What are complex numbers? | Essence of complex analysis #2

[@mathemaniac]

As seen in the beginning of the video, feel free to pause the video to catch a breath to absorb everything! Please consider SHARING this video to help more people if you find this useful / helpful! Of course, like, comment and subscribe with notifications on! If you are feeling generous, consider giving on Patreon as well: www.patreon.com/mathemaniac I have crushed everything into (roughly) 30 minutes to keep my sanity. I am going to rest a bit for now, and while I know that your comments wishing me to release videos more quickly are due to appreciation of my content, sometimes it did lead to stress (well to be honest, it is mainly due to myself: wanting to grow the channel means uploading more frequently), and so despite this video having a lot of “holes”, I still pushed ahead and published this video. I know that this video is not as advanced as the content that I am usually known for, but hopefully there is still something that you don’t know here. If not, consider this a refresher or a convenient compilation. The next video will be more into the complex analysis region, and more “advanced”.

[@heisenmountainb6854]

I just finished watching the video, and the fact that you call this video "not as advanced as usual" is frightening and exiting at the same time. Lul

[@jaycorrales5329]

Sure it works if you give motivation based purely on mathematical derivation, reasoning, definitions and logic, but I like the electrical engineering motivation better where the imaginary number represents field energy stored in the environment due to capacitive and inductive effects w.r.t. antenna gain patterns (and e^-2*pi*j*theta). Then there are real and physical equivalences, and reasons to not just think about the crusty old math theoretical ideas of complex #s.

[@mariamamart8086]

Very Good , i loved this class !!!!!! 👌🤙💪 👏👏👏👏👏

[@enbyarchmage]

I took a course of introduction to complex analysis a couple semesters ago. The teacher was one of the best ones that I've ever had and, until today, I thought no one could ever explain the beauty of complex numbers to someone with the same degree of mastery that he had. I'm so glad I was wrong!

[@mathemaniac]

Wow thanks so much!

[@RifatRahmanRimon]

Kudos to Mathemaniac for making this video of course, but I feel this video was inspired from Neadham's book. Probably the best book I came across in terms of visualization.

[@jowillll]

Can i have his name pleasee, can i find him/her on google?

[@Nick-lx4fo]

You can learn anything on the internet

[@ifyhu92]

What is your teachers name? Which college are you in?

[@sayandeb4444]

I'm doing my PhD on applied mathematics(optimizations) ... Even after all these years of studying, I find these explanations really helpful and fascinating.

[@mathemaniac]

Thanks so much for the compliment!

[@HerrFunnybones]

That’s crazy lol

[@PunmasterSTP]

How's your PhD going, or did you already finish it?

[@oosmanbeekawoo]

@@PunmasterSTP Don't ask these questions and just let him do his best!

[@PunmasterSTP]

@@oosmanbeekawoo Why not? If he doesn't want to reply he's under no obligation to do so.

[@benburdick9834]

Love that you were able to condense nearly half a semester of complex analysis into 30 minutes. If only this video came out half a year earlier; I would have had such an easy time in my complex analysis course!

[@mathemaniac]

Thanks so much for the kind words!

[@PunmasterSTP]

I just came across your comment and was curious. How have your studies been going since then?

[@benburdick9834]

@@PunmasterSTP Just graduated with a double major in physics and applied math. Now I'm going to grad school for physics.

[@PunmasterSTP]

@@benburdick9834 That's great to hear! Are you ultimately hoping to stay in academia, or go somewhere else?

[@benburdick9834]

@@PunmasterSTP We will see. I would like to, but academia is not without it's flaws.

[@Bbb78651]

Absolutely blown away by the high quality of this video. Really comprehensively summarized complex analysis fundamentals. I've been looking for perhaps a year for a really good series for complex analysis. This is undoubtedly the first and best that stood out. Can't wait to really wrangle the whole series. Great combo of visuals and proofs with a pace enough to keep your curiosity piqued. I felt like a young child learning what multiplication is for the first time.

