e to the pi i for dummies

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For this Christmas video the Mathologer sets out to explain Euler's identity e to the pi i = -1, the most beautiful identity in math to our clueless friend Homer Simpson. Very challenging to get this right since Homer knows close to no math!
Here are a couple of other nice videos on Euler's identity that you may want to check out:
• Math in the Simpsons: ... (one of our Math in the Simpsons videos)
• e to the pi i, a nontr... (by 3Blue1Brown)
And for those of you who enjoy some mathematical challenges here is your homework assignment on Euler's identity:
1. How much money does Homer have after Pi years if interest is compounded continuously?
2. How much money does Homer have after an imaginary Pi number of years?
3. As we've seen when you let m go to infinity the function (1+x/m)^m turns into the exponential function. In fact, it turns into the infinite series expansion of the exponential function that we used in our previous video. Can you explain why?
4. Can you explain the e to pi i paradox that we've captured in this video on Mathologer 2: • e to the pi i = -1 par... .
If you own Mathematica you can play with this Mathematica notebook that I put together for this video
www.qedcat.com/misc/Mathologer...
Thank you very much to Danil Dmitriev the official Mathologer translator for Russian for his subtitles.
Merry Christmas!

[@zavionw.8052]

i: Be rational! π: Get real! e: Stop fighting, you're gonna make me negative!

[@yassinghareeb5761]

Nicee

[@alexanderying1558]

i ain‘t rational though

[@zavionw.8052]

It is if you really think about it

[@donnacraumer8492]

LOL

[@turquoise770]

Why are they always dividing each other?

[@headshock1111]

I feel like I'm cheating on numberphile but this guy is good

[@krisztian76]

+Sam Parker They should do it first.

[@thelatestartosrs]

+Sam Parker numberphile dont talk about things like that too much

[@MarcelRobitaille]

Lol

[@OneZombieTrain]

cough cough, what got released today

[@rookandpawn]

This is the best explanation i've ever had. His teaching style is perfect for my homer brain

[@MaxDamage1984]

I seriously doubt Homer would be able to understand this.

[@alexandermizzi1095]

Why not

[@bargolyr8660]

​@@alexandermizzi1095 because it would make him very thirsty

[@andrewzhang8512]

@@mmath2318 I recognize this link!

[@maythesciencebewithyou]

All he needs to do is to remove the crayon from his nose

[@JacqueyQuacky]

spider pig

[@porphyrino2897]

"really awful to the power of awful"

[@rasterbate87]

"offal to the power of offal" is what I thought he said

[@kaet8333]

As a non math person that is my reaction when I see something like that in a calculator

[@CarrotSlice]

Thanks I now feel more stupid than Homer.

[@jmiquelmb]

Well, it's not about being stupid or not. You need to know several fields of math before trying to understand this, like complex numbers and some algebra tricks. Don't feel bad about it :)

[@dannygjk]

lol

[@fokkenhotz1]

playboy bunnies love carrots

[@graffitijunkiejfk]

Was just thinking, how would Homer et this when I can't...lol...

[@zandnothinga7367]

Carrot Slice Not feel You are

[@mdnpascual]

I like that quote "Really awful to the power of awful"

[@kaninchengaming-inactive-6529]

(a+b)^2 =a^2+2ac+b^2 (a-b)^2 =a^2-2ab+b^2 (a+b)*(a-b) =a^2-b^2

[@Miftahul_786]

@@kaninchengaming-inactive-6529 someone figured out the difference of 2 squares..

[@AirshipToday]

@@XeNoX_off I think it was a typo it should be a 'b'

[@aliasghargondal3787]

@@kaninchengaming-inactive-6529 In the beginning , there shouldn't be a 2ac , it should be 2ab. (Everyone makes mistakes don't feel bad)

[@aliasghargondal3787]

Sorry if I seemed rude

[@conordorney217]

I remember seeing this video 3 years ago when I really was getting into maths and science. I could never make sense of how this could possibly work. What's nice about this video is that now I'm studying chemical physics and have the maths to fully understand this. Kind of nostalgic looking back on problems that I once was unable to grasp.

[@Dustin314]

I've noticed this same thing with various math topics. As my education has progressed, I find that certain things I used to scratch my head over have become much clearer. It's a great feeling!

[@w451-qx3kx]

facts, things that i never possibly thought i could understand in comp sci are now my everyday formulas and functions.

[@alejrandom6592]

The feeling

[@pkrissz]

We are all getting old folks.

[@davidashbourn6342]

@@Dustin314 what tips can you recommend to someone trying to get to your level?

