e to the pi i for dummies

[@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?

[@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

[@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

[@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?

[@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.

[@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..

[@MilanMilan0000]

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

[@mattsoutback59]

"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

[@Tysm_for_1k_subs]

​@@rasterbate87 "reply to the reply"

[@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

[@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]

@Tom Petitdidier 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

[@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.

[@krissp8712]

@Hassan Akhtar I guess that's kind of the point of the video. If you redefine exponential function as that (1+1/m)^m function you get closer to not needing exponential functions at all, you just do the multiplication thing instead. 3blue1brown takes that as far as the series (x^m/m!) And then you don't need to know how the exponential behaves at all, you just do the repeated multiplication thing and add up the terms of the series. In my opinion it's even easier with x^m because the whole numbers there make it simpler than Mathologer's multiply 1.001^m approach.

[@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

[@playingadofai]

bruh

[@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.

[@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)

[@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.

[@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

[@DDranks]

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

[@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)

[@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.

[@ThisWasEpicness]

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

[@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

[@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 ?

[@Khyrberos]

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

[@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.

[@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

[@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!

[@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/

[@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!

[@prasadjayanti]

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

[@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.

[@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!

[@stickfiftyfive]

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.

[@samlawhorn]

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

[@mrmeowtv6248]

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

[@dr2353]

yeah me too

[@haikal9329]

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

[@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

[@JMZReview]

@@joaoneto6216 really

[@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!

[@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!

[@whreREtjk4ko]

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

[@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". -.-

[@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.

[@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 ratio: 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) Golden ratio: 1.61803... (125.35 x (10^-1) - 1.61803) x (10^3) = 10,916.97 Circumference of the Moon: 10,916 km. Golden ratio: 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 ratio: 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. Golden ratio: 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. By Gustavo Muniz

[@findystonerush9339]

WOW A LOT OF PHYSICS AND SCIENCE! numbers! LOL!

[@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

[@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,

[@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

[@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 ?

[@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!

[@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.)

[@carmelpule1]

This is a brilliant presentation. The value of this presentation is the fact that the compounding or incremental interest increase or (decrease) does not only apply to numbers operating in one dimension, but to vectors operating in an AREA where the compounding orientations and rotations are possible. Basically, the $1 dollar compounding in the bank is not a linear compounding but an angular compounding where ( i) is the "quadrature bank interest which rotates its compounding rather than the linear compounding of the magnitude of the $1 dollar in the bank. The compounding shown in this video is incremental in both magnitude and rotational, in which, under certain circumstances the compounding, in quadrature with the length of the vector, makes the vector orientate steadily and rotate with the tip of the vector forming a trajectory of a circle. the philosophy of compounding $1 dollar can be extended to compounding in multi-dimensions available in the space contained in the formulation of the application, including. line space, area spaces, and volume spaces. Here we dealt with compounding a line that exists on an area, hence it could include both compounding of its magnitude and the compounding in rotation which is quadrature with the magnitude hence the use of (i) and not only the growth and losses/decays as the $1 dollar in the bank! .

[@graemsheppard5846]

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

[@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

[@grulice]

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.

[@andrewrobertson444]

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

[@ASLUHLUHC3]

A great explanation of where e comes from. So, "e" does not literally represent the number 2.71828, whereby e^x means 2.71828 being multiplied by itself x amount of times. Rather, *e^x = (1+x/n)^n as n approaches infinity*

[@mandelbro777]

... but e can still be represented as a unique number, which represents "growth plus the growth of that growth, where the period of doubling is infinitely recalculated"

[@ASLUHLUHC3]

@@mandelbro777 Nah, as I said, it shouldn't be thought of as simply a number. Rather, e^x represents the function *(1+x/n)^n as n approaches infinity* and so this is why e^1 (or "e") equals 2.71828...