[@melontusk7358]

This is the most elegant method to visualize Euler's formula with derivatives. I've always tried to memorize the proof by writing down each Maclaurin series, which isn't as convincing. Thanks so much for your brilliant playlist.

[@vitalysarmaev]

You don't need to memorize it at all literally. Math is not memorizing rules: it's a language like music. The math grammar are principals, but not rules nor formulas; this grammar leads you through the ocean of cognition avoiding stones and low waters.

[@akash3478]

You seriously deserve so much more recognition. Please keep making these !!

[@mathemaniac]

Thank you so much for the kind words!

[@fedormedin8861]

5:52 had to watch this moment twice

[@muhammadarsamdanish6237]

Vsauce moment

[@zyansheep]

Vsauce has reached enough people to become a meme : )

[@pucek365]

The division rule of triangles is really beautiful. I found that you could divide both numerator and denumerator by r2e^io2, and then it all comes together nicely that one part of that fraction creates the right vector and the other creates real vector with length of one. Complex analysis is trully amazing, and thank you for bringing it in such accesible way.

[@mathemaniac]

Glad that you like the video!

[@TwentyFifthTeen]

i havent even finished the video and you're already doing a better job in 17 minutes than my prof has done all semester

[@doodelay]

I love that numbers have a natural smoothness in their relations to one another. Gives you confidence that there is always a beautiful solution in the wings

[@anastasiawalrave5715]

the channel is amazing. Never seen anything like this, helps to understand maths usage in scientific problems. The visualisations are one of the best I've seen. Thank you for not omitting 'easy' components and explaining everything thoroughly, it is vital for aged beginners like me

[@ProCoderIO]

Already got stoked about complex analysis by @3blue1brown and @blackpendredpen. To have found ANOTHER math channel that is just as excited is super awesome!

[@mathemaniac]

Wow glad that it piqued your interest!

[@rinzhler6922]

Also check out Flammable maths

[@AA-gl1dr]

Thank you. Seriously thank you. you’re helping so many people by teaching math this clearly.

[@mathemaniac]

Wow! Thanks so so much for the kind words!

[@SolunaStarlight]

23:15 THIS. I have needed this so bad. Like... I have known for a long time what sine and cosine mean and how trig functions work, obviously I needed to for my calc classes. But they nonetheless have always felt like a bit of an... arbitrary construction? Seeing this and realizing there is really a more fundamental mathematical basis behind them is just... so so so enlightening to me. I really can't get enough of complex analysis. This is so so satisfying and you are incredible at teaching it!

[@jpnesseth]

This video saved me from just giving up on Complex Analysis. You may have just saved my education! THANK YOU!!!!!!

[@mathemaniac]

Don't give up on CA! It is a beautiful subject if taught properly!

[@LosizakII]

This is an excellent video, I can only imagine the amount of work it took to produce it. Thank you for your efforts. Many students will benefit from this.

[@mathemaniac]

Thank you so much for the kind words!

[@michaelashcraft8569]

I am 70ys old, and, was NEVER taught these things in the "Blackboard Jungle" in which I was brought up in the 50's to 1970..Learn, folks, never cease to learn, or, seek for knowledge!!

[@joelcurtis562]

Great timing for me. Just started a self-study deep dive into CA. Love this channel, thanks!

[@mathemaniac]

Hope this will help!

[@PunmasterSTP]

I'm just curious; how's studying been going?

[@zstew7950]

Taking complex analysis this fall; Very grateful for this series!

[@mathemaniac]

Happy to help!

[@Sheikxlove123]

Okay, graduated math major, I never really understood why complex numbers are represented by as set of points, but damn this series is so comprehensive.

[@mathemaniac]

Thanks so much for the kind words!

[@josenieves309]

Why am I as excited about the rest of this series coming out as I was with the Mandalorian?!