[@meh5082]

“This guide can be understood by anyone that knows how to do addition, subtraction, multiplication and division!” *Uses exponents, graphs, imaginary numbers, funcions...*

[@eduardoandrescastilloperer4810]

NotValik he eventually arrives there but the function just uses those operations. In fact that’s what calculators do

[@nspoly]

question, do you use grammarly? cuz the last wrd is messed up

[@neurofiedyamato8763]

Well he uses addition, multiplication and division to explain these concepts.

[@PotatoSofi]

Well... I don't reeeeally know imaginary numbers, but I know functions and exponentials, so I can uderstand everything very well. BUT you're right. Someone that just knows the four basic operations wouldn't even understand the point of the first part of the video when he explains the 'e'. If you really want to SHOW what is e^(pi*i), it's somewhat easy. You could cut a lot from his explanation and turn it in a more "childish language", but he tried to explain in a way that he talks about everything, trying to put every piece of the puzzle in place and forming that more complete image of parts of the operations. The problem is: kids (and hommer) doesn't like a 500 pieces puzzles, they don't need a 720p image to "understand" something. Just give them a 40 pieces puzzle and draw some stick figures in Paint and they will be very happy.

[@hassanakhtar7874]

Sorry to be that guy but exponents are repeated multiplication (for natural number exponents)

[@walterrussell9798]

If Euler's spirit were around, he would be so very, very pleased with this explanation!! Euler was known for simplified, and many, theoretical explanations for any given math "puzzle" (like e). He was not arrogant, his explanations were not configured to be hidden or difficult. He wanted everyone to enjoy/understand/be in awe of a given math puzzle - what this explanation does for this viewer. THANK YOU!!

[@findystonerush9339]

lol

[@redandblue1013]

@@findystonerush9339 three years has this comment endured without reply, and all you can say is “lol”

[@PC_Simo]

@@redandblue1013 Struck me as odd, too. 🤔

[@-originalLemon-]

This isn't even simple, this is too hard on my brain.

[@maybehelper]

​@@-originalLemon-samw

[@JamesAda]

You would have lost Homer @ 0:11 after you said "Pie"

[@Mathologer]

:)

[@dinitroacetylen]

Mmm... Pie!

[@ultrio325]

.esrever ni epyt I

[@johnthomas612]

James Ada Has

[@WG-tt6hk]

He lost me too.

[@arttukettunen5757]

Different school subjects' levels taught by youtubers: Anything else: middle school Math: *University*

[@shayanmoosavi9139]

And that TRIGGERS me. #StopDiscriminationAgainstOtherTopics

[@shayanmoosavi9139]

@Floofy shibe yeah, advanced physics like quantum mechanics and general relativity are math heavy. Some of the math include differential equations, single and multi variable calculus, linear algebra, tensors and other things I'm not aware of yet. All of these subjects are fun, specially differential equations.

[@emilmullerv3519]

@*Floofy shibe* not really, omce you get into advanced maths and advanced physics you see the differences in techniques and thing being taught. It is true that they are deeply related, but the interest of a maths phd are very different than those from a physics phd

[@keepinmahprivacy9754]

@*Floofy shibe* Math is a language, so if you think of a Math degree like a Linguistics degree, then Physics would be more like a Literature degree. One studies how language works, while the other studies something else that is expressed in language.

[@maoad_dib]

@@keepinmahprivacy9754 you just earned a poetry degree

[@alejrandom6592]

It took me more than a semester of teaching myself calculus, geometry, trigonometry and algebra but I finally understood a "for dummies" mathologer video :D

[@arminyoung1169]

bruh

[@rosefeltch6313]

e^πi = i²

[@footlover9416]

@Eric Lee you know what he meant

[@footlover9416]

no I mean since the video is e ^ (pi x i) you could assume that u multiply before you exponentiate for this comment@Eric Lee

[@footlover9416]

ur right I didn't see he had the 2 I was going to give him the benefit of the doubt and say that maybe he copy and pasted the 2 and couldn't find anything else but this dumb as you can change this easily I agree@Eric Lee

[@rosefeltch6313]

The carat (^) is a symbol for exponentation, and e^πi is -1, so is i²

[@soup4001]

@@rosefeltch6313 they are talking about how it should be e^(pi*i) instead of e^pi*i because the latter would be equal to i*e^pi =/= -1 ... As long as we get the point i don't think semantics matter though..

[@MrHandsomeStudios]

When m goes to infinity, it comes closer and closer to a Pokemon ball

[@davidramadeen8129]

@@gcb642 Because Homer say m cannot be greater than your Pi.

