Euler's Identity (Complex Numbers)

How the Fourier Transform Works, Lecture 4 | Euler's Identity (Complex Numbers)
Next Episode: bit.ly/38qm6W7
Course playlist: bit.ly/2WyzWD4
howthefouriertransformworks.com/
In order to describe the Fourier Transform, we need a language. That language is the language of complex numbers. Complex numbers are a baffling subject but one that it is necessary to master if we are to properly understand how the Fourier Transform works. What is the imaginary number “i” and why it is so useful to us when dealing with the Fourier Transform?
This is the sixth in this series of videos which takes a new and visual look at the maths behind the magic of how the Fourier Transform works.
For a comprehensive and visually intuitive exploration of the Fourier Transform and its workings, I invite you to explore my book series on the Fourier Transform available at:
www.amazon.com/dp/B0BSJJ69Z1
Please help me finish filming the course by supporting me on Patreon: www.patreon.com/MarkNewman
Click below to subscribe to the Course's mailing list to receive an update when the next video is available, updates about the course's production, and a notification of when the full course is available to purchase.
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#math #mathematics #learning
Thanks for watching the video How the Fourier Transform Works, Lecture 4 | Euler's Identity (Complex Numbers)

[@gk10002000]

What people have to understand is how brilliant these guys were. They had no internet, few if any textbooks. They had to reason things from first principles, so much original. Just stunning

[@andraskovacs517]

They _wrote_ the textbooks & had them published; they published papers in mathematical journals, of which they had several; and some of them maintained extensive connections through personal letters.

[@ernst9100]

@@andraskovacs517 Thinkers existed in centuries gone by....they invented whole new concepts to explain reality of the times

[@boukharroubamediane119]

and above all, they did not have an electronic calculator! everything was done manually !! 😳 wow!!

[@chandradeepraut9306]

This is what I think ,we could also have been smart if we had no tv (in my childhood upto 6th) as I spent my childhood in watching tv and no critical thinking of sciences ,so I end up just learning what others had done but I started in 10th class inventing techniques to solve maths but it was too late

[@boukharroubamediane119]

Chandradeep Raut 👍

[@roger7341]

For a long time 0 didn't exist, and some people who stupidly claimed that nothing existed had their heads bobbed. Now imagine imaginary numbers. That was like claiming the earth wasn't flat.

[@MarkNewmanEducation]

It was Descartes that called them "imaginary numbers". An unfortunate name. Perhaps he might have done better to call them lateral numbers or something that intimidated the idea of their working in more than one dimension. That might have made them easier to understand.

[@purplrshadowyay]

​@@MarkNewmanEducation A good name could have been surreal numbers (S) or just the other name it has, complex numbers (C)

[@neilduran2335]

It is over 20 years since I studied the maths of Euler but this is by far the best explanation I have ever seen. I wish I had seen this video back then. Students of today have it a lot easily than years ago, when you were expected to just get it!

[@MarkNewmanEducation]

This is why I made the video. I was also just expected to know it and it frustrated me that I didn't. This video is part of a whole series on the Fourier Transform which I made for the same reason: howthefouriertransformworks.com/

[@AmikaofMan]

@@MarkNewmanEducation Thank you for the link and the video here for Eulers. Definitely makes more sense this way than the way my physics teacher tried to explain to us back in the day.

[@martlock1]

So beautiful that the simple identity e^(pi*i)+1=0 can link together the most important mathematical concepts (0, 1, i, e, pi) using the most fundamental mathematical operations (equality, addition, multiplication, exponentiation)!

[@BRORIGIN]

Equality is not an operation

[@mp0011]

Now I am waiting for Euler's Supremacy and Euler's Ultimatum...

[@MarkNewmanEducation]

+Niespotykanie Spokojny Rowerzysta haha... Apparently, I might have been wrong in stating that it was Euler who gave e the name Euler's number. It seems to be that the name was coined later although exactly when and by who, I don't know. Euler probably used the symbol e simply to denote "exponent".

