Euler’s Equation’s Beauty Explained

[@alithedazzling]

For more chalkboard videos including the next one: kzbin.info/aero/PLZZOG63zmCLE1eWvazftEMl8kMpXvSzst

[@goldCrystalhaze]

That is why math is wrongly hated. More than 25 years back, when I was a student, not even one teacher bothered to take 10 minutes to break down this formula to us.

[@danielweatherman2252]

Yah Same....none of this was discussed.....dropped out of math when this was starting.

[@CohenRautenkranz]

Respectfully, I would wager a guess that the sequence in which you learned the necessary parts simply connected these same dots much more slowly. For example, if you took a person with only a rudimentary knowledge of arithmetic and tried to show them this, they would (rightfully) struggle greatly. There are a mountain of foundational concepts which build up to this. It is an unquestionably elegant result - as are almost all great theories. Relatively, quantum probability, etc. can each be summed up on a single page - but take years to fully grasp. You just happen to be at the point in your journey where a Taylor expansion, derivatives, exponentials, imaginary numbers, etc. aren't like a new language - and good on you for that! Lots of talented educators have sought a path around that and there are good reasons that no one has really found a better way yet. (We will likely agree that there are better and worse teachers!) Cheers!

[@ZatoichiRCS]

You give teacher too much credit. That’s why he does this work. Teachers in general don’t have a clue. They mean well but meaning well is not understanding.

[@christopherward2748]

Teacher didn't understand it herself

[@treeross]

It is in Calc 2. You have to prove it using calculus so they don't even begin to explain it until then.

[@Сделайэтодома]

I have zero use for the information you're sharing here. But you make it so interesting that once I start watching I can't stop.

[@alithedazzling]

That’s the goal, to make learning fun!

[@happyhippo4664]

That equation always blows my mind. How can two transcendental numbers and an imaginary number combine to form an integer? Euler was a genius and my favorite mathematician.

[@Delan1994]

Thank you Dr. Ali . Keep up the great work, we appreciate you.

[@DD-gi6kx]

that taylor swift is multi talented

[@oddvenn]

If only that were Taylor Swift…

[@bradatherton9369]

​@oddvenn the remark was a play off of Ali's remark at 5:17

[@primajump]

I woke up in the middle of the night and couldn't get back to sleep. So, as usual, I began learning something new in mathematics. I find this time to be very productive in terms of absorbing new concepts. I came across this video and got curious. I am just a math enthusiast who is now retired and spends a fair bit of time catching up on mathematics. This made me truly understand why Euler is considered the greatest mathematician who ever lived. You made me appreciate the real beauty of Euler's formula. It felt like a detective story as I tried to stay ahead of you and guess what would come next. It was so much fun! You were quick, but I had no problem following along. Now, I kick myself for not pursuing this path earlier. At 67, it's not too late-at least for having some fun with mathematics. Keep up the great work, and thank you for taking me on this wonderful journey.

[@dagoff0309]

Hi, at 72 I share your enthusiasm for math. I suggest you find the channel 3Blue1Brown for awesome visual tutorials on many aspects of mathematics.

[@arlenestanton9955]

I, too am retired and do math as a hobby. Great to have found a kindred spirit!

[@life_is_beatiful1883]

Diving into mathematics is never too late. Its beautiful.

[@stephannikoloff]

Thank you for your videos Mr Ali. I'm like a kid at christmas, when I see that you have a new video out. Concerning the reason amateurs like the formula is perhaps that it illustrates all of the conceptual revolutions of mathematics through the history of the subject ... the discovery of unity, zero, negative numbers, irrational numbers, imaginary numbers, calculus ... and many more.

[@testboga5991]

This video is exactly what I've been waiting for a long time. Thank you very much.

[@ABOAdam-g8r]

I have ever seen like this explaining ما شاء الله عليك وفقك الله.

[@alithedazzling]

Thank you! Glad you like the videos

[@Stoicsnowadays]

Even though I am already happy with my major, (communications and systems engineering) you are hyping me up even more! Explaining in detail the how and why of each equation. Keep up the excellent work! Greetings from Greece

[@nasir_khan-16]

Nice explanation Ali Sir, I'm from Kashmir and in my last semester of Mtech at IIT Jammu i used Eulers equation for analysis of dynamic system subjected to perodic excitation and evaluating response of dynamic system. From your explanation i now understood why Eulers equation is used in structural dynamics course. Thanks for the explanation.