[@dymo7607]

it only goes half of circle apparently

[@soulswordobrigadosegostar]

Now homer gets it

[@donnacraumer8492]

LOL

[@camkiranratna]

Dæm bro didn’t notice that

[@math2693]

I never never ever thought that I might see as strong visual proof for this amazing formula as this guy's👏🏻👏🏻👏🏻 this is so 100% enough

[@ksbvddz229]

I am 14 and understood this. These videos are the types that remind me why I love math even tho my teacher is pretty bad since he spends the whole hour arguing with kids about eating in class and not teaching us. I love this video. Thank you for it :D

[@YilmazDurmaz]

after so many years in my engineering life, first time to see what it means to multiply two complex number in graphical explanation. thanks.

[@maythesciencebewithyou]

@oynozan Sen simdilik sadece lise desin. Bu lise konusu degil. Yilmaz mühendiz.

[@maythesciencebewithyou]

@oynozan ilginc. Türkiyede lisede karmasik sayilari ögretiklerini bilmiyordum. Teknik lisesidemi oluyor, yoksa her lisede konumu?

[@shantanusolanki3794]

It is easy to infer from the Euler's formula, e^(ix). And so, the demonstration here, is assuming what we have to prove. Well, not exactly, what we have to prove but a particular value of what we have to find.

[@hikarunakamura5582]

@@maythesciencebewithyou karmaşık sayılar standart müfredatta vardır ancak karmaşık düzlem üzerinden anlatılmaz. sadece i nin kuvvetleri verilir ( i^2=-1 tarzı bilgiler)

[@GyanPratapSingh]

this is probably the best, most intuitive explanation I have seen of Euller's identity. Really well done!

[@Mathologer]

+Gyan Pratap Singh (GyanPS) Glad you like this explanation and thank you very much for saying so :) Maybe also check out the two videos in the description (just in case you have not seen them yet). There are quite a few more nice points to be made about all this.

[@prasadjayanti]

I am a Ph.D physicist and working as a data scientist but never knew this simple intro to e. Thanks .

[@stickmcskunky4345]

Fantastic. I didn't quite grasp complex multiplication and I definitely didn't understand e^πi but this gives me a *solid* starting point. Much appreciated even 7 years later.

[@DDranks]

This was the best explanation for e^(pi i) = -1 I've ever seen, hands down.

[@mrmeowtv6248]

This video said "for dummies" so I'm here.

[@dr2353]

yeah me too

[@leofranklin84]

Absolutely brilliant explanation!!! I don't think even Euler could have put it as good as this. U even got the perfect Tshirt for this

[@MsAlarman]

The progress of computer graphics

[@samlawhorn]

Mathologer, you're one of the greatest explainers of math of all time. Loved this one!

[@VelexiaOmbra]

The end bit where we increase m toward infinity was beautiful.

[@Phoniv]

+Velexia Ombra great comment

[@ashrasmun1]

+Defendor mediocre comment

[@Zerberusse777]

shitty comment

[@U014B]

+Zerberusse777 f(comment) = lim- [x -> 0] 1/x

[@Phoniv]

funny,

[@ENCHANTMEN_]

14:20 Observe how as M approaches infinity, the endpoint of the series approaches OOH LOOK A POKEBALL

[@robinfrenzy]

so (1 + i π/33)^33 = pokeball ?

[@Kyrbi0]

Off-topic but HEY A PREQUEL FAN. Noice username & avatar. : )

[@gorgono1]

this formula is the secret of creating a pokeball

[@MilesQuickster]

Pokémon is just a bunch of Maths. Migrating from regions is a computed simulation

[@theinquiringminded6952]

Magnificent seeing it again so many years later and understanding more… bravo! What a great teacher!

[@madmaximus2836]

The first thing my Physics 1 professor said during his introductory lecture (after he greeted us) was: The universe is described by 3 numbers: 2 are irrational and the 3rd does not exist. He was referring to pi, e and i. Rest in peace, Prof Strauss ... Thanks for your cool lecture of the 3 numbers that describes the universe. This brings back so much nostalgia even after more than 30 years. It is Saturday night and I watched this video. Guess I'm still a nerd and loving it!

[@indigogolf3051]

Thank you. I watched this purely for nostalgic reasons. I forgot how much I enjoyed studying mathematics forty years ago.

[@Mathologer]

Glad this one worked for you :)

[@z3lop59]

kirvesmies can you still do it?

[@indigogolf3051]

Not a chance, unfortunately.

[@z3lop59]

kirvesmies thats sad. im thinking about studying physics where math is very present as well as you know

[@z3lop59]

One Thou Wou didnt get what you mean

[@yanwo2359]

EXCELLENT! WONDERFUL! Perfect! I have run out of superlatives. No distracting, annoying, background "music." No superfluous sound effects. No constant, senseless, movement of the presenter all over the frame. No distracting, constant, hand and arm motion. Excellent presentation. Excellent graphics. Excellent explanation. Excellent diction. Other math(s) and science videos on KZbin pale in comparison. Thank you very much! Jon

[@Mathologer]

+Yan Wo Comments like this make my day. Thank you very much :)

[@turun_ambartanen]

+Yan Wo you ran out of superlatives? you never started XD

[@markhollingsworth1725]

Agree. The presentation and production style is excellent.

