Polynomial approximation of functions (part 7)

[theminecraftchanel1]

i feel so bad for anyone who has to do this

[distractionxx]

Very enjoyable thank you

[bribribri56]

you have some great stuff here

[jjclassjj]

brilliant video

[alzahiriscool]

The funnest part is from 8:12 to 8:19

[edmundstoiberderdepp]

I almost threw up at the end.

[xXsoulcatcher]

You are amazing! thanks for your videos!

[stsrawmos]

best video from sal he was so stoked

[OniLinkPlus]

Euler's Equation relates the 5 fundamental constants:

e, the fundamental constant of exponential functions

1, the fundamental constant of multiplicative functions

0, the fundamental constant of additive functions

i, the square root of -1

pi, the ratio of the circumference of a circle to its diameter

This bizarre equation relates all 3 forms of arithmetic, imagination, and geometry.

I think we have discovered god, and his name is Euler. All hail Euler.

[ThaFacka]

@OniLinkPlus

+ this equation explains or defines the operation of "to the power of". even more amazing.

so how do we get prime numbers and the ceta function into play? are they "above" or "below" those cascade of related fundamentals?

[killyokoluvpaul]

what about Lagrange error bound ? am i missing something? I can't find that lecture.....

[bugfacedog44]

And I also just realized if you put 0 in for x it works too! THAT IS EPIC!!!

[bugfacedog44]

I watched this from start to end! That should be made into a movie!!!!

[UltraMaXAtAXX]

MINDBLOWN.

[sakuracheerio]

I'm the 111th who like this video XD~

Request: since I cant find any video about pi, will you make one?

[MartaAndrade]

 'this should blow your mind. if it doesn't, you have no emotion' lol amazing!

[ladicius]

why didn't i grow up with friends like you? i wouldn't even care that i wasn't as smart; the entertainment value of watching such a brilliant human is absolutely priceless. i just watched seven videos not knowing what 'factorial' meant, how to do a derivative, the funny n=0 to infinity above and below the greek letter...and it was still awesome. (i'm in precalc - i'm getting there) thanks a lot.

[ThommyTiernan]

I think I found a new religion... I shall worship e...

[DrLasker1]

seeing it all come together caused me to piss my pants

[caml17]

Euler was my calculus prof. He told us that he derived Euler's identity from symbols formed by his poop laying in the toilet

[Masandr3w]

We learned Euler's Formula last week in Calc 2, and this proof is incredible!

[56jmoney]

I've learned more from your vids, than in taking Calc I & II.

Great Job!!

[TheMikeSideGallery]

You can use Euler's equation e^ix=cos(x)+i*sin(x) to prove that 1/e=cos(i)+i*sin(i) and e=cos(i)-i*sin(i). Then you can use the maclurin series to prove that cos(i) is real and i*sin(i) is real. I think thats the wildest part of his theorom and the maclurin series.

[bluejimmy168]

Sal, your are my god haha. Can, you show us the proof please. I really want to know. thump up this comment if you would like Sal do show us the proof of the series when he have time.

[thelastwords]

goosebumps

[VELOCITYangus]

omg e^pi ~ 23 so i is the pi coeficient that transforms 23 into -1. My brain!!!!!!!

[zuppah]

you're making great videos, although I was hoping to hear someting about that remainder of the taylor series; 'cuz it's completely beyond me haha

[atown2k3]

Wow...this is proof of math and order to the universe...and yes sal i believe its only a glimpse of what the universe's order!!...Thanks, love mathematics!!!!!!!!!

[seniorbooboojuice1]

This is incredible I have goosebumps hahaha

An invisible order in the universe that human reasoning caught a glimpse of!

[GreenAms]

my teacher did all this in one breathe and wasn't even excited about it lol Which I found really weird.

[Werwolf2x]

i believe e^πi + 1 = 0 or e^πi = -1 is also called Euler's Identity and yes, it's truly one of the most beautiful and interesting relations between the 5 most important constants in mathematics.