[@alithedazzling]

Glad you enjoyed the explanation!

[@iWillRun_K]

Kashmir is such a beautiful place,

[@nasir_khan-16]

@@iWillRun_K yes indeed

[@DavidAKZ]

@@iWillRun_K is it safe for tourists ?

[@iWillRun_K]

@@DavidAKZ Its mostly safe., My grandfather visited recently , he enjoyed it. But yeah if you are visiting to any new place take caution regardless

[@riderpaul]

I love your clarity. It's been 30 years and I don't use it that often so thanks for the refresher!

[@alithedazzling]

It’s always good to revisit these ideas!

[@walterstevens8676]

Great video. Its nostagic, looking at these early maths building blocks...

[@MrBroseph19]

Beautiful. Keep this series going. Thank you!

[@louis-karimnebati6143]

I really like your demonstration, one of the best elegant one I've seen. You plugged more Beauty in e !!! Thank you

[@alithedazzling]

Glad it was helpful!

[@wus9472]

Outstanding explanation.

[@Tade.G.B]

Thank you Dr.Ali appreciate your clarity

[@DistortedV12]

Dr. Ali, you are the man. We are almost getting to path integral and schrodinger equation.

[@sangramd8675]

The Taylor Swift joke cracked me up😂. Particularly the way you put it Straight face 😂. Like your videos 🙏🏻. Subscribed.

[@demetrioskasabalis5536]

You are a fantastic teacher!!! You have an innate talent of communicating esoteric ideas with such a charisma!!! Congratulations!

[@osirusjacky6283]

Thanks so much for taking your time to do this!! Really nice and interesting to hear explanations with passion from a passionate to passionates !

[@alithedazzling]

My pleasure!

[@MrBsdninja]

You are a legend 🙏 thank you for taking the time to make such awesome videos.

[@Ananya-lq9vq]

This channel is gold❤

[@datakprosto123]

Ali! I think you don't understand how much you bring to this world! You are soon good at explaining!

[@alithedazzling]

Haha thanks!

[@samvanzyverden1749]

Thanks Ali. Looking forward to the Fourier transform video. Keep it up!

[@paulcoonce2493]

Thank you for that. Much better explanation than they gave me back in engineering school. Actually, I think they taught me that but by the time they crammed everything else into my head, I forgot. And didn't see it as clearly as I do with your teaching skills!

[@longbeachboy57]

Thank you for reminding me how I felt the first time I saw this 35 years ago!!! Awesome!

[@DolphinPain]

I failed precalc back in high school and always had a chip on my shoulder about learning math. I really hope you keep doing lectures past the fourier transform, because you make things so simple to understand.

[@subashp7925]

Excellent Sir, I was searching this explanation for many days and got it finally with great clarity, many thanks for your valuable efforts, please keep post videos like this it would be helpful for people forever 🎉

[@alithedazzling]

I'm glad it was helpful!

[@antoineaubert9403]

Very well done for your videos. I found them perfect and intuitive. You teach with real example (I love the video with the fingers being hidden to represent the imaginary numbers). You are awesome and keep going.

[@alithedazzling]

Thank you very much!

[@mohsinshawkat6531]

Great, Dear Ali. Most videos on this equation focus on the existence of the five constants as a sign of great beauty, but you brought to my attention an issue that seems new to me, which is that the existence of (J) is the cause of the rotational motion. You did something impressive. Thank you very much.

[@alithedazzling]

Well said!

[@MohammedImadMahmud]

Thanks a lot for an excellant explanation. Please continue to do more.

[@alithedazzling]

More to come!

[@bbuild9044]

Thank you, I am a student who tries to get a deeper understanding of maths and physics to answer questions I have that I cant answer now. You are helping me a lot

[@WASSIMGAMES]

absolute cinema dude you're goated frrr

[@blizzi1933]

Thank you so much man, love this intuitive explainations, Euler’s eqation really is beutiful

[@alithedazzling]

Glad you enjoyed it!

[@gamingjamari6898]

Wow really love the video and the explanation. Keep it up bro.

[@arclight4600]

I was waiting for this video! Thanks!

[@swizzbeats1212]

Love these videos, takes me back to university :)!

[@gavinscoville5424]

I am so exited for the Fourier Transform explanation. 10^2%

[@thomaskara2395]

I love your videos, you explain it all so well. Thank you!

[@alithedazzling]

Glad you like them!

[@ursulawestenberger3287]

Thank you! You're just amazing! Please continue! 👋

[@alithedazzling]

More to come!