[@itscky2007]

Explaining with the graph just makes it so much better to understand, thanks for the explaination!

[@colinholloway8076]

I got some of it. That will do for now. Love this explanation and REALLY like the graphics. So good to add some concrete explanation to the abstractions.

[@PotatoMcWhiskey]

I dont understand what went on in this video beyond a certain point, but at very least my awareness of my mathmatical ignorance is expanded

[@Mathologer]

+PotatoMcWhiskey Maybe just watch it a couple of times and maybe do some background exploring in between (reading up/YouTubing up on the basics of complex numbers might help). Also there are a couple of other videos on e to the pi i that are worth checking out and that explore different approaches. I've linked two of them in in the description :) Anyway keep watching videos by us and some of the other great math channels out there and I am sure you'll get there :)

[@aigen-journey]

+Mathologer I think the part about complex numbers might need a bit more explanation. If someone has never stumbled upon complex numbers the bit about multiplying them on a 2D plane is probably very confusing. Especially since one would usually associate a two axis plane with functions for example.

[@mohasandras]

This video is so awesome, I bursted out in tears of joy.

[@GenericInternetter]

Huh? Why?

[@damienw4958]

+Generic Internetter he uses simplicity to explain something so complicated that your brain would explode into a thousand pieces if you understand only a tenth of it!

[@DaysDX]

+Generic Internetter because understanding something that is already so beautiful and important in a brand new way is really really awesome!

[@AreYouGettingThis]

+mohasandras I was actually grasping onto the edge of my seat towards the end :D

[@iAmTheSquidThing]

I vomited with amazement.

[@dylan7476]

Wow that was really well explained. I have a few maths exams next week and I could actually understand and follow along with everything you said which is reassuring.

[@lolaharwood619]

Using the approximations and taking their limits to infinity visually has to be the absolute best way I've ever seen this proven! Thankyou, i understand it now much more than before

[@bonecanoe86]

So basically e gets you a formula, i puts that formula on a circle, and pi sends you halfway around that circle to -1. Is that the gist of it?

[@elementalsheep2672]

Yes, but the circle that the formula puts it on is complex, as in it exists in both the real and imaginary plane.

[@riccardopuca9310]

bonecanoe86 very interesting take on the explanation. I'd say the "circle" is already in e's formula (it's in dividing by "m", which gets you closer and closer to 1), but it is in one dimension only. What "i" does is putting it in 2 dimension, allowing us to see the "circle" we're used to (I.e., a bi-dimensional circle). In my opinion, this connection between e and the circle (and thus with pi) is all the more interesting as it is so intrinsic and unavoidable.

[@ddogworld4]

Riccardo Puca not even close

[@mrsecify]

Basically if you find the Taylor series of exponential function, cosine and sine function you'll see the connection and why e^i*x = cos(x) + i*sin(x), or in special case where x = pi e^i*pi = -1

[@MarkMcDaniel]

All mathematical constants can be expressed in formulae, pi included. functions.wolfram.com/Constants/Pi/09/

[@Mathologer]

For this Christmas video we set out to explain Euler's identity e to the pi i = -1, the most beautiful identity in math again, but this time to our clueless friend Homer Simpson. Very challenging to get this right since Homer knows close to no math! Here are a couple of other nice videos on Euler's identity that you may also want to check out: kzbin.info/www/bejne/j5qWk4djbZeCa9U (one of our Math in the Simpsons videos) kzbin.info/www/bejne/fJCTqpmsopWIpbc (by 3Blue1Brown) And for those of you who enjoy some mathematical challenges here is your homework assignment on Euler's identity: 1. How much money does Homer have after Pi years if interest is compounded continuously? 2. How much money does Homer have after an imaginary Pi number of years? 3. As we've seen when you let m go to infinity the function (1+x/m)^m turns into the exponential function. In fact, it turns into the infinite series expansion of the exponential function that we used in our previous video. Can you explain why? 4. Can you explain the e to pi i paradox that we've captured in this video on Mathologer 2: kzbin.info/www/bejne/iamYkIR9maugp5Y. Merry Christmas!

[@fahadAKAme]

+Mathologer how do you know by how much you stretch the triangles?

[@Mathologer]

+fahadAKAme One of the triangles stays fixed and then you align and stretch the other triangle with the fixed on as shown in the video until the touching sides are the same length :)

[@fahadAKAme]

Mathologer so for every x increase in length of side a,b(the side adjacent to the fixed triangle) there is an equal increase in the length of the other two sides? or is the increase in length is proportional?