[@us-Bahn]

They are coming! But they wouldn’t be if Euler had never been Bourne.

[@petrophilip2279]

LOL, Good one Ian McCutcheon

[@ruffifuffler8711]

Happy you didn't move stuff across the = symbol without a triple deep incisive prayer.

[@georgesealy4706]

Excellent!

[@J0nasBern]

To me what is beautiful is that you have a number with infinite and random digits that is related to exponencial growth/decay, then you raise it to the power of a number that we find impossible to solve and so we call it imaginary, and to another number with infinite and random digits that is related to circles and it's geometry, and then you add a single unit, probably the most basic number that we know, to all of this only to get what we call "nothing"

[@bayestraat]

You should also highlight how euler's identity is nicely shown with multiplication of complex numbers as vectors around a circle plot on the imaginary plane. And how to maintain symmetrical values working out the power spectrum density in FFT.

[@MarkNewmanEducation]

Thanks for the idea. Will look into it.

[@isaacrajagopal391]

It is not just beautiful "in mathematical terms," it is just BEAUTIFUL. Period.

[@MarkNewmanEducation]

Very true.

[@petrophilip2279]

I have seen and used the constant "e" in the study of calculus, complex numbers, infinite series, natural logarithms etc, but no one explained what the number is. This is the simplest explanation I have seen. It takes a special kind of a skill to correctly explain a complex concept in simple terms. Thanks Mark Newman.

[@NoferTrunions]

Explanation by example or picture are the best.

[@NoferTrunions]

@@oldlonecoder5843 Unfortunately, mathematicians are the worst at explaining from example or pictures - they do their explaining by showing how to manipulate equations. Physicists on the other hand are the opposite - they use sketches and examples regularly. A meaningful humorous comment on mathematicians is that once they prove a solution exists, they lose interest and move on to the next problem...

[@parmachine470]

e is also involved in some of the most beautiful comprehensible markings on a chalkboard.

[@Chad-qk1ig]

@@NoferTrunions depends what kind of mathematicians you're talking about. Historically mathematicians heavily utilised geometry.

[@dianedong1062]

A lot of people seem to think that math is boring, but for me personally, studying mathematics has been like discovering a hidden cave full of beautiful treasures.

[@likeitornot6821]

Likewise

[@MarkNewmanEducation]

I think that the problem is the way that maths is taught, and that it is treated as a totally abstract concept with no relation to the real world when in fact it describes the real world with precise beauty.

[@jaredhouston4223]

@@MarkNewmanEducation You hit the nail on the head

[@nanzhang9837]

I don't believe Euler named the number after his own name. From what I know Euler was a very modest man, he instead named the number e because it was the next available letter that was not already taken. Listen to the podcast of 'In Our Times' discussing this number.

[@thomashan4963]

I just want to say, there's so much efforts in making this video and I appreciate it. From animations to sound effects to historical facts and figures .. this is so much works.

[@BlackNSB]

That sure as hell is beautiful--especially because, as a student, I didn't understand why this formula was so special. Great video.

[@MarkNewmanEducation]

I'll let you into a secret. As a student I didn't understand it either. I just had to accept it. It wasn't until years later when I had to work with it that I found out what the link was when I had to research it for my work. Glad you enjoyed the video.

[@Frieza.exe.z]

8:10 "The brilliant thing about mathematicians is that . . . when they are on their way to some wonderful mathematical discovery, they don't let a little thing like "Numbers NOT EXISTING" stop them." Is it safe to say HERO ?

[@audience2]

It also links exponents, zero, addition, equality, the identity element under multiplication, and when expanded, trigonometry, division, factorials, and infinite series.

[@DejiAdegbite]

And also indirectly, logarithm.

[@jeanpaulniko]

My first ever comment in 10+ years of watching KZbin. Mark, you nailed it! This video has me feeling ecstatic. You have shown me the connection between sin, cos, i, e, and π as presented in Euler's famous identity. This reveals the deep foundation that underlies all of classical math and ties everything together. Now I have seen the light! Thank you so much.