[@OmbarkAmber]

i just want a teacher like u...in my college for every subject i study...from you i got to know about the Imaginary number (j), (e) and e with power (j.theta) and why the hell it represent cos (theta) + jsin(theta)....thanks man! 👏

[@alithedazzling]

Glad you're enjoying the explanation!

[@OmbarkAmber]

@alithedazzling yeah...those understandings are very necessary while studying electrical subjects...u know... electrical signals⚡

[@sarak8576]

As a high schooler I’m genuinely surprised that you had me interested in math! I can’t understand most things ur saying but I really enjoyed listening to you explain with passion and drive. Im so excited to get to uni so I can study engineering and hopefully start understanding what you’re all about 🤍

[@alithedazzling]

That's the spirit! I'm sure you'll be an amazing engineer.

[@Aryanz101_Origial]

I am from 7th grade and I love your videos and how much in a simple way you explain🤍

[@farshaddehqani3502]

Brother you got a gift for teaching. That's rare

[@exo-580]

how many think this channel is so cool? for me it has captured my attention and im really appreciating this man for his intuitive understanding of his audience ,i love it keeep it lit ALI! SO COOL

[@alithedazzling]

Glad you're enjoying the videos!

[@bordercut1]

the way he writes infinity is breathtaking.

[@mouadtouir6515]

very good explanation , thank you Dr Ali

[@CATPLANET24]

Excellent video! Thank you for this information.

[@Yassine-sami-1]

You are providing clarification between the lines keep going please

[@anrdhx]

Great video! You explained everything really well 👏👏👏

[@alithedazzling]

thank you!

[@azeemishaq8240]

I am very excited for the next video

[@MIsam-fv9kb]

Great video thnx for sharing

[@enetniss]

I love these videos! Thank you for taking the time to explain it so clearly. What helped you the most when trying to understand the math for engineering?

[@alithedazzling]

if something doesn't make sense don't just accept it, keep asking why

[@Lumpyspaceranger]

Really beautiful to see the breakdown of it :)

[@alithedazzling]

Thank you!

[@treeross]

Wow, thanks Ali this is literally a free Calc 2 lesson! Also, you should hang your necklace backwards while you record. It keeps jingling against your mic. If it's against your back we (probably) won't hear it.

[@alithedazzling]

Good point, will keep in mind!

[@jaredhouston4223]

I'm always thinking about this identity, in the back of my mind. It's important to our physical world in some way.

[@Satiric_cat]

5:21 You😄 sunvagun... Here, have a thumbs up!

[@PaulAnca-k4f]

An excelent teacher. Please do not stop, up to be PhD also.

[@teberer3246]

Its amazing how powerfull taylor series is

[@AndyKidd-o4x]

Great video. I'm always trying to get my (UK) university students to appreciate math, in engineering, where math is primarily a tool. But no less beautiful

[@gushubbard7645]

A few observations if I may make so bold: Firstly, the reason that mathematicians prefer e^j.pi + 1 = 0, and see amazing, almost mystical wonder in it, rather than e^j.pi = -1 (which is another form of the statement) is because it involves the five most important numbers in all mathematics. 0 and 1 are the basis of our numeration system (see Peano) and that point is lost in this video. Secondly, the transition from a numerical statement involving only constants (e^j.pi + 1 = 0) to a variable function (e^j.theta etc) involving the variable theta is made without any explanation of what is happening. Thirdly, The idea that 0! = 1 is totally counter-intuitive to most learners and needs some explanation. If 3! = 3x2x1 why should 0! =1? The reason is that mathematicians define it to be 1 to make the system consistent. If n Choose r is n!/r!(n-r)! then n Choose n, which of course must be 1, is n!/n!(n-n)! = n!/n!0! and so 0! must be defined as equal to 1. I have no wish to be rude Ali and applaud what you are doing, but teaching an audience which can give feedback during the session as opposed to comments afterwards, is quite different to giving a lecture, and gives insight to where learners get lost.