[@Mathologer]

+fahadAKAme Yes, all sides are scaled by the same factor resulting in a triangle similar (in the mathematical sense) to the one you started with.

[@alexharkler]

So if M=(pi)xi, then doesn't x have to approach negative infinity on the imaginary number line? When M=(pi)x, m approaches infinity as x approaches infinity, but when M=(pi)xi, m approaches infinity as x approaches -i(infinity). So does this mean that -xi and x both approach the same number as x approaches infinity? (Please correct me if I'm wrong, I haven't taken a math course in 6 years)

[@evansiegel1732]

Brilliant! I always just thought this was somehow a mystical union of pi, e, and i. But now it all comes together. This is really what math is about!

[@kwxutube]

Thank you, indeed the best explanation I've seen ever, connects everything and demystify the formula!

[@LiborTinka]

I want to live in a world where this has more views than a hip hop song about some guy's new Lamborghini.

[@mishadoomen8372]

Libor Tinka you think numbers are more interesting than letters?

[@LiborTinka]

Misha Doomen I just found the relationships explained by such a powerful formula much more intellectually pleasing than an empty song of basically three words dominated by "yo" and "bro"...

[@blovlh6258]

Libor Tinka But they aren't dominated by yo and bro... Still young people mainstream though so it works.

[@mishadoomen8372]

Libor Tinka but that's not true actually. If you would watch a video from KSI about his lyrics on his song 'lamborghini', you would see that he did put a lot of effort into his lyrics.

[@LiborTinka]

Misha Doomen You're right - I am not completely honest with the comparison. I just wanted to express sadness over societal values, where an overpriced transportation device used as status symbol matters more than intellect and knowledge.

[@BryanLawlor]

Brilliant! Watching the magic at 14:20 changed my life. I have always been mystified by this identity but never understood how you get it. Thank you so much!

[@JD-fi4nk]

Wonderful video. Thank you for helping to put this mystery in layman's terms.

[@isaacwolford]

You did an absolutely beautiful job of explaining this in a simple and beautiful manner! This is the way math should be taught.

[@Mathologer]

Glad it was helpful!

[@vmaier3000]

This is extremely beautiful! I’m an engineer who loves math but it was many years after graduating college that I came back to math to go beyond formulae and try to ‘internalize’ stuff that I knew by heart! Thank you!

[@haikal9329]

“For ddUMmIeS” Me: “huh, you underestimated my power” *15 minutes later Me:...

[@chekovcall2286]

(: IKR? At least he gets credit for my repeated views.

[@JMZReview]

@@chekovcall2286 WHY DID U WRITE YOUR SMILY FACE BACKWARDS its :) not (: *I HAVE OCD*

[@joaoneto6216]

@@JMZReview :j

[@alwayswinning7282]

@@JMZReview (: /: [ :

[@JMZReview]

@@alwayswinning7282 i want to die

[@Christina.Anne.]

I’ve been studying Euler‘s formula in addition to Euler‘s identity, and this was so incredibly helpful! Thank you so much!

[@andreasandreotti4492]

Thank you sir, it could not be more pedagogical presented. I like mathematics because it has always given me peace of mind. I often try to I solve problems on algebra, geometry and trigonometry in order to get away of everyday concerns. The fascination of it is to reduce a complex problem into the four main calculation forms as you wisely pointed out. You call them tricks, I call it maneuvering, or favorable manipulation. Being a Greek I always have the Aristotelian logical categories in mind that help a lot in mathematical thinking.

[@xakkep9000]

When you stretched triangles on the complex plane, I say: "WoW!!!! It's awesome!" I had never ever seen that math like THAT. Beautiful!

[@IoEstasCedonta]

Huh. I've got a couple degrees, but I never actually was taught the triangle trick.

[@WattSekunde]

Wow. Me too!

[@NoConsequenc3]

The worst thing is that I've always played with shapes on graphs to do things but my teachers would get mad at me and tell me to do it with numbers... yet this guy uses the shapes and I FINALLY UNDERSTAND WHAT IS GOING ON and it pisses me off that I was told this was not "the way"

[@davidwright8432]

Your teachers were cautioning you so strongly because there are many occasions in which diagrams can be seriously misleading. On the other hand, when they work, they do so beautifully and all is clear! Being able to spot which situation you have - informative or misleading - comes only with experience. and, of course error; but the only way to detect and correct the error, is by using symbolic (algebraic) arguments, not diagrams. so we're all cautioned never, ever, to use diagrams - or, also as kids, get in a stranger's car. Either can lead to unhappiness!

[@PR-qe1zn]

I think the problem is that unless you're using a graphical program, you still end up multiplying the other sides of the triangle by some scale factor in order to scale the triangle, and it ends up being the same work basically.