[@Greenwood394]

When you represent multiplication as adding the angles, it is very clear when you use the exponential polar coordinates but i decided to try multiple 2 complex numbers in terms of sin and cos (before using eulers formula). After using foil and grouping some terms, i was amazed to see that it produced the angle sum identities for sin and cos. I loved this cause it just shows the consistency of mathematics. Just beautiful

[@pasta_fanatic]

Such a comprehensive definition that really looks at the complex plane in such an intuitive way. Looking forward to this series!

[@mathemaniac]

Thanks!

[@Arces6]

As a french student, it's really hard to understand all of this even when i saw it during lessons. However, it's really interesting and teachs way more than a "traditionnal lesson". I enjoyed to watch it and put a like for the work :D. Have a nice day and i hope you'll get good reviews.

[@amineaboutalib]

qu'est ce que tu comprends pas ?

[@mnada72]

Amazing approach and excellent demonstration. Thank you very much. Waiting for the rest of the series.

[@mathemaniac]

Thanks so much for the compliment!

[@wiktorm3838]

I've watched some other complex analysis courses, and quite frankly this one is the most appealing one, especially with large amounts of visualisation, whcih is really useful for remembering all the stuff

[@blunistark6555]

I couldn't ask for a better explanation on complex numbers! Thank you so much!!!

[@ConnoisseurOfExistence]

I wish one day I have the time to thoroughly examine and understand all of this...

[@mukeshkamble5533]

Actually I realize mathematics is beautiful rather than hard thanks a lot, you changed my perspective!

[@mathemaniac]

Glad this makes you think so!

[@MusicEngineeer]

very good video! but may i suggest that for visualizing complex functions as 3D plots, you could use some sort of shading, if possible? the flat color of the plot of the arg(z) function at around 4:30 made it a bit hard to see what's going on

[@mathemaniac]

Thanks! This video has to be left as it is because I can't edit this on KZbin, but for the future videos, since I will introduce what coloring means when visualizing complex functions, I would use coloring a bit more.

[@ImNEVERSarcastic]

Your videos are gorgeous. Not only in their visual presentation, but they also remind me of the beauty and creativity that permeates the field of mathematics. I realize they must take a lot to produce, but from the bottom of my heart thank you for taking the time to do it

[@mathemaniac]

Thanks for the compliment!

[@ZsoltCseresznye]

Great video, thank you! (At the demonstration of multiplication (z1 and z2) by transforming the triangle on complex plane, I really missed to see the similar triangle transform for demonstrating the multiplication of Z and its conjugate.)

[@mathemaniac]

Glad that you enjoyed the video! I considered making the similar triangle transform but for some reason that I couldn't remember, I chose not to do it...

[@wqr0805]

Really looking forward to seeing the whole series of it

[@kentkeatha9250]

I'm first time watching this channel and very satisfied with the content of this video most of the video in youtube just showing parts that tested in exam but your video is truly explain about the math thx for your effort

[@mathemaniac]

Thank you so much!

[@angel-ig]

Best explanation of complex numbers I've ever seen.

[@mathemaniac]

Thanks for your compliment!

[@bilubilu2481]

That crash course is amazing, keep posting more

[@mathemaniac]

Thanks for the appreciation! As said in the pinned comment, I would like to rest a bit before making videos again. So wait for a bit more time.

[@pinklady7184]

Nice, this tutorial has left me with a swollen brain. I have learned much here. Thank you very, very, very much.

[@mathemaniac]

Great to hear!

[@inverse_of_zero]

hello, uni maths educator here 🙋‍♂️ i loved this, i've already started recommending it to my students! my only gripe was your use of arg and log instead of Arg and Log to denote the principal values of the functions for the latter, other than that, superb work!

[@PunmasterSTP]

I'm just curious; what classes do you teach, and how have they been going?

