Calculus 6.11 - Euler's Identity

[@StevenTorrey]

Of all the explanations for a very abstruse topic, this video was one of the clearest.

[@gentlemandude1]

Your humility in the face of Gauss's pronouncement regarding Euler's identity is so humble. It gives me hope.

[@Mrius86]

It's December and I am depressed, drunk, lonely, no girlfriend, no job. Watching this amazes me, takes some of the pain away.

[@kingscross4233]

Are things better now?

[@Viewer2812]

@@kingscross4233 We will never know.

[@matthewc6342]

Thank you. Your presentation was very easy to understand & enjoyable to watch

[@ramsarangan1701]

Wow. This is way better than derivation/proofs of many of the great math youtubers.

[@Tenuki2]

WoW. Nice video. Cool topic. Yet the exact reason why this creator earned my Like was his humility at the very end. My respect to you, sir.

[@comprehensiveboy]

This is a great video. The fact that you admit you can't see intuitively why its correct at the end just increases the impression of being an honest teacher. You actually take the majority of viewers along in clear steps unlike some of the show offs on KZbin who miss out steps or claim intuitiveness when that would be unlikely. You've made my day and reminded me why I like math, like I like boxing, I'll never be a champ but so what.

[@Infinitesap]

I thank you for your work. I would like to call you Little Euler. This is the first time I have seen this explained in about 12 minutes in a so clear and concise manner that I actually understand this. I don't know if you have been observing this but it's the rotation and magnitude of the numbers that makes it rotate from negative one into zero.

[@TonyRecord]

Brilliantly explained, Derek. Truly one of the best presentations on the topic I've seen.

[@zivanayimwaenza8498]

Thanks a million man,the only video that made me understand this concept in all my search for 2 days.

[@1matth3w1]

i love you, your vids are so well made, so clear it just makes me all tingly inside because im picking up the info so fast

[@stm3252]

The best and simplest derivation of Euler's Identity.Thank you Sir.

[@kalvin90210]

This is such a lucid and elegant explanation of Euler's Identity. I am with Derek Owens in not seeing the equation as obvious, as Gauss had once suggested. I wish I saw it's use as obvious however, then making me a first rate mathematician.

[@engman6]

Very nice vid, crystal clear explanation to one of the most complex formula in my life :) Now I really appreciate the beauty of this formula. I didn't before

[@isambo400]

Best explanation I have ever seen. Only basic calculus required, nothing fancy at all. I can actually show it now instead of just trusting people with crazy hair on the internet

[@derekowens]

Glad you liked it! And yes, the crazy hair is not actually helpful. I often think the crazy hair people are trying to make a tenuous connection to Einstein. "See my hair? See that? Don't I look smart? Yeah, crazy hair, so I must be smart!"

[@iV1n5]

Spectacular!!! I never saw the derivation of this formula. Now it finally makes sense! Thanks!!!

[@jacobvandijk6525]

I like the calmness of your explanation(s). Well done. Nice identity!

[@YoZZgAtLiii66]

thank you for the great backround information of all those great persons... and this formula is just a stunnig one which is facsinating myself since the first time i've seen it... kinda got goosebumps while you were talking about it, at the end of the video. greets from the university of technology; kaiserslautern/germany

[@howmathematicianscreatemat9226]

Well Derek, if that isn't the new Mr. making math feel colorfull again! :-) You explanation style feels as clear as water with empathic human interaction. It's joyfull to watch it because you use colors and take the time break your thoughts into parts so that people really see where it comes from. What also feels thrilling almost like the white shark movie is the story about Euler you told people here. It feels so attractive to also start becoming creative in maths because one feels like wanting to participate in Euler's ingenuity. That's because the way you tell it, it makes people paint the picture of Euler writing down his new ideas with total joy for the supreme queen he researches about. It draws one into this wonderfull world full of patterns. I'm actually a colleague of yours but honestly not into software systems. And I feel the desire to help people with my strengths too. So, if you tell me which software you use here, then, as soon as I become successfull with it, I'll write about your support and connect with you, so we can help even more people appreciate our subject ;-)

[@swedishstream]

Very relaxing presentation!