[@lukapopovic5802]

IoEstasCedonta Does anyone know why the triangle trick works ?

[@narrotibi]

I watched this 2015 and a couple of times in between and right now. One of the best videos ever!

[@wailingalen]

I’m not a chem or math major or profession but I’ve always been fascinated with visualizations on math. I love this stuff. Sometimes I have to watch it a few times to wrap my head around it. But this guy does a good job of breaking it down

[@wailingalen]

I just watched this again and just had an ah ha moment and really understood it more

[@whreREtjk4ko]

Beautiful explanation, far clearer than the usual "stretching and rotating numberlines" explanation.

[@monolyth421]

e^(πi) = cos(π) + i*sin(π) = -1

[@JatinSanghvi1]

That's right. But now you need to explain why the equality e^(πi) = cos(π) + i*sin(π) holds true to Homer.

[@monolyth421]

That's how I understood it at first because I took Euler's formula without question. The video's explanation is much better.

[@adamolesiak6528]

Well, e^(πi) = z => z= x + y*i, now just draw the z on the Re/Im axis and draw the connection between z and (0,0) (r). Now some simple trygonometry and we get cos(fi)=x/r ^ sin(fi)=y/r for all z points except from (0,0). So now we got z=e^(πi)=r(cos(fi)+i*sin(fi).

[@JatinSanghvi1]

@Adam, A point z in complex plane can be represented either in terms of its real and imaginary parts (x + iy) or in terms of its magnitude and phase [r (cos ɸ + i sin ɸ)]. I could not understand how this is relevant to the discussion. Anyway, z can also be represented using Euler's formula, z = r e^(iɸ)

[@monolyth421]

It would be easier to remove the crayon from Homer's brain, then he could understand anything!

[@GiI11]

This blew my mind on so many levels. It made higher arithmetic finally become relevant. Thanks!

[@hamidkh5488]

Thank you very much professor. I believe this is the best way to show how this formula works.

[@chrisgreen_1729]

Very nicely explained! I've always found it difficult to understand Euler's identity intuitively - amazing that Carl Friedrich Gauss said that "immediately understanding" Euler's identity was a benchmark pursuant to becoming a first-class mathematician.

[@dannyspeagle10]

Thank you, MATHOLOGER... This video is perhaps the most brilliantly simple explanation of a seemingly impossibly hard topic. It's not perfect, but it is damn amazing.

[@CameronCajun]

This was super fascinating. Best explanation I've come across. Very succinct.

[@MrFlaviojosefus]

Absolut fantastic!!! This is how a great teacher of Mathematics should speak. Even if I am doing something else while hearing to this video I've got impressed by it.

[@Dhirallin]

You lost Homer at 0:50

[@UnchainedEruption]

lol XD

[@toka266]

No at 0:00

[@irrelevant_noob]

Rajie Music indeed, and i would like to add a bit more : especially since he misstated those *_BASIC_* facts about i : 0:49 "square of minus 1", skipping over the "root" bit, and 0:51 "i squared is 1", skipping over the "minus". -.-

[@robertodelier9999]

Pi: "GET REAL" i: "BE RATIONAL" Me: *no comment*

[@ZerDoxXie]

roberto delier you just commented therefore you lied

[@robertodelier9999]

@@ZerDoxXie *gasp*

[@ZerDoxXie]

@@robertodelier9999 :O IMPOSSIBRUUUUU

[@ZerDoxXie]

@@robertodelier9999 xDDD

[@heisenberg4703]

e: join me and together we'll get-1

[@skoff7628]

I really want to thank you for the great videos. I am a huge fan! As a business student I was really happy to see you do a video covering continous compounding. I would be more than happy if you could do a video solely covering this topic in more detail. Sadly I never got this explained, it was always just a given which one has to except. A thing I cant handle really well in mathematics. Das wäre mir eine große Hilfe und glaube ich für sehr viele Zuschauer interessant. Nochmals herzlichen Dank! Ich wünsche eine besinnliche Weihnachtszeit :) LG

[@physicself]

Another great video. I especially like the geometric explanation of complex number multiplication! Always did it algebraically, but this similar triangles way is much clearer!!

[@MarufSajjad]

This gotta be the greatest explanation for this equation, hats off

[@MayankRamnani1802]

This is by far the best explanation I have seen, and that unit circle visualization was very good! You earned a subscriber.

[@pinklady7184]

Last year, I didn't understand e^pii. Now I do, completely. Thank you for this tutorial.

[@pedrosanchezpalma4443]

This explanation is the golden glue for mixing intuition and maths with regards to the e^pi*i=-1 formula. Many Many thanks for it. This glue will be there for ever.

