Euler's Formula and Euler's Identity

[akondude]

So apparently I watched this 9 months ago and didn't realize the significance of this equation...

After a year of calculus and re-watching this.. all I can say is Mind-Blown.

[Cadabena]

I've watched all the videos leading up to this, and the ending of this really is more gripping than any film in the cinema I've seen. My eyes widened, and I was left staring at my screen unable to look away with a huge grin that wouldn't fade. Sal, you really do prove why maths is so amazing and inspire us to do it, and I salute you sir.

[trecool212]

What video comes before this one?

[GenericCoder]

MAN THIS IS Amazing. its pretty amazing how people came up with cosine and sines and trig functions to explain just ratios between triangles, and in the end it became all connected to something more bigger than its acctually is I find that mind blowing.

[iseefurtherthanyou]

awesome!!!!

[Jackieception]

In my opinion 0 is best number cuz is not too big, not too small.

[boldy97]

8:50 What happens when we throw in pie? :D

[raoulhery]

2 people have no emotion

[MrTelaviv123]

I find this identity very exciting and disturbing at the same time. Mind blowing.

[texymex1]

@ 11:00, No, 1 is not a cool number. It's the loneliest number.

[rogleo14]

Sal's excitement + many previous comments make me feel like I'm not the only person in the world who finds math so exciting. I get weird looks from people when I describe problems as being "fun" or when I get excited over a problem I hadn't been able to solve.

[Theklstuff]

Thank you for reminding why i love maths.....

[andremeIIo]

I would die of happiness if he had used τ (tau) instead of π (pi).

[ATypicalUsername1]

I can't stress enough how much I appreciate you pronouncing Euler correctly, great video regardless.

[DavidsonLoops]

Sal, this has to be at the very bottom of the Khan maths knowledge tree. Or is there anything below this o_0

[TehTyg13]

@DavidsonLoops, no there's an entirely more complex topics, 3d and vector calculus, differential equations, and integral transforms available from Khan. It's amazing how many subjects Khan can cover while still being helpful and incredibly knowledgeable about all of them.

[thec00kiem0nst3r]

OHHHHH

[8644371]

I've found my religion.

MATH!

[xe3tec]

thats pure math porn!

[proko77]

Thank you so very much for deriving this series of videos of series of derivatives of.......(n).

It's the first time I've managed to wrap my head around Euler in a way that allows me to see just how mind blowing this identity is. I owe you dinner!

[projectbrumaire]

x=a=b-x THUS GOD EXISTS!

[stormcorrosion]

God damn, everything makes sense now!

[hbk6211]

MIND. BLOWN. 

[pepteamsergi09]

Isn't it Any negative # raised to an even power = +positive #????? So why is i -? when raised to an even power (ix)^2= -# doesnt' make sense.... sorrry.. Nice video anyways..

[boredbigmac]

@pepteamsergi09 i is not a "negative" number, its the SQUAREROOT of "-1". You should review comlex numbers, merry christmas! :D

[CrayZDude2]

e^gad!

[CrayZDude2]

that is literally one of the awesomest things i've ever learnt even though i have no idea what it actually means

[therealjordiano]

it all makes sense now O_O thank you khan xD

[PoviIas]

How come, pi is 3.1415... and 180 degrees at the same time?

[PurpleBandit3000]

@PoviIas it is 3.14 radians and 180 degrees

[lifeDotGov]

@PoviIas pi in radians is 180 degrees lool..

[TheBagoomie]

My mind has been blown. Really blown.

[dumbum007]

1 person has no emotion!

[mixidiot93]

so crazy $%&*^&*^@$!@$ this makes me happy

[elidrissii]

Yes, that did blow my mind. THAT WAS F**KING AWESOME!

[bballgirlsl981]

love it!! i got this excited when i saw it explained in lecture first time too lol. great explanation!

[PaulsDigitalSelf]

Eee!

[shawnwilliams77]

I've had to have seen Euler's formula a hundred times from different places and I still get this excited feeling in me every single time!

[ElectroMagneticWeak]

I love how he sounds so excited; so am I!!

[mdwael]

what level of education do I have to be to understand euler's formula? I understand basic calculus and college level calculus.. it seems an interesting topic.

[zeusm0de]

"And if this does not blow your mind... You... You have no emotion." Best math comment ever by a teacher. You rock Sal!

[mrmusicfresh]

@zeusm0de I laughed when he said that ! SO AWESOME.

[EclecticSceptic]

Who or what the hell is CombatEx?

[mtksbctk]

@EclecticSceptic some starcraft nerd

[EclecticSceptic]

@mtksbctk Is that an ongoing joke or something? Or is he here, I can't find him!

[nubsonnn]

CombatEX ++ respect for you

[wilson6951]

Combat-EX is nerd... expected.

[meroftw]

EZ

[Tassadarmaster]

Combat E to the X

[ArbitraryOrangutan]

Combatex I am proud

[turbis]

Starcraft 2 is all maths

[kTeachings]

Combat-EX CALLS IT A TRICK!