[@williejohnson1669]

How could you have just done such a great lucid succinct derivation and NOT see the beauty of this equation? His formula e to the ix=cos + isin is essentially a complete consolidation of trigonometry. His identity is a complete consolidation and distillation of Hamilton's quaternions. That is e to the i pi=i squared=j squared=k squared=ijk=-1. And no i is not just literally imaginary, i as a quaternion has amazing powers. His identity represents an entire unique branch of mathematics.

[@derekowens]

@borissman i is the square root of negative 1. It is sometimes called the "imaginary unit". In other words, it's a number that is not a real number. Some understanding of "imaginary numbers" or "complex numbers" would be a prerequisite for this video.

[@Bhamilton-ws4go]

This is the best explanation I've seen.

[@derekowens]

Hey, that's interesting. I think the answer to your question lies in the part where you raise e^πi to the power of 2. Raising a complex number to a power actually corresponds to a rotation in the complex plane. Starting at zero, rotating halfway around gets you to -1 on the Real axis. That's (e^πi)^1. Twice that much rotation gets you back around to the 1. So 2πi is in fact equal to 0 in this case, because it is an angle of rotation, and 2π radians is the same angle as zero!

[@georgepapandreou906]

@shakeenbake If am not wrong z = cosθ + isinθ can be derived if we remember that r has gone away because r=1 in the unit cycle, also remeber that with i is symbolized the complex number (0,1) so according to the complex number addition and multiplication we have: cosθ+isinθ =z=(cosθ,0)+(sinθ,0)(0,1) =(cosθ,0)+(sinθ*0-0*1,sinθ*1+0*0)=(cosθ,0)+(0,sinθ)=(cosθ,sinθ)=(1*cosθ,1*sinθ)=(r*cosθ,r*sinθ)=(x,y) where x,y are real numbers and (x,y) a complex number written in the formal form.

[@michaellewis4636]

So Many Metaphors in this equation, it radiates beauty!

[@derekowens]

@shakeenbake Not a dumb question at all! In fact, that's a great question. You can include the constant and make a more general case. In this specific case, we are confining ourselves to the unit circle. The more general case will allow any number in the complex plane. I should make another video showing that...

[@ElizabethRobillard1]

'separate the variables' dude u r a prize, thank u for sharing

[@VeryLazyAngel]

That was amazing, you are awesome. Great job!

[@derekowens]

Right. I'm not saying that 2πI = 0. Rather, I'm saying that e^(2πi) = e^0, because 2π radians of rotation lands you in the same spot as zero radians of rotation. I'll look at the complex logarithm article also - thanks for the pointer.

[@derekowens]

@chris69666 Well, I don't go into the applications in this video. The point of this video was simply to derive the equation. It does have applications, though. It is useful for deriving identities, and solving differential equations, for example. About it being amazing, the fact that a connection between e, pi, i, 1, and 0 can be stated in such a concise and succinct manner is amazing to most people.

[@kendler09]

great video, good expanation! it's truly the most beautiful equation in mathematics

[@solcarzemog5232]

I've looked at many proofs but yours is the BEST, the clearest and the most elegant... do you have more cursus on calculus ? Thanks!

[@imnecessaryevil3879]

+Carlos Gómez-Orellana I don't think that this explanation is the best.

[@solcarzemog5232]

imnecessaryevil Which is then?

[@imnecessaryevil3879]

Carlos Gómez-Orellana *watch?v=-dhHrg-KbJ0*

[@peggyfranzen3741]

I like Professor Euler because his understanding of the measurement of infinity, (calculus), has been awesome. Thank you

[@1hassan1100]

thanks Derek.I almost quit math due to gauss remarks about this formula but you talked me out of it

[@derekowens]

I think I understand your question. The 2πI is the exponent. You are correct that 2πI is simply an imaginary number, along the Im axis. But e^(2πI) is equal to 1. In other words e^(2πI) is the same as e^0, because 2π radians of rotation brings you to the same point as zero radians of rotation.

[@derekowens]

Well, that's a good question, and I don't think I'm inside Euler's head on this one. In fact, I think he actually did his derivation using an infinite series. The formula rightfully bears his name, but the particular derivation presented here is, as far as I know, not his original derivation of the formula.