[@h4ck3r211]

I saw this video a couple of years ago and I didn’t understand, I just realized how good it is, it would be very nice that you explained why the complex numbers multiplication has that geometric interpretation, but thank you :)

[@awfuldynne]

You can write a complex number a + bi as r(cos θ + i sin θ), where r = sqrt(a^2 + b^2)-the distance between the complex number's coordinate (a, b) and the origin (0, 0)-and θ is the origin angle of the triangle with vertices (a, b), (0, 0) and (1, 0) (same triangle as the video). θ = Arctan b/a for complex numbers with a positive real part. Add π or 180° if a is negative. (you don't need to know these functions; I'm mostly listing them as "this is how you translate complex numbers from 'Cartesian' grid coordinates to 'polar' circle coordinates.") For reasons I'm not sure how to explain, cos θ + i sin θ = e^(iθ), so when you multiply two complex numbers together, their angles add together. (the explanation I got involved Taylor series, which requires calculus). Adding the angles is represented by the rotation of the triangle needed to line its base up with the side of the other triangle opposite (1, 0). Multiplying the lengths together is represented by stretching the triangle, and it works because the triangle base had a length of 1 before we started stretching, so if the triangle had side lengths 1 and z, and we stretched the whole triangle so the 1 became a w, the z would have to become a zw or we'd have a distorted triangle. (I can't always tell which bits are easy or hard, so feedback is useful if I skipped over something I needed to explain) Supplementary notes: θ is a Greek letter called "theta". It's commonly used as a variable representing angles. Trigonometry: Picture the unit circle on a coordinate grid. Starting at (1, 0), travel θ units counterclockwise along the unit circle. You are now at (cos θ, sin θ), where θ is an angle measured in radians. tan θ = sin θ/cos θ. Arctan x is also known as tan⁻¹ x; that's an inverse, not a reciprocal. To keep Arctan x as a single-valued function, its range is limited to angle outputs in the right half of the unit circle, which is why I said to add π "if a is negative", which describes angles on the left side of the unit circle. (x, or tan θ, is undefined for θ = ±π/2, since a = cos θ = 0 for those values that are neither left nor right) exponents: e^x * e^y = e^(x + y). The Taylor series for e^ix can be separated into a real polynomial with even exponents and an imaginary polynomial with odd exponents. The former is the Taylor series expansion for cos x. The latter is the expansion for sin x. So e^ix = cos x + i sin x

[@pyramidteam9961]

@@awfuldynne Good explanation, I don't think Homer would know what you're talking about though

[@eirdonne_]

@@awfuldynne that went waaaay over my head as a 13 year old. ...what... what language do you speak of?

[@awfuldynne]

@@eirdonne_ I think my main point was to say, "For one of the standard ways to express complex numbers, 'rotate and scale' is a natural way to geometrically interpret multiplication", as an explanation, but I tried to explain the explanation which still needed an explanation because everything requires _some_ background knowledge and then I lose my point amid the rabbit trails. To be fair, that first paragraph or section is _how to convert_ a+bi into Re^iθ, making it harder to follow.

[@graemsheppard5846]

"It's real magic happening about to happen. Ready to go for magic?" My favourite part of the video :) 14:18

[@augumentum]

That was explained very well with the unit circle and the x I coordinates with the formula. I was able to visualize the process with the triangles. Thanks

[@neurofiedyamato8763]

This was amazing, you really explained this very well and simple. Yet also entertaining.

[@Darkness2179]

Genius. Tyvm for this explanation. It's beautiful, really.

[@NathanTAK]

"for someone who can only do +-*/" "Reminder: i = sqrt(-1)" Waaaait...

[@Roescoe]

how about "remember that i*i = -1"

[@adlamis]

I prefer to treat i as a sign, like the negative sign -. In this format the formula is "e to the i pi equals negative one", always remembering that it's "i pi with a little i".

[@amritpatel3794]

Love it. Absolutely genious way explained

[@andrewrobertson444]

You are a pedagogical genius. I literally laughed out loud when I saw that triangle multiplication. Brilliant.

[@dr.drakeramoray789]

the only thing i understood is that im more stupid than homer :(

[@General12th]

You're more stupider than homer. :)

[@lukapopovic5802]

Seymore Butts Precednice nismo znali da volite matematiku

[@Shibzzeg]

Just return to this video in a few days - it'll be easier to understand

[@dr.drakeramoray789]

moram, zajebase me zadnji put kad sam prodavao rakiju

[@Oraclethingy]

Have a good night's sleep and come back! Einstein liked to think of a problem before he went to bed, and it helped him think of a solution in the morning.

[@mwerensteijn]

Thank you, greatly visualized and explained!

[@0xaronfan]

I think this is the most easily understood explanation, thank you.