I mean, there's Euler's number, Pi, the imaginary Unit i, then the real Unit 1... put that all together and the result is ZERO, or nothing, so to speak!!!

Think about it for a few minutes ;)

Also i'd like to know what Euler must have felt, when he found this relation ^^

[ThePz0ut]

@Werwolf2x

It might not seem very intuitive, but I think I found a video explaining the intuition behind Euler's theorem

watch?v=qpOj98VNJi4

[xenoepist]

I found an explanation on youtube for eulers formuula . It is pretty much what WKREPELIN mentioned on his comment. But I do agree with Sal that this explanation is something we created as a tool for analyzing imaginary numbers. We created an imaginary plane and graphed complex numbers. But I do not think that fully grasps the meaning of this formula, but I could be wrong!

[Taowhr]

Euler's formula is the most beautiful formula in all of Mathematics.

[clebsar]

Its just an amazing, incredible and beautiful proof of how we are limited and tiny, when compared with Universe. All these math manipulations lead us to the idea that we're underneath something bigger and bigger, beyond the dreams and visions of human beings. Thanks for demonstrations, Sal. You have the gift! God Bless you!

[ryanguy6789]

Incredible! And Justin Beiber's Baby video has 256 million view, while this is stuck at 8,000. This world really is quite strange, Mr. Khan.

[budman533]

Burrr my goosebumps have finally arrived...thats amazing

[RickRubik]

Sal, I know you think that "e" is a very fascinating number, but I actually find the concept of infinity to be as fascinating, if not more. I find it hard to fathom what infinity is because we live in a society where many things are quantified in finite numbers; my mind has a hard time excepting why the limit as "x" approaches infinity of (1/infinity) is equal to zero.

[JohnNumberThree]

This, believe it or not, has helped strengthen my faith in God a little. Something this amazing (the fact that these factors are all related) can only be explained through something higher than the human mind, like you've said. I know it may seem weird, but I wish everyone could come to know and understand these things, then maybe it would help them to see the Light of our Universe a little better.

[mauroprovatos]

mind = blown to pieces!

this deserves my channel's background

[rakurai19]

mind

blown

[J0EH3AD]

The most amazing coclusion in mathematics is Central Limit Theorem! You got it wrong. :-))

[xenoepist]

@J0EH3AD In no way the Central Limit Theorem is more amaizing than Euler's Formula. If you think about it, it makes sence that the mean of several samples approaches the center of the function. For example, if you throw a dice 4 times, it is highly unlikely you get all 1's or al 6's, but it is more probable you get something in between like, 2, 4, 6, 5. Do that several times and you'll get a center spike on your graph. But Euler's formula makes no sense at all, however, it is true. Amaizing!

[TheLivirus]

My math teacher didn't give Euler's formula this much attention. I guess all mathematicians have different taste hehe. He's on the other hand really amazed by the chainrule, which is proven useful in many unexpected contexts.

[supremum1111]

thanx

[rinwhr]

I think he pronounced Euler wrong lol

[Nelyami]

yeah, Euler = "Oiler" - like oily olive oil.

[reik2006]

there are a lot of people who can write this derivation down. But there are very few which are able to communicate the fascination for it. Sal you rock ;)))

[morvind]

cosh(x) + sinh(x)=e^x

cos(ix) - i*sin(ix)=e^x

[RoguePianistTetsuo]

Me: THIS IZ MADNESS!!!!

Salman: THIS IZ MATHAMATICZ!!!!!

[olkomat1]

Actually, it's pronounced "OYler" (like 'oiler'), and NOT "YUler" (even though that's how it LOOKS when it's spelled). But still, this video is presenting such a majestic mathematical concept, and both the equations @ 7:30 and @ 9:30 (as well as others) are often called "Euler's Formula". He was an important figure in science, and when you're important enough, you're allowed to name stuff after yourself. Euler was just really important, so he was allowed to name a lot of things after himself.

