my students a quarter century ago!!! Beautiful :-)
Brilliant. Absolutely brilliant. This is excellent teaching, as we've come to expect from your channel. Thank you!
people uploading lectures, and videos like this on youtube are literal gods
Watching this again after just having done some problems involving resonant circuits, where Tau is used as a time constant, I just had the revelation that this is actually not unrelated to this Tau = 2*pi idea! Freakin beautiful.
You help more people with these videos than you could ever know!
Kurzgesagt AND 3Blue1Brown uploaded today! This is fantastic!
Also, I can't neglect to say, very elegant proof and brilliant explanation!
They should do a collab
As an electrical engineer, I learned AC circuit solving through complex exponentials (phasors and impedances), so I'm really excited to see this video.
(Also, the pace of the video relative to the usual 15-20 minute vids reminds me of the first video on the channel.)
This is the single greatest video and explanation I´ve seen on this!
Finally I understand why it produces a circle! :)
This is beautiful! Has to be the best way to show e to the i pi = -1 that I've ever seen! Thank you!
You are an incredible instructor. Thanks for what you do.
It would be awesome if you did a series on modern numerical methods. Finite elements (for for elliptic problems) add Runge-Kutta methods for hyoerbolic and parabolic problems, etc... With your unique talent you could make this fascinating topic accesible to millions of talented young people who might chose to pursue a career in applied math. Thanks for all the great work!
As someone who has been in science long enough to see this identity everywhere, I always hated that the only explanation I ever got was "It works because of Taylor expansion", with the demonstration separating the terms of the Taylor expansion of exp into those of cos and sin. It's just super abstract, like "It works bc of these random formulas that converge to the result after infinite steps".
Now I just learned about this way of seeing Euler's formula, was mindblown, did a quick YT search to see if anyone I watched had talked about it before...
And of f-ing course 3Blue1Brown did it. I'm kinda mad I never saw this video and also not surprised that you would use this representation that really feels like something you would find on this channel.
Such a cool way to see this otherwise very arbitrary-looking identity!
Why am i coming to this after watching Animation vs Math?
You have the best visuals for any channel on UA-cam I've ever seen. Outstanding work!
This without a doubt is the best explanation I've seen of this yet.
I love the different way you explain, very careful with the concepts and very detailed, which is good for those who are graduating in the bachelor's degree in the area, in addition to the illustrations are fantastic, congratulations, I follow you always - I am an undergraduate physics student in UFRN- Brazil
this is lovely to watch and listen to. thank you
Who else got this recommended after seeing animation vs math?
That was an awesome way to think about this. And in such a short form makes it easy to process. Thanks heaps, and can't wait for the next video.
Its always so fascinating when you explains it from another perspective
Watching 3b1b makes me feel like I am unlocking the unused 90% of my brain.
Bad move buddy.
So you believe the founder of Scientology?
The idea that we use only 10% of our brain is nonsense, and has no basis in science. The fact that a lot of people still believe this baffles me.
Exactly
haha exactly
Booyah! Fantastic explanation, especially for something that’s usually so abstract, and one of the few likes I’ve ever given to a video (not because I don’t enjoy a lot of stuff, but because I just don’t usually see their importance).
What a wonderful breakdown of that pesky identity that stumped my visual undersanding for so long. Incredible video, incredible channel!
Just wondering who came for Alan Becker
I did.
Present!
Hi
exactly
Best explanation yet!
This took my math lecturer about 30 minutes to explain, but you've done it in just 4 minutes!
This is the best explanation yet. Congrats
I was researching e^ix= cos(x) + i*sin(x) after watching Lockdown math and wondered why there seemingly 'just happened to be' a neat connection between two quite different things: circles and e.
From what I understood on Wikipedia and forums, there was an explanation that involved the Maclaurin series of e, sin, & cos, and whilst it is a proof, it didn't satisfy my question. But then this video made the connection obvious (with the help of another 3b1b vid about e and calculus) and I can now say that I understand e^iπ=1 :)
3Blue1Brown can you do a video about the 7 millenium problems?
huge demand from my side
He'll prolly solve them ez lol
he did 1 on the riemman hypothesis
Then he'd accidentally solve them, and where's the fun in that?
This explanation is amazing; so clear and concise. Keep it up!
Genius, genius, genius brilliant explanation. I am moved by your ability to convey intuition so perfectly. This is the greatest
Amazing! I never thought that there would be such a simple explanation to such a complicated equation. You are the best maths teacher!
Alan becker brought all of us here!! 😅
The best tuto on math on YT-many thanks!