[@hklausen]

Thanks for the exelent video. Its the only video here on KZbin about Eulers formula where I understand a little :-) Your other videos is good too.

[@Edthebanjo]

Here's an explanation from wikipedia: The exponential function ez can be defined as the limit of (1 + z/N)N, as N approaches infinity, and thus eiπ is the limit of (1 +iπ/N)N. In this animation N takes various increasing values from 1 to 100. The computation of (1 + iπ/N)N is displayed as the combined effect of N repeated multiplications in the complex plane, with the final point being the actual value of (1 +iπ/N)N. It can be seen that as N gets larger (1 +iπ/N)N approaches a limit of −1.

[@sakeekawsar4003]

very useful video..thanx a lot

[@derekowens]

Ha! I see your point. I've been hoping for decades that it would become obvious to me. But I still think studying math is worthwhile. You can still be a great mathematician, even if you are not as great as Gauss. Gauss was perhaps the greatest of all time, and there is only one Gauss. But others can still be great.

[@fortytwo6257]

Great video, the graphical explanation was very helpful!

[@ultravidz]

This video just had it all. Favorited. Subscribed.

[@-IIya]

I mean if e go forever and pi go forever but they so alike each other, its truly beautiful

[@tunicana]

Thank you for this proof Really you are a good teacher . thank you 100000000 of times

[@kolkatarockerz]

Blown away..a real beauty...just subscribed it..thanx a ton.. :)

[@FineFlu]

It means that numbers are complex. Sometimes they are imaginary, and sometimes they are real, but they are always complex. They are infinitely complex/convoluted until you are able to bound or constrain their degrees of complexity(how real a number is or how imaginary it is). This extends to philosophies of life because anything that is it not explicitly real is trivially imaginary. And anything imaginary is trivially real. Ultimately all things are complex. It is why your fear is irrational, it is a fear of something that might exist or might have existed, but does not and is thus ultimately useless(infinite). @derekowens

[@peggyfranzen3741]

Euler, great mathematician... I had no idea he was blind.My God, great man! How did a man,as he was, continue his work?Thank you.

[@uday8792]

He is great

[@greekpotatodanger]

When you integrate at 8:20, why do you not have a constant added to the result? (+c)

[@Bsblob]

generally when you integrate both sides of an equation you don't write the "c" term because they cancel out, its a little bit hand wavy but the math works out.

[@bp56789]

Brosef Stalin No they don't. They could both have different constants, so you should put one constant on one of the sides.

[@michaelwilson457]

Because, in this case c=0. When lnz=iθ+c, if you sub in z=1(=cosθ+isinθ) => θ=0 then ln1=i0+c => 0=0+c => c=0

[@Bsblob]

but wouldn't you have a c on both sides after integrating. so you would have c=c, which is inconclusive.

[@emilioskarwan4599]

C does not cancel out. I think C is not much of an importance here because everything takes place in the Unit Circle.

[@mortenrobinson]

It can also be stated in a different way where the number infinity occurs instead of Euhlers number: (1+i*pi/infinity)^infinity+1=0 Since we can't do infinite iterations this could instead be stated as: Lim((1+i*pi/N)^N, N->infinity)+1=0

[@fromMouq]

Beautiful presentation. Thank you.

[@rpg32tamu]

I am not sure that it is completely 'obvious' as Gauss would say but if you have a pretty good grasp on logarithms it makes sense. Consider ln(x) for x>0. This function contains all real numbers; for x

[@Paleoint]

Truly remarkable. Many thanks!

[@LaureanoLuna]

Real plane and complex plane are two different forms of representation. They are what they are simply by definition, being required only the existence of a bijection between points in the real plane and pairs of numbers, on the one side, and between complex numbers (of the form a+bi) and points in the complex plane, on the other.