[@MarkNewmanEducation]

Amazing. So happy to have helped and thank you for making my video the first video you commented on in a long time. Any suggestions for future videos you would like to see would be gratefully received.

[@MorganBW53]

I watched this video awhile back and did not comment. It occurred to me that this presentation helped me to connect a couple of dots which enhanced my understanding. I actually had to spend a bit of time to find this video again but I felt the need to say thank you for your time and for the explanation.

[@MarkNewmanEducation]

+Benny Morgan my pleasure. Really happy to have helped you.

[@bronzekoala9141]

This is the best mathematica axplanation I've found so far on KZbin for anything.

[@jimrogers2579]

Mark Newman -- wow! Sorry I missed the release of this video almost 4 years go. Beautiful! This is a stunning exposition of an often spoke wonder I had never grasped. Your explanation left me gobsmacked twice. Holy cats man -- great! I will share this unfolding to help my students understand - and point them to this video. Can't wait for more! Extremely well done.

[@mtolympus-worstplayerever7924]

I have watched over a dozen videos on Euler"s identity, and this is the most clear and straightforward.

[@ApenstaartjeYT]

Sometimes I wonder if the internet made us numb.. Back in the day you were kind of 'forced' to think. Just look at this absolute beauty.

[@GrandTeuton]

I think this is the finest maths video I've seen on KZbin - and I have sought out many. Well done!

[@pfp3438]

What an excellent explanation of one of the most used fundamentals in the real world. You did a far greater job in under 15 minutes than my lecturers did over hours of classroom time. Well done.

[@JH-ho3qu]

I saw various equations named Euler's method or formula, I was so confused about what Euler's formula is. This is the best video I found to clear up my confusion. Thank you very much!

[@meneyahelzegeyetesfaye8340]

Aside the brilliant minds behind the formula, your presentation is also "beautiful" and very structured. No wonder, it is almost 2 in the morning and am wide awake!

[@Ch4dBR]

This is the best explation about Euler's identity! Thanks.

[@DeadCatX2]

Another beautiful aspect to that representation in particular. It involves precisely one each of addition, multiplication, and exponentiation, while also being set equal to 0, as is the custom when solving for roots.

[@whatever5575]

The beauty of the formula is that it says so much in so little space and in a simple and elegant way. That's what good literature or well written instructions ought to be. Say it simply.

[@manamsetty2664]

You done something to me in 8 minutes which many people could not do. Thank you

[@ernstboyd8202]

you can consider e^ix to be (e^i)^x. then imagine e^I , (e^i)^2 , (e^i)^3 ... as a special case of a spiral on the complex plane that stays on the unit circle and advances 1 radian (57 degrees) each time similar to (1+i)^1 which is 2^.5 long and pointing at 45degrees. then (1+i)^2 = 2 units long at 90degrees = 2i which is 2*( cos(90)+I sin(90)) and (1+i)^3 is (2^.5)^3 long at 135degrees etc.

[@jacquesjutras1635]

This gentleman Mark is a very good teacher he is a master.

[@MarkNewmanEducation]

You're most kind.

[@lylecosmopolite]

Euler was perhaps the most productive mathematician of all time. The number e is named in his honour. It is to Euler that we owe much of our basic understanding of infinite series and complex analysis. But Euler had a weird flaw -- his proofs always fell well short of the standard laid down by Cauchy, Riemann and Weierstrass, and often were wrong even by the relaxed standards of the 18th century. But no result ever published by Euler was ever shown subsequently to be wrong. Everything he claimed to be a theorem in fact was, even though his proofs were never rigorous and were often downright wrong. Euler was perhaps the most spectacular example of history of mathematical intuition. Euler derived his eponymous identity using the infinite series for e^x, and his proof was largely correct.

[@hazemsaeed6372]

you have no idea how much i loved this video... beautifully explained...

[@evierma]

Brilliant, simple, elegant. The video is a piece of art.