[@inverse_of_zero]

@@PunmasterSTP I work in co-curricular mathematics support at two universities. It's going 'ok' I guess. Things really took me nosedive the last couple of years, and are only (very slowly) starting to pick back up this semester.

[@PunmasterSTP]

@@inverse_of_zero I see, and I'm sorry things took a nosedive. I am assuming it was mostly COVID, and I hope they keep picking back up.

[@igor-sukharev]

I'm watching this video to improve my English skills. Both your pronunciation and tempo are just perfect.

[@mathemaniac]

Didn't know my videos have this purpose, but in any case, glad that you enjoyed the video

[@igor-sukharev]

​@@mathemaniac I've already had some basis of higher-school maths. In addition, the visual part of your videos corresponds to the sound one well.

[@akash3478]

At 16:00 , so multiplying a complex no. By any real no. Simply means to adjust the 'r'(magnitude) without changing the argument of the complex no. And also multiplying a complex no. With another complex no. Which has r=1 means to add the arguments of the two complex no. And the 'r' remains same. And the combination of these two would simply mean the operation of multiplication of complex no. Edit : similar explanation for division as well

[@mathemaniac]

Yes, though originally I was thinking something related to similar triangles, a little more geometric explanation.

[@Infinitesap]

I really admire your insights. Very beautiful and simple - yet very complex. How do you do these visualisations? I need to learn that to investigate for myself. Please make a video about that also. I'm so happy that I found your channel. Been watching 3b1b for years. This is the same quality. I really love it - thanks again

[@mathemaniac]

Thanks for the appreciation! For all these questions, please see the description.

[@einsteingonzalez4336]

So I've heard of complex numbers and it's all thanks to the cubic formula. Now that I've seen it, it's sad to see teachers not going though in-depth of the topic. I just hope that they teach it one day.

[@mathemaniac]

Mathologer made a video about cubic formula!

[@SV42165]

The VSauce tone made me laugh. Because I had the exact problem there while solving qs and I heard that arctan is the possible answer, I immediately thought, “or is it”? And you played the music.🤣❤️

[@drywall4310]

As an aspiring mathematician, it's so amazing to see more advanced branches of mathematics explained so eloquently! You did a superb job. However just a minor detail, around 5:47, wouldn't tan(theta) be a / b and not b / a? Since tangent is Opposite / Adjacent and "a" is the opposite side, and "b" also has the same length as the length between "a" and the point of origin on the Real axis? Thanks again, you did do an amazing job

[@marzinsouad2728]

Je tombe amoureuse des nombres complexes, c'est magnifique, I love your video very much.

[@gradf8678]

You are late bro. I just finished the complex analysis. But I will watch this whole series because of the animation and explanations. Good job ❤

[@mathemaniac]

Thanks!

[@Cole-ni8ib]

I really liked the part explaining complex numbers to the power of complex numbers around section 3.6

[@aashsyed1277]

Very exited! :D i enjoy other videos as well :D

[@divine6104]

16:00 Start with the rotated triangle at 15:47- I'll call this T1- and imagine the stretched triangle that you want- T2. Now, draw a line through the middle of both triangles. The bottom half of each triangle are similar triangles. I'll call the hypothenuse of T2 as k. Hence, the ratio of 1 to half of the magnitude of z1 equals the ratio of the magnitude of z2 to half of k (you can see this if you draw it out on a diagram). Ie: 0.5k:|Z2| = 0.5|Z1|:1 0.5k/|Z2|= 0.5|Z1| k=|Z1||Z2| The magnitude of the vector z is just r. Therefore, k=r1r2

[@hanzade8385]

There is no need to divide the triangles in half T1 and T2 are already similar triangles because T2 is a proportionally strechted version of T1. Therefore if you label the bottom left side of T2 as k you can just directly write k/r2 = r1/1 Which you can rewrite pretty easily as k = r1r2

[@oxbmaths]

Good introduction to complex numbers with very nice animation and illustrations!