[@DARCIOSILVESTRESABBADIN]

great demonstration made me remenber all things to use with my students at college tanks

[@joaquinherreros8083]

first, integration carries border constants, so the explanation is not perfectly rigurous. second, about the intuition of the imaginary number, is dificult to find because the square root of minus one is not intuitive, but is a really cleaver construction when used in this formula: e to the pi times i means a dephase, a change of angle introduced in a linear system, as is commonly used in phasors to represent algebraically different vibrations with the same oscillation period but with different amplitud and phases (widely used in electricity), or more clear in light beams, when you can see changes in direction of light paths given by lenses, that could be represented by e to i times the angle involved in the change of direction of the beam, for example, when concentrating light with a magnifying glass

[@Gorund92]

Joanna Lada You don't need the absolute value when you're working with complex numbers. In fact, any negative number has a log in the Complex Field. Actually, the notion of negative or positive number does not have meaning if you're talking about a complex number, as C isn't an ordinated(?) set. mor on wikipedia's complex logarythm page As for the constant, you can easily assume both of them to be 0 and still not be in error. BTW, loved the video. I only knew a proof for Euler's Formulas starting from e^(i*theta) and involving it's taylor's series. this one is so more elegant.

[@NavalKishoreBarthwal]

what is the significance of taking the derivative...?? @ 5:54

[@SuntzuDragon]

Im not 100% sure but I think it was so that went he took the integral of dz/z he would get ln(z) and could put a base e without any complex explanation.

[@makarlock]

your handwriting is very pleasant, you should make a math font out of it

[@Dyslexic-Artist-Theory-on-Time]

This is an invitation to see an artist theory of the physics of ‘time’ as a physical process. In this theory Euler Identity e^iπ+1=0 is interwoven into the dynamic fabric of space and time of our Universe! With time being formed by the spontaneous absorption and emission of light. Time is an emergent property coming into existence photon by photon. The 1 in Euler Identity represents 1 photon oscillation and the zero represents zero time the moment of now within an individual ref-frame.

[@darkraiandmanaphy]

clearest video on euler's identity!

[@turtle207]

darkraiandmanaphy unclearest and not presize.

[@mataburt]

Great video. Really helped me shake off some cobwebs on my math understanding.

[@fcarriedo]

Beautiful how the formula arises. The one question that I have is why would Euler derivate it just to integrate it again (what would be the motivation for it?), since they are both opposite operations. Similar to adding a number and then subtracting it, you would be left with the original one. I would love to get an intuition on why they don't end up with what you started up with. Just for playing around and see how you can reaccommodate things and see how you end up?

[@timevampire83]

euler is a cycloptic genius!

[@barissannan2731]

finally got it!!! thanks a bunch...

[@modzio131]

Now everything is clear. Thank you very much!

[@idnumber002]

this is cool coz i watched the derivation on khan's channel and they're very different but really cool thanks!!

[@rpg32tamu]

What I also wanted to mention, before I ran out of my 500 characters below, is that from this perspective we can define the imaginary number, i, in a different way instead of the square root of -1. What base logarithm will give y = 1 in the formula below? The answer is e^pi ~ 23.141. So we can say... logb(-1) = i, where b=e^pi~23.141.

[@dekippiesip]

Nice proof! I was familiar with the proof using the Taylor series of e^ix and the proof of this along that road.

[@Dyslexic-Artist-Theory-on-Time]

Do you think Euler Identity e^iπ+1=0 could be interwoven into the dynamic fabric of space and time of our Universe? With time as a physical process formed by the spontaneous absorption and emission of light forming time as an emergent property coming into existence photon by photon? The 1 in Euler Identity represents 1 photon oscillation and the zero represents zero time the moment of now within an individual ref-frame. This is an invitation to see an artist theory of time

[@shuangyuqiao]

Well, after 1 and a half years, I finally realized why it is obvious.

[@franschan]

Great Job, Thank you.

[@asadmajeed7740]

Superb sir its very helpful

[@user-ke3wp7cn1i]

very clear explanation, tq :D

[@euva209]

Nice but upon integrating, there's a constant. However given that the constant leads to z = Ce^(theta*i) and since z = 1 when theta = 0 in the unit circle, then C = 1.

[@1996sagark]

If you use Tau (i.e. 2pi) instead of pie in the equation you actually get equal to 1 instead of -1 that is more something to admire

[@社區青年服務隊]

Euler is very amazing & creative......