[@Ensivion]

Ehh this is pedantic, e is a number it has a position on the real line and behaves exactly as a transcendental number does. When your higher level math says e^matrix, it's short hand for writing out the series expansion for the exponential function. When considering e^i(something) you're just showing a complex number, just written nicely. e = 2.71... is as much a number as pi = 3.14... We don't know all the digits, but we treat it exactly like a number. Do you extend this reasoning to pi?

[@javiersaneiro6412]

e is a number on the real line, but one of the things this video shows is the algorithm to calculate e. This just means that e is computable. If you think carefully you notice that at most basic levels all mathematical operations that computers can do are just the 4 basics arithmetics operations. Any other operation like exponential, factorials, or root must be expressed on those 4 operations, but also the digits of irrational and transcendental numbers like pi or e require an algorithm that reduces it to those operations. In this case (1 + 1/n)^n is the algorithm that makes it possible to compute e. That means that beyond the know transcendental numbers, there is an even bigger unknown real of non-computable numbers, numbers that can't be expressed in the form of an algorithm and so we can't calculate the digits of those numbers.

[@jmwild1]

My math analysis teacher in high school gave us a great story/mnemonic for remembering e to 15 decimal places. Andrew Jackson was the 7th President, served for 2 terms, first elected in 1828, and was allegedly involved in a love triangle. 2(terms).7(th President)18281828(elected in 1828 for 2 terms)459045(alleged love triangle).

[@Mathologer]

Nic one :)

[@Amir-wf2if]

why alleged love triangle is 459045?

[@noa_1104]

Amirabbas Askary The angles are 45° 45° and 90° :)

[@theroboticscodedepot7736]

It's easy to remember 2.7 now just tack on 1828 and again 1828. Do you need more than 10 digits?

[@kylemorman7284]

This will help me remember that Andrew Jackson was the seventh president.

[@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.

[@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.

[@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".

[@warrenvwilson]

This actually is a good visual explanation, and is probably the best approach for folks who aren't familiar with Taylor series. One tiny mistake he made is @0:51 where he said "i" squared is 1. He meant "i" squared is -1.

[@aaronleperspicace1704]

Yus

[@MarufSajjad]

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

[@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.

[@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.

[@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.

[@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

[@floridaman6982]

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

[@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.

[@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 :)

[@Lavvysuperstar23]

NO WAAYYY math can be really cool sometimes

[@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.

[@800Bathurst]

+Lucas M i like lamp

[@stephencarlin6273]

Same

[@einstien311]

Pokedex entry 132, brah.

[@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!

[@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?

[@Cinqmil]

I'm not gay. But I just fell in love with this bald man wearing glasses and a funny T-shirt.

[@DiCasaFilm]

So is e to the Tau i equal to 1? Or should I say is e to the 2Pi i equal to 1?

[@Mathologer]

Yes :)

[@DiCasaFilm]

+Mathologer I actually got something right! Due to your really great explaining skills. Keep up the good work!

[@elvispepely4778]

so does e to the 3pi equal -1? and similarly as you keep going up in distinctive whole numbers, would you switch between -1 and 1 as you keep wrapping around the circle?

[@xavimontero5295]

Yes, but not to "3 * pi" but to "3 * pi * i" - don't forget the i. It keeps doing circles forever. in general: "e^(alpha*i) = cos(alpha) + i*sin(alpha)" so it keeps doing circles around.

[@subsnovideoschallenge-go2op]

Consequently, e to the npi=0. Correct?

[@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

[@manuelmenar]

how does he know wich part of the image is selecting with his finger in that green screen? .-.

[@attitudeadjuster793]

He is probably using a projector.

[@want-diversecontent3887]

Traitor How dos his hands go behind the image then?

[@attitudeadjuster793]

I assume he is using a projector for being able to see what he is doing, and overlays the same image in post-production for quality.

[@tucker8676]

Based on that he looks off to the side a lot when interacting with animations I’d guess he has something like news stations; likely a few monitors showing the live feed.

[@becomepostal]

In some of his other videos you can clearly see it’s a projector.