[@taripar4967]

This makes WAY more sense than some of the other videos I’ve seen on this. Thank you.

[@lillysumathi7825]

So in simple terms, the value of the function of e raised to ix at pi rads is -1. That's mind blowing

[@MeButOnTheInternet]

This is beautiful. I've never seen it explained so clearly

[@screenflicker1]

I agree that the Euler's identity is beautiful but so was this vid. Hard work went into this!

[@MarkNewmanEducation]

+screenflicker1 thank you. I really enjoyed every moment of it.

[@MadScientyst]

Truly BRILLIANT expositon!! The introductory theory & linkages to the Trigonometric Series were so cleverly explained, that I think even an average Math student would find the topic both fascinating & understandable. I also noted the emphasis on demonstrative exposition with the aids of Graphs, etc. Please keep up the great work & tutorials....a new subscriber!

[@chineduokabechinedu9126]

It so profound and amazing. It has been long I encountered all these signs in maths, like sine and cosine, + - ÷ but right now I got understand how all those signs came into existence, with help of your explanations here. Thanks

[@stmellion1]

brilliant, thank you Sir... i was "taught" this badly over 30 years ago... i get it perfectly now...

[@TheKellydelight]

Beautiful explanation i have never seen such a nice presentation skills . God bless

[@MarkNewmanEducation]

+Unique Family thank you

[@garvitprashar3671]

This is really amazing!! Sir, I would like to thank you for rekindling and making the beauty of intuition work so well. I chose mathematics in my high school years because I was interested in it but just being told to cram formulae and pressurized to just score good grades deteriorated my interest but videos like these is what will keep me going so thank you.

[@milestoitaly]

An absolutely wonderful wonderful "experience". Like a time-capsule back to math classes when everything was proved. All that is missing is the smell of chalk dust! Thank You Mark.

[@ENBOmniGaming]

There is a great feeling when you understand a new math concept - I finally understand e and its relation to sin/cos after this video. Excellent work please keep it up !

[@MarkNewmanEducation]

I SO know what you mean. For years this stuff was incomprehensible to me. Then I got this project at work (I'm an electronics engineer) which was all to do with complex impedences and all things "imaginary" that I HAD to understand in order to get the job done and finally, after much research, I got that Eureka moment that I am so happy to have helped you arrive at as well, where everything just fell into place. Then I really wanted to make sure I understood it properly and the best way to do that is to try and teach it. I am now busy working on more stuff as we speak which I shall be putting into an online video course about the Fourier Transform. I have published my research notes for the course at: www.themobilestudio.net/the-fourier-transform-part-1. I'm posting progress reports on the course on a Facebook page facebook.com/TheFourierTransform/ which includes snippets from some of the videos that will be in the course. I'm just putting the finishing touches to the video on "phase" and will post an extract from it on the facebook page (and probably here on my channel too) during the next week or so, so please stay tuned. Thanks so much for your comment.

[@Bobby-fj8mk]

@@MarkNewmanEducation - good post - i might be imaginary but it gives you real results when you play with complex numbers & impedance's in electronics. It's all to to do with time - relating one voltage to another by a phase difference. When you do Fourier analysis you don't only get the amplitudes of the various harmonics you get the phase differences coming out of the equations. It's about time domain & frequency domain. We couldn't calculate anything much with using i. It is very powerful mathematics.

[@pwnmeisterage]

EEs describe phase impedance as _Z=x+jy_ ... I guess easier (visually relates to Euler's identity) when graphing with polar coordinates but why isn't _i_ good enough?

[@lockdowntrendings1207]

I don't say this to every explainer or professor or technologist but I think it's suits you well. "You are real intelligent"

[@sohrabnabi-zadeh7558]

Thank you Mark! I am Enlightened and Delighted. It's great to find you!!