[@mathemaniac]

Glad you liked it!

[@rektwatermelon6746]

Never clicked a video so fast..amazing work as always

[@mathemaniac]

Thanks so much!

[@elia0162]

Congratulations, now you have a new subscriber

[@mathemaniac]

Awesome, thank you!

[@VanDerHaegenTheStampede]

Complex calculus is a well-established field with a wide range of theorems and techniques. On the other hand, calculus on the double/split-complex plane and dual plane is less developed and has specific applications and limitations. While some basic calculus operations can be defined in these systems, they lack certain properties and theorems present in complex calculus. Could you discuss hypercomplex numbers in more videos?

[@samharrison9126]

Amazing video! I decided to take a crack at Riemann a couple months ago just out of curiosity. Lol to be clear I didn’t think I’d solve it I just wanted to know what was so hard about it. This video started me off on my track to developing intuition for the Hypothesis. Super motivated, great animation and explanation! Absolute blast to watch

[@gamistry2947]

Good luck on solving it!

[@badlydrawnturtle8484]

It took me a bit to figure out what was throwing me off about this video, but I finally placed it. You always intend the natural logarithm throughout the explanation, but in some of the formulas, you use "log", which I was reading as log base-10 because you were also using "ln" in those same formulas. Consistency in symbols is important.

[@mathemaniac]

I have never said base 10, though. Unless otherwise specified, any time we use log without a base, it should have the natural base. (Perhaps its a convention issue here.) log in this video always refers to the *complex* logarithm, while ln refers to the *real* logarithm, if that's more clarifying. This is because almost always when defining complex logarithm, we say log z = ln |z| + i* arg(z), so this is, again, perhaps just a convention issue.

[@badlydrawnturtle8484]

@@mathemaniac The issue, apparently, is that the convention for denoting the complex logarithm conflicts with the convention taught at lower levels that "log" with unspecified base is assumed to be base ten. If you're introducing a new concept, as you are here, the conventions for annotating such a concept should be explained explicitly, especially when they conflict with other conventions.

[@gigaprofisi]

I believe there may be a mislabling of the two axis at 2:30 With the way you labled (a, b) and measured the lines below, it should be at (b, a), or the axis are mislabled. This occurs several times and just bugged me a bit. Amazing video!!

[@Jaylooker]

Thanks. Needed the refresher

[@Roadshot1]

In the first minute and a half you have answered a question highschool and 2 university professors neglected to answer me

[@pamdemonia]

so interesting! so clear! the visuals are amazing! so is Needham's book.

[@mathemaniac]

Thanks!

[@boukharroubamediane119]

Your videos are nicely clearly and well explained !!your efforts in preparing your videos are very grateful. Thanks very much. I subscribe, like and share. Good lucks.

[@mathemaniac]

Thanks for the kind words, and liking, subscribing and sharing! It really helps!

[@boukharroubamediane119]

Mathemaniac you are welcome sir😊

[@mmek10]

damn i got hooked, what a nice video, waiting for the series of trigonomic functions and the branch, thank you sir

[@mathemaniac]

Thanks so much for the appreciation!

[@yedukrishnan6183]

Thank you so much. You explained this so wonderfully. 😊😊

[@benradick1489]

Is there any resource you would reccomend which is more formal that I could use alongside your course? I'd love to watch your videos, gain the intuition, then formalise my understanding I.e. with a textbook. (I'm so hyped for this series :)

[@saulberardo5826]

Search for Visual Complex Analysis. A classic (if a remember well, he's mention this book in his last video). I can't recommend this book enough

[@duchoang1759]

For a standard recommendation (not as visual as needham), try Complex Analysis by Freitag and Busam or Complex Analysis by Ahlfors

[@benYaakov]

I would be glad if you please reveal the proof of that geometrical rotation in multiplication and division. I am a new learner and beginner to complex numbers.