[@adiramrakhani]

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

[@enzila468]

I don't think homer would understand this, but it was a REALLY cool take on the identity.

[@theinquiringminded6952]

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

[@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.

[@Gastogh]

6:31 "So it's like, really awful to the power of awful..." Now you're speaking my language!

[@CalculatedRiskAK]

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

[@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.

[@emilyemily9953]

The cat meowing in the background is a beautiful touch

[@santolok7662]

Thank you for interesting content! And for russian subtitles! 😍 Would you like to analize maths in musical theory? There are two great intonations. - *Just intonation* (natural ratios of sound freqs) is about _rational_ numbers (3:2, 4:3 etc). For example, 300 Hz & 450 Hz = 4:3, or quart. Just intonation is the most harmonic but for a concrete key (tonality) only. Another key can sounds very poor. - *Equal temperament* is about _irrational_ numbers, {freq = 2 * octave}, and "octave" is the only common musical interval in JI and ET (2:1), excluding not very interesting unison (1:1). Octave has 12 half-steps (do, do#, re, re#, mi, fa... etc). Octave repeats to the left and to the right on a piano. It can be illustrated in polar coordinates system as the logarithmic spiral! I was surprised to investigate this simple and beautiful thing! x(t) = -2^t * sin(t * 2pi) y(t) = 2^t * cos(t * 2pi) "t" is quantity of octave (can be any real number) from basic to any tone. In visualization: Octave = 360⁰, half-step = 30⁰. For example if basic tone is 300 Hz then quart will be 300 Hz & 400,45195625... Hz. What? Why second tone freq is not exactly 400 Hz? Because 300 * 2 ^ (5/12). There is 5/12, it is about "quart = basic tone & 5th semitone of 12 from basic". All modern music is based on ET and it is just good approximation to the natural sounds order. But ET ("E" for equal) gives the same musical intervals between different keys (tonalities). Any half-step in any key is 2^(1/12). 2:1 = 2^(12/12), octave 3:2 *≈* 2^(7/12), fifth 4:3 *≈* 2^(5/12), quart etc... Sorry if my English looks poor. Thank you for reading!

[@narrotibi]

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

[@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

[@oneofthesixbillion]

Yikes, you're equating me with Homer. It's fun anyway, thanks. I'm inspired now to relearn what I learned in the early 80s.

[@KASASpace]

Wait... This means that e^ipi = i^2....

[@Carlos244]

KASASpace wow...

[@samburnes9389]

And e^ipi/2 = i

[@samburnes9389]

Militar Zombie e^(ipi/2)=i e^(ipi)/i = i

[@justanotheruser6468]

Sam Burnes well i like my pie with cherrys....

[@hellothing]

e^2πi = e^0? lol e^1.5πi = -i idk anymore :p

[@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.

[@TheaForg]

So would e to the power of tau times i be equal to 1 ?

[@diozavgemd2521]

+Joshi ThePoshi Yes

[@TheUntamedNetwork]

+Mohammed Vates pff, just wait until the Irrational celebration days come around, then we'll see who's laughing as I tuck into TWICE as much pie as you. hehe.

[@cameron1729]

+Joshi ThePoshi From the video you should be able to understand that Pi gets you halfway around the unit circle, therefore since Tau = 2*Pi, Tau must get you the whole way around, i.e., back to 1.

[@TheaForg]

***** Yeah that was what I was saying and I just asked for someone to verify or falsify it.

[@Vulcapyro]

+Joshi ThePoshi e^(iτ) = e^(i2π) = e^(iπ)*e^(iπ) = -1 * -1 = 1 ez

[@StevenTorrey]

A good explanation. Pi as half the circumference of a circle.... but it takes several viewings to really get it....

[@neronampo5200]

All I wish that I had you as my maths teacher. It would be an honor sir

[@010falcon]

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

[@Darkness2179]

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

[@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.

