Euler's formula & Euler's identity | Series | AP Calculus BC | Khan Academy

Euler's formula is e__=cos(x)+i_sin(x), and Euler's Identity is e^(i¹)+1=0. See how these are obtained from the Maclaurin series of cos(x), sin(x), and e_. This is one of the most amazing things in all of mathematics! Created by Sal Khan.

AP Calculus BC on Khan Academy: Learn AP Calculus BC - everything from AP Calculus AB plus a few extra goodies, such as Taylor series, to prepare you for the AP Test

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[TheHalalPolice]

"if this does not blow your mind, you really .. you have no emotion" - Sal Khan  

[mulimotola]

I always come back for the last sentence

[Mswordx]

My calculus III teacher skipped this when we went over Taylor approximations.

Yeah thanks, teach, I didn't want my mind blown anyway

[thebattleof]

What an absolutely SAVAGE ending "If this doesn't blow your mind you have no emotion". SAVAGE

[awesomesoccerfan]

This is, hands down, the greatest video ever.Having said that, I have just had an infinite series of goose bumps :P

[Markus]

"If this doesn't blow your mind... you have no emotions."

Spot on! :D

[sibsbubbles]

Wow. As an aspiring math major, this really is quite amazing. I've heard of this identity, but never was exposed to it fully either. Just did the series of this formula. For someone who didn't even have a 4 function calculator to aid in even basic arithmetic, but discover something so profound is pretty mind blowing. Also, to anyone going through what I went through in Calc 1: keep going. Don't do what I did and get defeated by it. Self-doubt is hard to conquer, but SO worth doing. Even if it is a struggle well... that's kinda the point actually hehe. Calc II was actually pretty cool. Calc III was a lot like "one, " but you get to learn about new spaces and ask bigger and bigger questions after that. Wasn't until after my intro to abstract math and proofs class that I started to see how deep this subject can really go... and to appreciate how much work this had to have been to discover the first time back then.

[marcrogue]

if this doesn't put a smile on your face you don't like math!!!

[fuzzybuns]

"if that doesn't blow your mind...you have no emotion"
if Euler were here today he would be laughing his ass off

[jeremycorrea]

This is the best video on the internet ever lol

[sarahfarag]

when a math equation makes cry out of joy :')
*it's so beautiful *

[XCyclonusX]

You forgot another cool mathematical identity found in Euler's Identity...infinity.  Eulers identity is only true as the series goes out to infinity. So infinity is also a part of Euler's Identity. 

[kimthi]

This is the most mind blowing thing i have seen in my entire life, like maths has its own universe trying to relate completely unrelated topics. After watching this if someone still thinks math is not fun then he has no right to be on this planet. Thank you Sal, long live maths, long live Khan academy

[manofculture]

"If this does not blow your mind...you have no emotion"...haha don't you worry, I was fanboying all over the place when I noticed the connection with "e", "cos(x)" and "sin(x)".
Let alone e^(pi*i) - 1 = 0 XD XD XD.

[LoOpzOops]

All these years taking courses... and I finally see the beauty and awesomeness of mathematics. Thank you very much Khan!

[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.

[cembilaloglu]

this is the most beautiful thing i have ever seen in mathematics

[bautrey]

I don't get why this video does not have more views....

[ratixyz]

14 people out of the 204, 394 viewers (as of November 15th, 2014), neglecting multiple views, have no emotion. 

[ZeeshanAli-nmeq]

was gonna skip this video, until I saw the comment section. Watched the whole video, and waw! I seriously had goosebumps. this explains the question, why Sal always gets amazed whenever e comes while solving any problem...