[@animowany111]

The atan thing is exactly why atan2 exists. It's more commonly used in software for converting to polar coordinates in general, not in complex numbers specifically, but it would fix the issue here.

[@franzkern4208]

This is a great video. Thank you very much. Keep on teaching us math.

[@mathemaniac]

Thanks for the compliment!

[@anuman99ful]

Can't wait for the rest of the videos, they are great!

[@mathemaniac]

Thanks!

[@xyz.ijk.]

That was outstanding!

[@mathemaniac]

Thanks!

[@parveensheoran2368]

A very great intuition it is.

[@mathemaniac]

Thanks!

[@aashsyed1277]

Are there gonna be more videos? If yes then i am exited! :D

[@mathemaniac]

Yes, there would be, but as said in the pinned comment, this wouldn't come as frequently as you might expect.

[@aashsyed1277]

@@mathemaniac Ok.

[@sinasoltan.m4859]

Thank you for this video. Please make a video about the intuition of convolution. As an electrical engineer, I have always memorized it without knowing what does it actually mean.

[@mathemaniac]

Glad that you enjoy the video, but for now, I myself don't have the "intuition" unless you are talking about those on the Wikipedia page, in which case, making a video on it wouldn't make sense when a brief browse of Wikipedia does the job. Will consider convolution if I can think of a more direct intuition.

[@lateefahmadwanilaw8948]

Excellent!!!

[@aashsyed1277]

i love the vsauce moment!

[@qrubmeeaz]

Brilliant!! Thank you.

[@victorribera5796]

In my set theory book of university it introduces the euler formula usong the fact that modulus multiplies and argument sums when multiplying two complex numbers, and setting the fact that this is similar with exponentials.

[@cuttingcut1321]

I needed this video tutorial. Thank you so much

[@mathemaniac]

Glad to help!

[@AJ-et3vf]

Awesome video! Thank you!

[@paulhbartley8030]

I always look forward to the tritone.

[@JojiThomas7431]

It is interesting. I didn't know about De- movire's theorem to Euler's formula. Nice presentation. Eagerly awaiting the video on complex functions. Though you shouldn't feel stressed. Take your time to make it perfect(as perfect as something can be). Regards Stay safe!

[@mathemaniac]

Thank you so much!

[@gearstil]

Best explanation ever!

[@mathemaniac]

Wow, thanks!

[@PunmasterSTP]

I’m always unsure when I pronounce that name, so thanks for helping me practice it some more. Also…that was a truly incredible video!!!

[@SimonFrank369]

The German name "Euler" sounds same like the English word "oiler". And that´s also what Leonhard Euler did: he lubrified mathematics in so many ways...! ;-)

[@PunmasterSTP]

@@SimonFrank369 Oh snap, I’ve used the word “oiler” to help me, but never thought about how he lubricated math!

[@dariyashka]

Hi, Thank you very much for the video, I don’t know if you are answering, but I would like to ask in what application can you make such graphs? thank you in advance)

[@saifullahashique]

Pls upload rest of the videos quickly 😍

[@mathemaniac]

I know this comment is out of appreciation of the content, but as said in the pinned comment, don't expect me to upload videos that frequently.

[@awildscrub]

You're just like 3b1b's little brother. Great video!

[@mathemaniac]

Thanks!

[@brihaspatiangiras9081]

Love your channel.

[@mathemaniac]

Thanks!

[@jursamaj]

15:28 Regarding e^(x+y)=e^x*e^y, any math student would have study ordinary exponential functions long before getting to complex analysis. And e^(x+y)=e^x*e^y is just an individual case of the general rule that a^(x+y)=a^x*a^y.

[@mathemaniac]

This is just for rigour's sake: e^(x+y) = e^x * e^y is well-known, sure, but this is only a *property* of the exponential function - not directly from the definitions described. But what I am saying is that this property can be derived from the three definitions described.