[@CoranceLChandler]

Beautiful! Inexpressibly and inexhaustible beautiful! Astoundingly and undeniably wondrous! Didn't understand a single, solitary syllable he said but I want to. Time to get back to learning math

[@artha3283]

AMMAZING!!👍👍 !! this is how a story is told and a lesson is learnt 👌

[@tarunsharma209]

This video really made me understand how beautiful Euler's identity is

[@veerk3494]

this explanation much better than other videos that try to explain euler's identity by rotations

[@cayezara8110]

The most simplified illustration. Amazing! Beautiful!👏

[@agytjax]

Great Video. You might as well add that this is the only equation that connects all the important mathematical constants - e, pi, i, unity (1) and zero.

[@sacredsanctuary420]

this is the best explanation I've seen so far :)))

[@MarkNewmanEducation]

That's very kind of you to say so.

[@ayoubmounadi2142]

Sacred Sanctuary i share the same opinion with you 😃

[@wickrenner488]

Yes, a very good explanation. I finally understand Euler's Identity, huge revelation, thanks.

[@sophialopez8024]

Maybe that's because you limit yourself to 'religious' sources..

[@michaelzhu6952]

Agree!

[@selimedelice3131]

A very nice,clear and comprehensive video. Thanks for the preparation and share👏👏👏

[@yarabamba]

The best explanation of Eulers identity.

[@Learner..]

Offc 🌸 it's beautiful the two fundamental constants e and π comes in a equation along with an imaginary number

[@ketchup143]

omg i've been looking for a video that explains what all these terms mean for so long. thank you for your elegant teaching!

[@erictko85]

WOW. Just discovered you through this video. You are a fantastic teacher of mathematical concepts. Thank you.

[@techno2371]

"They don't let a little thing like numbers not existing stop them" This is why math is amazing, the limit is how far you are willing to push it!

[@MarkNewmanEducation]

Very true!

[@SpacePoolNoodle]

Thank you! I was frustrated because most videos did not show why e^ix = cos(x) + i sin(x), this made it very clear.

[@hsenagrahdeers5468]

This right here, ladies and gentlemen, is what high-quality educational content looks like! I can't thank you enough!

[@MarkNewmanEducation]

You're most welcome.

[@violaisreallycool]

Hey, I just want to let you know that I liked the cute visuals and the explanations, they really made it easy to understand how truly wonderful that identity is! I also liked how your tone of voice wasn't too demeaning but in a gentle explanatory way conveying the ideas and concepts :)

[@MarkNewmanEducation]

Awesome, thank you!

[@snavarro7713]

The formula IS a wonderful solution. The 4 concepts are combined all together and zero is appearing. That is Amazing, that is elegant, that is math.

[@samaptisapui951]

Wow!!! Divinely beautiful explanation...thank you so much

[@martindupuy2071]

Clever and clear explanation of the formula makes this video a good one :)

[@Lawliet82]

This is basically a mix of all we learn in highschool but in one single calculation, absolutely madness bro

[@samanthabanya1349]

This is the best explanation of this that i have seen

[@bryanbowen4193]

Euler's number isn't just beautiful. It literally describes reality to perfection.

[@ucctgg]

What is reality?

[@bitchoflivingblah]

totally understood this - thank you so so much for your clear explanation.

[@adventure_digger]

Wow! I just understood so much I never got before. Thank you mate! Well done

[@shinigamisteve5607]

I remember when my Calculus II teacher taught me this. It blew my mind

[@dinoscythe6335]

It all makes so much more sense now. Thanks :D

[@rockmdii]

My mind is totally blown right now!!!!! Great job Mark!!

[@seahooh]

What a beautiful explanation it is!

[@SassanRohani]

The most beautiful formula in Mathematics explained in the most beautiful way in this video. Thank You!

[@esuchil]

Other than having that ugly ass Pi in it, and needing to subtract 1 for no reason, it's pretty meh.

[@MarkNewmanEducation]

You're welcome.

[@MarkNewmanEducation]

...but PI is SO useful. It crops up everywhere!! I use it a lot in electronics and filter design.