[@LaureanoLuna]

I see what you say about rotation, but things don't seem to add up. It seems I cannot write 2πi = 0, for it would follow that i=0, which is false. It also seems I cannot divide 2πi by 2πi, on pain of getting 1=0, as riley shows. Aftrer googling it, I see riley's derivation goes wrong when he assumes ln(e^(2πi)) = 2πi. That's wrong because the complex exponential function doesn't have an inverse function, due to the rotation issue you mention: see complex logarithm at wiki.

[@Jizer0]

Nice, thanks for taking the time.

[@lerubikscubetherubikscube2813]

Thank you sir. Very clear explanation :D.

[@MA-bm9jz]

i did it with the complex logarithm of -1 then used e^x function and did e^(log-1)=e^(ln1+ipi) and so on

[@nandakumarcheiro]

This indicates logarithmic spirality that oscillate between magneticfield and electricfield phase conjugated to produce logarithmic function inversioned from exponential function becoming logarithmic negative and positive functions in producing an amplification of energy by pi dynamics operating in between even in understanding the neutrino spirality out of negative and positive energies. Sankaravelayudhan Nandakumar on behalf of Hubble Telescope Research Unit complementing the Lambert function in between complex number and exponential power function.

[@MaryMiscarriage]

Yeah I lost it at about 4 minutes when you pulled the i, seemingly to me, out of nowhere. And I so hoped that this video would help me understand the stuff I got myself into for my maths project. >.< I hate maths when it's not just making properly explained excersises in the book :'(

[@ziboyang2056]

Ok I might be quite late but can someone explain me how you deal with that dz and d(theta) staying lonely at 8:20. Cause I thought in order to really give you the derivative both have to be very small like zero but he dealt them like they would equal 1...

[@sszushi]

Why is the imaginary axis drawn perpendicular to the real axis? What was the original motive for drawing as such? Who was the first to do it, Euler, Gauss, Argand, or...?

[@TerryGiblin]

Dear Derek, I really enjoyed the video, thank you. Guass was a mathematical genius, so don't be put off by what he said. You are born a 'statistician' or not. - But there is more than one to skin a cat. Fairy's are imaginary, Complex Numbers are NOT. Real Numbers are a subset of Complex Numbers The Real numbers line, and in particular, the point e^πi-1=0 in a complex plane, is the equivalent to, the path of a 'singularity', at the center of a Black Hole, at infinity.

[@aepassion]

excellent video!

[@waabu1]

My hero.

[@jsprite123]

Nice explanation. Can someone provide a practical application of this formula?

[@derekowens]

very nice!

[@MarceloDezem]

Well done video.

[@Bjowolf2]

The more you use it, the more obivous - and easier! - it gets. Do NOT let it scare you ;-) Your mp3 files and mpeg ( divx, mp4, flv .... ) files indirectly make pratical use of stuff like this ( via Fourier (cosine) transforms - time domain to frequency domain (spectrum) ).

[@nybotheveg]

truely amazing

[@fcarriedo]

Awesome videos BTW.

[@JSprayaEntertainment]

did he continue any ancient egyptian maths or work on something like the works of Fibonacci or jnhm and the fibonacci matrix grid of infinity ??

[@MrJohnMarbles]

Thanks, great video!

[@flamingpaper7751]

Wouldn't the integral of dz/z actually be (ln|z| + c)? z could be be positive and would make the absolute value pointless and the + c could be 0, but are there times where it isn't?

[@-IIya]

it's also cool that 2 irratinal nubers gives you a whole nuber

[@georgepapandreou906]

Formal representation and operations are a necessity to understand complex numbers. i^2=-1 is super easy to understand under complex operations, look: i^2=i*i=(0,1)*(0,1)=(0*0-1*1,0*1+1*0)=(-1,0)=(x,y) cause y=0 we can write it as a real: -1 so i^2=-1 for short, STRICTLY under complex multiplication THOUGH! An other case is when we have a complex root in an equation we must evaluate the polynomial STRICTLY under complex operations so it results to zero: (x-root)*...*(x-root)=0

[@MS-cj8uw]

Sir ....Thanks for you .... be sure that you are also significant mathematician ...

[@pithikoulis]

thanks derek! you rule!!