[@jonhakula8242]

If we make the square root of negative one or i, the imaginary unit, the field that represents the reference frame taking the measurement and place it above the stress energy tensor in Einstein’s field equations (Ruv - 1/2Rguv = 8piG) instead of an expectation value (^) of a probability distribution (from quantum mechanics), doesn’t that solve the measurement problem and quantize the time? The field, which could be called the time, measurement, conscious, or tachyonic field interchangeably is quantized by the reference frame taking the measurement and consciousness and those that have it becomes the metric the universe uses to measure or experience the time. Consciousness is the non-local universe and also the non-local hidden variable or “spooky action at a distance” that is collapsing the wave function of the energy system it is observing. Is this not the only way Einstein’s theories of relativity, quantum field theory based on the standard model of particle physics, evolution by natural selection, and Gödel’s incompleteness theorem can work and be compatible with each other? It is not a theory of everything but a step that has to be taken to understand ourselves better. The act of measurement is the same thing as the universe measuring itself. The tachyon, consciousness, and the imaginary unit all appear to have the same characteristics and relativity becomes the fundamental piece of quantum field theory being the quantization of this field, or time. The reference frame taking the measurement is simultaneously in the past, can anticipate the future, and take measurements in the present to take the time. It is not only there simultaneously but before anything that is moving through spacetime, including photons, making it faster than the speed of light. If you also look at the ‘beginning’ when all of the energy and time in the universe was condensed to a single reference frame, that is where the entanglement of all things happen, and this non-local reference frame (before time) measures itself to create the time. For a tachyon to measure itself or slow down to the speed of light, it would require an infinite amount of energy to do so, creating conditions similar to what we think of as the Big Bang. If this was the case, then everything (all energy) would fall under one wavefunction. Perhaps Euler’s Identity represents a moment in time (e^ipi + 1 =0) that can be derived from Einstein’s field equations through the imaginary unit i. Consciousness would then be something that does not emerge from spacetime but fundamental to it. Gravity becomes the curvature of this particle or energy, which we will call the tachyon, measuring itself and consciousness (the reference frames that have it) becomes the metric that the universe uses to measure or perceive the time. Simply there is no time without consciousness. Even if consciousness was emergent from spacetime, what configuration of particles, made of unconscious ingredients, creates a conscious one? Even if you figured out that a specific configuration of particles creates a conscious experience, how would you be able to verify that? You would have to be the experience itself, no? This relates to Gödel’s incompleteness theorem in which the fundamental truth can only be experienced not described by words or numbers. That which cannot be described by words or numbers is fundamental, it is assumed. Consciousness is fundamental. If consciousness is emergent, how did the universe know to assemble itself together in a specific structure that would inevitably lead to a conscious experience? What is the probability of that? That has to be zero, correct? How can the universe and spacetime work any other way if these theories are correct? The question I have is does this field choose which perspective of itself it wants to experience or is it random?

[@DavidAKZ]

"If consciousness is emergent, how did the universe know to assemble itself together in a specific structure that would inevitably lead to a conscious experience? " - because you are observing it this way.

[@EmmanuelMatabaro]

Amazing! I will check also the sinus and cosinus Taylor formulas

[@King_Baldwin_IV_777]

Amazing explanation mate 👏 🗿.

[@jimmypotekev1804]

super helpful! please do a video on the fourier transform

[@PaulAnca-k4f]

From Colombia. congrats!!

[@mahdizavvar7288]

you are a good teacher

[@ryanwaldt1710]

Great video 👍

[@alithedazzling]

Glad you enjoyed it

[@rpocc]

Will wait for the Fourier video but I noticed that in this video one counter-intuitive thing is explained via other (possibly even more) counter-intuitive thing because Taylor’s series are good for calculating value of functions with some degree of precision but it completely doesn’t explain what ratio of triangle’s sides has to do with adding powers divided by factorials, so that still looks like magic. In other words, there is still question, what comes first: powers of i or rotation of the θ. I think geometry comes first.

[@m_c_8656]

"i"'s curl to the right and "j"s curl to the left

[@AhmedIsmail-z4i]

Please explain the devergence theorem and stokes theorem

[@samarbaisoya608]

Brother love your video❤ Please also explain more about taylor series and about how that sin and cos come 15:04

[@matthewsheeran]

Question I have is how many possible proofs of Euler's equation are there? My engineers gut tells me they will be few and far between and will likely map to and look similar to the Taylor series above, even if they are from out of another branch of mathematics. It aint calculus but there ain't no substitute for intuition as it sure sounds to me like Euler himself just found it just through fiddling and intuition!

[@syn2142]

This is so cool Ali! Do you know if Euler actually gained his insight for his formula by playing with the Taylor series?