[@vikasvishwakarma5263]

e^(i*pi) means you have rotated the complex number 0+i to 180 degrees. Because in polar form it is written as cos(pi) + isin(pi) and it is -1 :)

[@Snow-qt7pn]

This was the most beautiful explanation of the most beautiful identity.

[@shezanahmmed5582]

I've got so interested. I want so many videos and knowings about mathematics like this

[@SuperKnowledgeSponge]

I think this is the only guy that ever clearly explained Euler's identity on youtube that I could actually understand. wow.

[@7rgrov198]

@@jacksmetana7932 then what did he spend 14 minutes doing?

[@bluegtturbo]

As a maths teacher I have to say this is one of the best and clearest explanations I've ever seen! Well done!!

[@TheNabob57]

good thing you can handle math, your English is atrocious. I depise the hip hopping, ghetto-fying of MATH to maths.

[@bluegtturbo]

@@TheNabob57 How dare you! I am English (as in I live in England - there's a clue right there. Maths is short for mathematics (implicitly plural hence the 's' at the end)). Unlike you lot of ex-colonials, we know how to use the English language correctly. That's why we use the correct construction 'If + past tense' rather than 'If +would' as in 'If I knew' as opposed to the stupid American construction 'If I would have known'...

[@CommanderSp00ky]

It baffles me that even in a community related to mathematics and science idiots and douchebags are thriving to ruin everything.

[@stuartdryer1352]

I'm not a mathematician or even particularly good at math but that is amazing. I've seen that identity many times but never knew where it came from, and never imagined I could understand it if it was explained. But it is simple which makes it even more beauriful.

[@jerzyzajaczkowski8537]

It comes from Maclaurin series expansion of functions e^x, sin(x) and cos(x) followed by e^(ix)=cos x + i*sin x, where i=sqrt(-1)

[@gerardomoyano9504]

I've used this for years but never thought of proving it, my mind has never been blown this way.

[@MarkNewmanEducation]

Amazing. So glad I was able to help. The proof of course is not mine. If only I was that cleaver! It is a special case of a Taylor Series called a McClaurin Series (see en.wikipedia.org/wiki/Taylor_series)

[@carlosalexandreFAT]

The association of the main numbers in the field of mathematics with each other, reflects numerical sequences that correspond to the dimensions of the Earth, the Moon, and the Sun in the unit of measurement in meters, which is: 1' (second) / 299792458 m/s (speed of light in a vacuum). Ramanujan number: 1,729 Earth's equatorial radius: 6,378 km. Golden number: 1.61803... • (1,729 x 6,378 x (10^-3)) ^1.61803 x (10^-3) = 3,474.18 Moon's diameter: 3,474 km. Ramanujan number: 1,729 Speed of light: 299,792,458 m/s Earth's Equatorial Diameter: 12,756 km. Earth's Equatorial Radius: 6,378 km. • (1,729 x 299,792,458) / 12,756 / 6,378) = 6,371 Earth's average radius: 6,371 km. The Cubit The cubit = Pi - phi^2 = 0.5236 Lunar distance: 384,400 km. (0.5236 x (10^6) - 384,400) x 10 = 1,392,000 Sun´s diameter: 1,392,000 km. Higgs Boson: 125.35 (GeV) Phi: 1.61803... (125.35 x (10^-1) - 1.61803) x (10^3) = 10,916.97 Circumference of the Moon: 10,916 km. Golden number: 1.618 Golden Angle: 137.5 Earth's equatorial radius: 6,378 Universal Gravitation G = 6.67 x 10^-11 N.m^2/kg^2. (((1.618 ^137.5) / 6,378) / 6.67) x (10^-20) = 12,756.62 Earth’s equatorial diameter: 12,756 km. The Euler Number is approximately: 2.71828... Newton’s law of gravitation: G = 6.67 x 10^-11 N.m^2/kg^2. Golden number: 1.618ɸ (2.71828 ^ 6.67) x 1.618 x 10 = 12,756.23 Earth’s equatorial diameter: 12,756 km. Planck’s constant: 6.63 × 10-34 m2 kg. Circumference of the Moon: 10,916. Gold equation: 1,618 ɸ (((6.63 ^ (10,916 x 10^-4 )) x 1.618 x (10^3)= 12,756.82 Earth’s equatorial diameter: 12,756 km. Planck's temperature: 1.41679 x 10^32 Kelvin. Newton’s law of gravitation: G = 6.67 x 10^-11 N.m^2/kg^2. Speed of Sound: 340.29 m/s (1.41679 ^ 6.67) x 340.29 - 1 = 3,474.81 Moon's diameter:: 3,474 km. Cosmic microwave background radiation 2.725 kelvins ,160.4 GHz, Pi: 3.14 Earth's polar radius: 6,357 km. ((2,725 x 160.4) / 3.14 x (10^4) - (6,357 x 10^-3) = 1,392,000 The diameter of the Sun: 1,392,000 km. Numbers 3, 6 & 9 - Nikola Tesla One Parsec = 206265 AU = 3.26 light-years = 3.086 × 10^13 km. The Numbers: 3, 6 and 9 ((3^6) x 9) - (3.086 x (10^3)) -1 = 3,474 The Moon's diameter: 3,474 km. Now we will use the diameter of the Moon. Moon's diameter: 3,474 km. (3.474 + 369 + 1) x (10^2) = 384,400 The term L.D (Lunar Distance) refers to the average distance between the Earth and the Moon, which is 384,400 km. Moon's diameter: 3,474 km. ((3+6+9) x 3 x 6 x 9) - 9 - 3 + 3,474 = 6,378 Earth's equatorial radius: 6,378 km. Orion: The Connection between Heaven and Earth eBook