[Mysticcplz]

combatex told me to come here.

[CMBrannstrom]

I have no emotions.

[onarass]

Combat-EX wtf is this.

[Dabills0831]

@onarass You need to know this kind of stuff when you go to two universities.

[bigtopbollocks]

And if this does not blow your mind. Sal Kahns vids and explanations will!

[lolololort]

Oh shit, I just came in my pants..

[tautologicalnickname]

this becomes even cooler with TAU !!!

[kentran9]

LOL from @11:15 . YOU HAVE NO EMOTION!

[TheBeardedThinker]

badass explanation

[elpresidio]

If we all had a high school and college professors like Salman Khan we'd all be math geniuses. Fortunately we can have some of his mathematical magnestism rub off on us with the magic of YouTube.

[punkerIII]

11:15 hahaha.

[Gatoradecup]

0 dislikes. beautiful

[someonep93]

As I guess many of you know, there are people arguing that pi would be better defined as the ratio of the circumference and the radius of a circle (2pi).

Plugging in 2pi for x in Euler's formula: e^(i*2pi)=1

Mind blown? ;)

[ilias18]

that is my next tattoo!!!

[Geo877]

@zelos88 meh.

[BostonBear]

One more thing, I absolutely agree that this equation SAYS something about the way the universe is constructed. It's a metaphysical construct on par with Maxwell's equations and the physical universe. More so perhaps; a parallel universe might have different physical laws but Euler's identity I believe would apply in any possible universe.

[BostonBear]

I was just telling people about Euler's Identity a few months ago. I posted a challenge on a pal's site: give me an equation with 2 transcendental #'s, an imaginary number and a monad and get it to all equal zero. Nobody on his page full of musclegeeks could figure it out. I'd like a t-shirt with it. I loved your appearance on Colbert and LOVED it when you said if that equation didn't BLOW your MIND you have NO emotion! GENIUS! Nobody I know appreciates its implications.

[ALoreMiles]

After watching the last 4-5 videos, my mind was truly blown. I'l going to step out of my house and stare at the stars now. O.O

[warvad]

This is the absolute coolest shit, and Khan's delivery is priceless.

God bless you Khan.

[oisin678]

When I first saw Euler's Identity I became obsessed with finding the proof. You taught me something I probably wont have to know for years in 6 videos. Thank you so much I'll do my best to watch everyone of your videos.

[dheeraj54]

@zelos88 i think he was trying to show us a more intuitive approach which is commendable since there arent many who encourage it, u should not worry so much about this method

u should appreciate the fact that he`s trying to encourage mathematical intuition

[dheeraj54]

i have a sort of method to show this:

since all complex numbers are representable as { |r|(cosx+isinx) }

lets say f(x)=|r|(cosx+isinx) then by demorgan`s theorem f(x).f(y)=f(x+y)

and f(0)=|r| f`(x)=(f(x+h)-f(x))/h (limit h->0) ( first principle )

f(x)((f(h)-1)/h)=f(x)(f`(o)) so f`(x)=f(x).f`(0)

solving the diffrential equation ln(f(x))=xf`(0)+c ,c is a constant

=> f(x)=e^(xf`(0)+c)

=>|r|(cos(x)+isin(x))=e^(xf`(0­)+c)

now we can use know values of f(x) to find f`(0) and c.i hope this is okay

[alreadytakenthe3rd]

I got "goosebimples" :)

[Artifactorfiction]

I Love this

[blackwest69]

his hand writing is pro

[jjkul1]

!#%^#$&REWSGDGW##^@#^YHBGFDKMJ­%UT@T#QAFFFTQ#T

THIS IS SOOOOOOOOOOOO AMAAAAAAAAAAAAAAAAAAAZING!!!!!­!

#@%&#I(*UYTEHRGA#FWV#$T%^UN&EU­A#CA@#R#Va$tb^%yn

[SonOfNye]

That last line was priceless.

[MeursaultBandini]

This video proved Khan's passion to me.

[chocolatebubbles91]

AMAZING! thank you for this video!

[midgeamoo]

That's brilliant.

Here's me being taught about Euler and his Eulerian graphs in discrete maths, and thinking it's all a bit boring, then I see he came out with THIS? Why the hell are we looking at the stuff he's done for graph theory and not the most interesting thing I've seen in maths ever? D:

[akondude]

Although it is not technically an identity...(something where you can plug in any value and the equation still holds true - there are no variables here)

it is more of a formula.

Should not have Euler's identity in the video title.

[pochankitty]

@zelos88 Enlighten us, you pompous shit-sack.

[akondude]

@pochankitty nice one :P

[xkriolox]

I do have emotions, but no longer a mind...

[RTRVII]

thanks for this video i'm student civil engineer this is so basic but oh so important. Thank you thank you thank you!!!

[Ferrus91]

Well now the Calculus videos are being worked on I hope we get videos on the divergence theorem and stoke's theorem, which are gaps in my calculus knowledge.