I’ve watched so many videos explaining this and I’ve never understood but this one is so clear and I understand it so well for my little background in math as it is.
I thought at the end you'd show how, if you project the circular motion in the complex plane onto the real axis, you see simple harmonic motion. This could help high school students to relate better.
Btw I absolutely love this videos! Short but original and always make your concepts click no matter how much you think you know on the topic. Keep inspiring 3Blue1Brown!
Underrated comment
This Chanel deserves all respect as possible
This is just incredible.
Simple, yet easy to understand.
Thx for your great video again
Is there a love button on UA-cam? There should be. A like is not enough.
Patreon is your love button
There's three. There's "subscribe", "share", and the link to Patreon.
no need to kiss his ass, you won't get a creator heart
how ins-'π'-ring
As-'τ'-nding
'τ'tally agree with that τ is as'τ'nding τ say otherwise is des'π'caple.
The Major That was awful
it's not a good science/math video unless someone makes awesome puns about the concepts involved
(-1)^1/2 186282mps what you did there.
watching your videos makes me feel similar to how I feel after meditating for a long while. It's so calming to know that it all makes sense somehow.
And that is the first video that makes me actually understand it. Great content!
Finally, a video about this beautiful, elegant proof!
Perfect use of the Engineering theorem for the pi approximation
I can't even put in words how brilliant that explanation was!
Almost didn't watch this, but then I saw your update with that guys tweet. This did not disappoint.
Good job 3Blue1Brown...here's a pie 🥧
The last bit, when you pointed out that e^iπ does not mean multiplying e by itself iπ times (even though this would have been the case if the exponent had been an integer, like e^3 = e*e*e) made me realize that it is the same thing that happens when you find an analytic continuation for a series. For instance, ζ(-1) = -1/12, but this equality does not mean that 1+2+3+...=-1/12, since the series for the Riemann Zeta function is only defined for numbers greater than 1. I had never realized before that even with simple arithmetic, such as powers, you can already get a feeling for the rule "you cannot always interpret new things using your old knowledge", because sometimes a blind substitution can be misleading.
That cleared up a whole lot more
ζ(e^iπ) = -1/12
never realized this before
That makes sense simply because the multiple of an imaginary is still an imaginary in the form ki. e isn't powered to anything in the real axis.
I've always seen Euler's formula explained using taylor series, but this is sooo much cleaner.
Great video 😃
I am glad people like 3Blue1Brown exist on UA-cam! I could learn all day and not have to pay $50K per year at college/university, which is now nothing but venue to buy a certificate that you have a BA/BS!
first i thought a random video about e pi. why would i care, saw that often enough. then i noticed 3blue1brown and thought "im sure i dont know enough yet" 'click'
i wish i had this channel when i was doing engineering. kids these days spoilt af
I'm 12 and I tried to ignore this comment
Wow you are such a miracle being able to do engieering even without animations wow such impressive
@mario mario same
Ok boomer
@@ericvauwee4923 fr tho these videos are kinda nice to watch but for the exams and rigorosity in university they don’t really help tbh. Boomers now pretending we can watch 3 UA-cam videos and get our engineering degree lol
Wow, elegant explanation! As always, thank you for great videos!
when you turned it 90 degrees, I was completely blown away. incredible stuff
I was Tortured by calculus, now because of your channel I'm starting to love it.
I never liked e to the i pi, mostly because it seemed everyone was just so amazed over a statement. you could just accept it and integrate it into your math, or you could not. what I am more interested in is the proof, and what it means, but when a while ago several different youtubers started making different explanations for why e to the i pi equals -1, I was never quite satisfied. They always seemed to make some jump in comprehension. This jump is the same one that you just smoothed over beautifully, and I am in awe of how this problem that has been bugging me for like 2 years now has been explained in such a short period of time.
Still I have more questions like if the only way mathematicians can explain or compute complex exponents is through e or if there is some more generalized formula or explanation for complex exponents. Also I wonder if there is an explicit definition for what exponents mean in relationship to the complex plane, beyond e. This promise of another video excites me, because it might answer these questions where others did not, but even if it doesn't, I remain in awe of your skills in explanation, as well as very grateful for your videos, because I don't know where I would be if it wasn't for your essence of algebra and calculus series.
Euler's Formula is the generalized version for complex exponents. e^ix = cos x + i*sin x. Basically think of x as an angle in complex space with the same logic as discussed in the video. It's all very magical how so many separate things end up relating.
This series deserves the youtube subscription