[@MadScientyst]

Thank u for this post friend....I intend to check these relationships out!

[@carlosalexandreFAT]

@@MadScientyst 👍👍

[@howardebenstein3204]

Apparently a higher authority uses the metric system.

[@talatdhk]

Most beautifully presented!!!

[@beasthunt]

Me winding down from my signal analysis class..... Can we all take a moment to appreciate the graphic when they introduced "i" next to the guys droopy right eye? Well played.

[@citehfan4585]

That was amazing. Never thought I would ever say that about maths. You have a real gift in explaining complex ideas in a straightforward way. It's inspiring.

[@MarkNewmanEducation]

Wow, thanks!

[@curtpiazza1688]

This is Excellent! Very well explained and illustrated! ❤ 😊

[@dollascience7808]

This is the best explanation I ever had except that one explanation when I was in 4th standard.

[@nandakumarcheiro]

By an inspiring discussion I have come come across phase conjugated dynamics of gravity lense operating between a dimness and brightness in between which will be applied in my future research also. Thanking you Sankaravelayudhan Nandakumar.

[@ashaydwivedi420]

lovely video! we just learnt about e^ix = cis x in school, and i was dying to find out how the heck do you raise an irrational number to a complex power? and i think that the use of series to solve this problem was just beautiful. gotta love how this equation relates e,i,pi,1,0 - some very famous and well known/fundamental numbers among math lovers

[@slightlychaotic8988]

Thanks for that excellent explanation, and for all the gracious and logical followup replies you gave below.

[@MarkNewmanEducation]

You're welcome.

[@MissingPleiad]

One of the best videos on youtube

[@SNROPQUEYR]

Absolutely Brilliant $ Fabulous - Only now (i) understand Euler’s (e) Thanking you all - for this superb documentary - !!

[@talramot2631]

It's lust amazing! I've finally understood this subject.

[@The_Ass_]

Euler was so obsessed with complex numbers, he based his identity off them

[@mdabusaqibalamansari9979]

This is the best video on the euilers identity

[@m.g.phadnis5073]

Superb clarity. Thanks Professor.

[@grittayasrinaphasawadi817]

Excellent explanations and presentation!

[@Atmosnofear]

Beautifully explained, thank you.