[@alithedazzling]

It's impossible to know for sure, but to me this is the only way you can mathematically derive in way that makes intuitive sense -- so I will vote with a strong yes

[@franciscodanconia45]

It was a different Era…

[@JRoadzReegz]

Just chilling in st peterberg checking out some Taylor serifties

[@aniketpachaury]

I understand this video and the previous one too, but why is e used so ubiquitously in other equations where we need an exponential, but are not concerned about oscillatory behaviour or the rate of change of the exponent? For example, in semiconductor physics, we can equate the carrier concentration at one point in space to another point in space by e^(the potential difference / thermal voltage). If carrier concentration is exponentially dependent upon potential, why does the exponent need to be e, and not any other constant like 2 or 3?

[@michaelcolby3632]

Great video 👍🏼, would you ever consider creating a video regarding internships for college engineering students (electrical and mechanical)? Thankyou.

[@theuselessman69420]

no way I wanted to learn how this worked for 6 months and this videou poped up my man really explained it in a way my monkey brain could understand I only had a proplem with how sinus and cosinus taylor extensions other than that even a high school student can somewhat understand

[@246fara]

Thanks for the wonderful videos Still wondering how intuitively connect exponential phenomenon to the periodic one. Some visual and intuitive examples Thanks again

[@biswambarpanda4468]

Long live sir.

[@omaro7015]

Can you do continous Convolution or Fourier Series?

[@Original_Moi]

love the explainations, could you do one on the Taylor Series to begin with?

[@cay1484]

I don't know if here is the correct spot to ask that but I don't know any explatory physics channel so here I am. My question is about electrical potential energy, in my physics book it says "when opposed 'q's gets closer the energy lessens and if they move away the energy increases" well my understanding of this is if you get two opposed particuls which pushes each other and pull them apart more they should push more and more till infinite but in infinite they don't have any force on each other. I thought about in (EP= k. q.-q/d) the - meant the pushing for event instead of pulling but if we give + to pull and - to push isn't that vectorial but electrical potential energy is skaler... That's all I'm not sure I expressed myself correctly but I hope someone has an answer for me

[@scathingpoo2477]

The derivation is cool but doesn’t really explain why exponentials are relating to cyclical in different spaces.

[@EVERYTHING_XYZ1234]

Sir can you pls ans my qn ... If we calculate the divergence of vector r/r³ then it's zero ..but if we draw then we will see that there are outward facing vector fields present and where divergence can't be zero... Pls tell me if you can.. or make a video

[@DuckForPope]

From one Ali to another, thank you so much for these enlightening videos. - Ali the unremarkable

[@alithedazzling]

Haha glad you enjoyed it!

[@quinn-bg3ok]

maybe do fourier series before fourier transform

[@datakprosto123]

I really though that Taylor series was invented by Taylor Swift (since i dont know the guys full name), till you said it was a joke. Thanks for clarifying)

[@abdulazizmisheal8883]

The problem for electrical engineering students they don’t take a course for complex math which lead to a big problems of understanding there courses

[@unes8307]

But complex maths is supposed to be taught in high school around grade 11 and 12, at least this how it's going in Morocco

[@abhishekgrover-pv1ic]

Nice explanation

[@anissbenthami]

Can you do trigonometric functions?

[@eofirdavid]

While this is one of the proofs for Euler identity, it is just a technical way of reaching it, and I don't think it gives the real intuition behind it. The main idea behind the identity, is that you want to convert movement over the circle (which is 1-dimensional) into the position on the (2-dimension) plane. We have very simple ways of modeling these two systems: for the circle we can just measure the angles, and advancing by some angle is done by addition. For the plane, we have the incredible complex numbers, where rotation is done by multiplication (by numbers of length 1). So basically, you are looking for a function that converts addition into multiplication, and in a sense, the only such functions are exponents. By the way, here the basis of the exponent is e^i, but you can choose other bases to get other "rotations" that look more spiral like. In any case, while the Taylor series is an important tool to study such functions, the main interesting property of exponents that make them so important here, is their ability to convert addition to multiplication, namely e^(x+y) = e^x e^y.

[@LucharPS]

Excellent!

[@nishan375]

Great video. Could you explain how the Taylor series was derived.

[@sergek6943]

Wonderful. Thank you

[@azeemishaq8240]

Great job keep it up

[@diegohernandez8104]

could you make a video about taylor series? where do they come from? your explanations are amazing and very easy to understand

[@marcosfernandezsoler268]

Fucking mindblowing. Cant believe it. I love learning math. Thank you for sharing it.

[@alithedazzling]

Glad you liked it!