[@albertstern3006]

By Euler's identity: e^(iπ)=cos(π)+i•sin(π) cos(π)=-1 and i•sin(π)=0 Therefore, e^(iπ)=-1

[@carmelpule8493]

This is an excellent explanation. The manner in which (1+i*(pi/m) is written then the spiral will only spiral out and may be settle on the circle but it will never spiral in, It will rotate in the other direction for negative pi. For the spiral to decay in magnitude then using that philosophy one needs to introduce the complex fequency r+ i*pi/m ao e^-r will see to the spiralling in and e^ipi will be responsible for the rotation. Hence e^( r+jpi) or e^(r+jw)t for a fully rotating spiral effect in most engineering functions.

[@mkb6418]

Genious explanation.

[@chorthithian]

great video with beautiful visual stimuli to understand the math going on in the background :) so in a more fundamental level, how was Euler's identity first developed? What approach did they first take? Thank you ;)

[@Mathologer]

Here is a great book about Euler's formula and identity press.princeton.edu/titles/9438.html It's got a lot of the backstory. Glad you like the video and thank you very much for saying so :)

[@laemmeelagi]

What happens if m is a negative number?

[@andreas23456]

You are calculating the limit when m goes to +inf, why do you need negative numbers?

[@jmiquelmb]

NEVER DO IT. The universe can collapse

[@redvel5042]

If m goes in the negative direction, you will get e for a simple reaso. Let n = -m Then (1 + 1/m)^m = (1 - 1/n)^-n = 1/(1 - 1/n)^n And what is (1 - 1/n)^n converge to? Well, it's 1/e. And so the limit of 1/(1 - 1/n)^n as n goes to infinity is 1/(1/e), which is equal to e.

[@reeshav4955]

awful boi, just awful

[@tanvirkaisar7245]

We assumed that m=n*pi, but when calculating e^i*pi should not m (the denominator and the power) contain i causing a problem to our assumption that m=integer number?

[@vwcanter]

Another way to picture e^ix is to remember that e^x can be defined as the function which has a rate of change equal to its value at every point. So the rate of change of e^x is e^x, and the rate of change of e^kx is ke^kx. Therefore, the derivative of e^ix is ie^ix. In other words, it is changing everywhere at a rate equal to its value, but in a direction perpendicular to its position. (Multiplication by i is identical to a 90 degree rotation). Motion defined in such a way is circular motion, by definition.

[@farrankhawaja9856]

yes this is what Grant (3b1b) did

[@patrckhh20]

I must be much dumber than Homer.

[@Alpenjodler1]

Does this mean e to the power of 2*pi*i is 1 since it will wrap around all the way?

[@alexparker7791]

Alpenjodler1 yep! and if you think about what's happening with the unit circle, it might make sense that e^(i*theta)=cos(theta)+i*sin(theta)! since π is halfway around, you can think of theta as a measure in radians. cos(theta) gives the x-coordinate of the point on the unit circle at theta radians, and since we start at x=1 when theta=0, swinging halfway around would take us to the opposite side of the circle where x=-1. similarly, sin(theta) gives us the y coordinate, which happens to be 0 every time theta is a multiple of π :)

[@Alpenjodler1]

Alex Parker I think I finally understand that. Thanks :-D

[@Doppelsaft1]

Alpenjodler1 Also, e^2πi = (e^πi)^2 = (-1)^2 = 1... isnt it great when it al fits?? :)

[@gearhat8506]

Yep, it's very simple and logical to think of that way right? It's one of the many reasons some people prefer to use 2pi as a constant rather than pi since 2pi describes the full circle and a single turn around it.. It's much more intuitive and easier to understand in almost all areas where pi is used. 2pi is called "tau" 𝜏. Look it up if you are interested it might help you understand more things more intuitively like goniometry ;-) P.S. using pi or tau is a preference though.. Some people like it and some (really) don't.. It doesn't really matter since it's all just built on the same principle. Working with tau just clicks quite easily with many people still learning maths since it makes everything seem more logical. But if someone is very used to pi and has been working with it all their life it might seem counterintuintive. :-)