[@floridaman6982]

I never understood e so well until i watched the first 2 minutes of this. Thank you

[@adiramrakhani]

This was brilliant- Homer may have not followed along but this surely was great and really useful for me

[@Anders01]

Really good explanations! I especially like the bank to e analogy and that multiplying complex vectors of length 1 lead to only rotation.

[@roundbunny]

I majored in math and i still found the video so useful!! Thank you for your great explanation ❤❤

[@CalculatedRiskAK]

This explanation is brilliant. It gave me a great visual intuition for what e is and how e^(i*pi) = -1.

[@DerGully]

I love how your arm becomes transparent whenever it's in front of the math you explain. Pretty cool skill, I want that too.

[@atulanand9292]

His figure has been edited to be behind the presentation and the opacity if the presentation has been kept high.

[@saurabhkurichh]

A lot of efforts goes into creating such videos .. thanks a lot once again for all the efforts

[@kraevme]

Thank you for subtitles, this is a very interesting math video

[@johannsebastianbach3411]

Your video was so great that I subbed you immediately !!

[@lucasm4299]

How are you Bach? I didn't know you liked calculus?

[@theresamay4280]

+Lucas M Well, he would have been about 19 when Newton fully published and explained his notation for calculus, so it's possible.

[@vinay0429]

+Lucas M i like lamp

[@stephencarlin6273]

Same

[@einstien311]

Pokedex entry 132, brah.

[@mkb6418]

Genious explanation.

[@drziggyabdelmalak1439]

Wowee!!! You, or anyone , will not probably read this now as the years have passed. Thank you for that, Mathologer. Brilliant and clearly explained.

[@yashagnihotri6901]

I just loved your intuition of the complex numbers ! Truly fascinated . The strecth -multiplication , and introduction of a constant "m" in place of "pi" was really a new and brilliant idea for me . I am eager to know that did you just figured it out all by yourself? Please let me know about the historical background of the astounding explanation that you just gave in the video.

[@chocolatecrud]

I finally get it after like 3 hours lol

[@Mathologer]

I'd say 3 hours well spent :)

[@chocolatecrud]

Mathologer Absolutely, helped me understand intuitively

[@tipeg8841]

chocolatecrud prout

[@010falcon]

This actually makes sense Studying physics and never understood how the concept of e to the power of i worked

[@MrVoayer]

What a lovely explanation ! Great job ! Well done !

[@jonathonfrankel5338]

I was in your class last year! I randomly found you on KZbin :D

[@Mathologer]

MTH1030?

[@jonathonfrankel5338]

Mathologer Yes that was it. I wanted to do the class the beauty of mathematics in nature too

[@Mathologer]

That other unit runs in first semester in 2017. Meybe I'll see you there :)

[@timetraveler1203]

I am in your MTH1030 class. Love your lectures so much. Really make me think about maths in a new way.

[@adammiller6299]

@@timetraveler1203Where does he teach?

[@ronald7821]

Wow, you are probably the best mathematician/math teacher I've ever come across. You kept me intrigued during the whole entire video and the explanations are so clear. Love your videos!! :D Just out of curiosity, how did you make those background animations, they are awesome!

[@Mathologer]

Glad you like the videos. I make the animations using various pieces of software: Mathematica (e.g. for the red spirally pictures), Apple Keynote (for the overall slideshow), Adobe Premier, Illustrator and Photoshop for all sorts of detailed stuff :)

[@ronald7821]

wow the editing must be a lot of pain. thanks for the amazing work :)

[@brochestedbs]

Thank you. A superb explanation.

[@bmurali5128]

Wonderful visual!

[@mhelvens]

Nice video. But the bit starting at 5:57, where you suddenly have _m_ instead of _nπ_ because "that will also get you there" feels a bit hand-wavy. Maybe clearer if you explain you're going to increase _n_ in steps of _1/π._

[@Anon54387]

Hand wavy? What in the world does that mean?

[@irrelevant_noob]

Anon54387 means a figurative waving of hand(s) in that superb style that Obi-Wan exemplified with his "these aren't the droids you're looking for". ;-)

[@jye-mings.2476]

Michiel Helvensteijn you get to infinity both ways so it doesnt matter

[@irrelevant_noob]

Jye-Ming Serres yes, that's basically what Mathologer said in the video. To make things a bit more formal, i think the explanation would rely on the idea of limit(s). And if one knows how a limit works, it basically says that for a given context there's a point (a natural number) after which something interesting happens (in this case, the result gets closer to the value of the limit). And now hopefully it's clearer why an always-increasing (and necessarily unbounded!) sequence of irrational numbers works just as well as the sequence of natural numbers: if there's an irrational point past which the property holds, then there's a natural number for which that same property still holds. QED.

[@jerryweatherby1171]

Pirate's Piggy I also felt like the substitution went unexplained (hand-wavey.)