[@jursamaj]

@@mathemaniac But my point is that adding exponents is a property of exponents in general, and thus doesn't *need* to be justified in this specific case. If it weren't true, you wouldn't be doing exponentiation.

[@zaek2144]

@@jursamaj But it is never proven that e^(z+w) = e^z*e^w for z and w both complex numbers. It wouldn't be too hard to prove (although it depends on the definition of complex exponentiation you are given).

[@jonmichaelgalindo]

Complex numbers: "But first, let's spend 90% of our time talking about all the issues caused by attempting to use rotations and angles for anything at all."

[@WolfrostWasTaken]

So true!!!! Like, wtf???? Why do mathematicians like to torture themselves like this!?

[@jonmichaelgalindo]

@@WolfrostWasTaken 3d graphics transform function accepts angle inputs. I have to take vr controller vectors and use arctan to compute angles for it. Then, the function uses sin and cos to compute the EXACT SAME INPUT VECTORS, and pastes them into the matrix... I seriously don't even know what to say. :-|

[@yijiaocao2847]

How subtle it is!

[@KimHengTeo]

18:41: The visual for division has a minor mistake which causes the arg(quotient) = theta 2 instead of theta 1 - theta 2. The mistake is at the rotation: "rotate it so that the side originally corresponding to z_2 is aligned with the positive real axis" should change to "rotate it so that the side originally corresponding to z_2 is aligned with the side originally corresponding to z_1".

[@mathemaniac]

The triangle is between z_1 and z_2, and so that angle in the triangle is theta_2 - theta_1, and so after rotation, the argument of the quotient is the negative of that difference, hence theta_1 - theta_2.

[@jozefgoethals6867]

So introducing complex numbers by inventing V(-1)(the square root of -1) is a way to become a plane of numbers instead of a line with ai+b where a&b are a pair of real numbers and i a imaginair number.So to have a cube of numbers(3 dimensional set of numbers) we can invent 1/0=j then you have a 3 dimensional field of numbers a+bi+cj where a,b&c are a trio of real numbers,i=V-1 and j=1/0

[@Cole-ni8ib]

I really like the visual comment at 1:32

[@morchel332]

you are the best man... really love your videos, gets my dumbass leanring/doing maths and even having fun while doin it :P if i someday have some spare money ill gonna tip lmao

[@Darakkis]

Incredible video! Thx a lot

[@mathemaniac]

Glad you liked it!

[@yos2413]

What textbook do you recommend for learning complex analysis?

[@mijmijrm]

"i" is only mind bending if you are an accountant and think mathematics is about bean counting. If you're a navigator and think mathematics is about manoeuvring on a surface, then "i" (quarter circle rotation = half of half circle rotation = (-1)^(1/2)) is totally natural.

[@johnchessant3012]

d/dt [exp(it) (cos(t) - i sin(t))] = i exp(it) (cos(t) - i sin(t)) - exp(it) (sin(t) + i cos(i)) = 0 so exp(it) (cos(t) - i sin(t)) is constant; at t = 0 it equals 1, hence exp(it) = cos(t) + i sin(t). q.e.d.

[@mathemaniac]

Yes this is indeed a proof, rigorously speaking. The reason why I didn't include is that for this proof, you kind of already know Euler's formula, and just try to verify it.

[@shashanktiwari4209]

Great video!! loved the explanation about n-th roots of unity

[@mathemaniac]

Thanks!

[@muhittinselcukgoksu1327]

super, thank you so much...

[@hoodedR]

The last function is very intriguing. I see no other comments guessing what it is. Could you give me some hint as to what it is?

[@mathemaniac]

It is one of the functions mentioned in the video. That's probably a hint, but not too big as to spoil it.

[@adarshkishore6666]

Its got to do with trig functions

[@razieldolomite698]

Im like, 80% sure the ending graph was complex sine.

[@adarshkishore6666]

@@razieldolomite698 Yup exactly