[RTRVII]

@Ferrus91 yep very fundamental and indeed very important. You really need to fully understand them if you want to go into maxwell's equations / electric theory !!

[dannyboy12357]

love the last line...

[Schindlabua]

On a related note: Check out Michael Hartl's 'Pi is wrong! / The Tau Manifesto'

Makes Eulers Identity even a little cooler. And also a lot of other things.

[DeluxeWarPlaya]

Euler was a genius.

[spamdude1]

Didn't you cover this in another video a while ago? I can't find it though. 

[DeluxeWarPlaya]

@spamdude1 It was a part of Sal's introduction video to both his TED talks. The audience laughed when Sal said "if this does not blow your mind, then you have no emotion"

These recent videos appear to be remakes of the original used in the TED talk intro.

[hedonism13]

e to the power of tau multiplied by the imaginary unit equals 1+0

It's marginally more beautiful, because you don't need to use pi... which is a pretty non-beautiful number...

[ladicius]

insanity baskets. that's straight up wizardry. 

[potatofos]

mind blown



[sanjor8r]

about time you did calculus again.

[Enlight3nd]

I think this formula is missing Kelvin!

[RTRVII]

@Enlight3nd its missing the speed of light !!!

[thisiswebbmaster]

Isn't it "YOU-ler"?

[QuantumMaths]

@thisiswebbmaster

No it's actually "OIL-ler". I pronounced it incorrectly for a couple years lol...

[OniLinkPlus]

This is the most amazing equation in the history of mathematics. It combines the 5 basic constants into a single expression:

e - The fundamental constant of exponential functions

1 - The fundamental constant of multiplicative functions

0 - The fundamental constant of additive functions

pi - The fundamental constant of geometry and trigonometry

i - The fundamental constant of the imaginary plane

[PoketoMtg]

I'm glad I found the reason why this happens before I start Calc 2 in the fall. (I also like to know the proofs before the class it taught so I can keep my test scores up). I've always wanted a proof and I thank you for showing it.

[Waranle]

Beautiful!.....Sal please do a number theory course :)

[akondude]

I guess I have no emotion..

Honestly I think it would be cooler if the i wasn't included in the identity..

i is completely made up and makes absolutely no sense -.-

[quiksilver99]

@akondude All numbers are 'made up'. We chose base 10 we chose what certain values mean. We are simply giving a value to root -1... Thats not weird at all. We can a value to 3.14159.... and 2.78 or whatever e is.

[VCube100]

@quiksilver99 2.71828182845904...

[akondude]

@quiksilver99 Ok sure all numbers are made up... but i is just making up new numbers for the numbers that are already made up...

[voutasaurus]

@akondude: Doubting the significance of i is on par with doubting the significance of pi.

All numbers are abstract, and only have physical relevance in constrained circumstances. Many very useful models of phenomena, in physics and elsewhere, use complex numbers to represent aspects of the model. Hence, they are useful, and I believe that Euler's identity makes them go further than useful, into the realm of truth and beauty.

[PedroGynVibes]

@akondude It's completely made up indeed, but it makes sense... it gives a meaning to what otherwise wouldn't have any. The number of possibilities that arised from that meaning are numberless.

[QuantumMaths]

@akondude

BULLSHIT!

Cure your ignorance and emotional apathy right now by reading "An Imaginary Tale" by Paul J. Nahin, or acquire a copy of Needham's "Visual Complex Analysis". or consult an online course fully detailing the amazing magic and utility of the imaginary unit i.

[BoxStudioExecutive]

@akondude you're pretty dumb, how is i a made up number? It's pretty well understood. If you multiple i by i, you get -1. if you add i to i, you get 2i. if you subtract i from i, you get 0. If you divide i by i, you get 1. it follows the same rules as every other number that exists, if it were made up, then it wouldn't.

dumb.

[akondude]

@BoxStudioExecutive Suck my dick faggot.

[BoxStudioExecutive]

@akondude Bro, your dick is made up.

[akondude]

@BoxStudioExecutive Cool story bro.

[glooDortil]

I've been waiting for this video for a long time. It delivered on levels chemical substances struggle to attain.

[MoGaDeX]

My math teacher actually said that it was Euler's Identity that made her be a math teacher in the first place.

[THETYGUY77]

@MoGaDeX dude!! no way my teacher said that too......euler's a basterd >.<

[angel209056]

i wish this was up when i took calc II last semester!!

[HoorayBier]

Euler's identity has been called beautiful. It uses the 5 most important constants in mathematics (e, i, 1, 0, π) and uses exponentiation, multiplication, addition AND equality.

If I ever had to write a proof of why Euler was a badass, it would be this identity.

[TheLiberalSoup]

@HoorayBier

Its also sooooo great because you can derive trig identities from it! I love when i can do that!

[Andnextwehave]

mind blown khan, thanks for the videos :)

[rinwhr]

I have no emotion...