[onlyAerik]

Would that be the same guy we heard about in my discrete math class? where we drew diagrams with dots and lines and found... shit I don't even remember.

anyway's nobody has actually spelled it. iuler? iuoyler?

[izi2bhappy]

you succeeded to ecite!

good job dude!

[lowgangsta]

Truly Amazing

[gobberpooper]

hahahahhahaha.

"This should blow your mind. If it doesn't, then you have no emotion, I will just judge you."

The Holy Trinity of e, i, and Salman Khan

[Thymonico]

O.O

I convert to mathematicism!

[rishkaal420]

@Thymonico more like i convert to SALism :P Youtube is like our bible right here and the language is Mathematics! (no offense to any christians due to my reference of the bible. This just reflects on my exuberant enthusiasm for mathematics when we actually find something amazing. Once again, no offense to you all :) )

[alexgordonepic]

"The holy life of the apostles, and of the other primitive Christians, appears to me an

irresistible proof of the truth of the Christian religion. If true happiness consists in

union with the Supreme Being, which it is impossible for a moment to doubt, the

enjoyment of this happiness necessarily requires, on our part, a certain disposition,

founded on supreme love to God, and the most perfect charity toward our neighbor" -Leonhard Euler in his "Letters" p 507

[wiseravi2000]

God is imaginary. Every society in history had some sort of "god" so is the Christian theory that everyone before this new religion is "burning in hell." Christianity doesn't make any sense. Sure there is order in the Universe, but the idea of Christianity is primitive and dying.

[alexgordonepic]

" infinitely more worthy of the almighty Creator, who formed it. The government of

this universe will, likewise, ever inspire us with the most sublime idea of the sovereign

wisdom and goodness of God. -Leonhard Euler in his "Letters", p. 382)

[80amnesia]

one of your best vdos

[arthenor007]

3.177 views... oh poor mankind..

[gimmeanickalready]

Great Videos! However I have one question:

PI in this context is just defined to express the angles (as angles given i radians). I'm sure you have a good explanation for this, but why couldn't we just write?:

exp(i*180deg) = -1

[santimda]

Because in maths we use angles as plain numbers, so they MUST be in radians, which is like having no units... so, forget about "degrees", they can be used in another context, not this.

Hope this helps..

[gimmeanickalready]

@santimda: My point was that what is used in math (pi) is defined so, and if one wants to be moved by mind-blowing truths one must try to separate what in math is defined by humans and what comes from "nature", to avoid circular logic. But this might be a topic for philosophy or at least philosophy-oriented mathematics.

[hideki9999]

Ok, it just blew my mind! I'm now cleaning my brains from my bedroom's ceiling...

[Bomberofdoom]

So...a relationship between exponential equations and trigonometry....

Which form complex numbers, or Euler forumla.

Oh, hang on.... PI was defined for all circles, i for all polynominals, e for exponentials.....

So founding/finding these defenitions eventually have a combined meaning to it all, is that it?

If so, that really is insane....

[Password1andUsername]

Yeah this stuff is just completely fucked up.

[myselfable1]

I am impressed. I would never have put all that together. I look forward to some analysis videos.

[jhanstra]

Speechless...this is why I love math

[lisinka3]

This is so...I'm here shaking, scared and laughing at the same time.

This changed my world view.

Euler was quite a genius.

[moreoriginalusername]

gg god. gg.

[sharvil1993]

OH!!!

MY!!!

GOD!!!

This was the BEST math class I have EVER attended!!!!

[startvideo2007]

o god !!!! now that is really metaphysical

[vosabre]

This... is... amazing...

[polariss0i]

See this just proves god is like us, an irrational dreamer :)

[UndeadTheta]

beautiful ^^

[pmvaziri]

hahah best part is when Sal says "This should blow up ur mind, if it doesn't then u have no emotions, i will just judge you" hahah

[Ren520]

i love the conclusion O.O!!! i was like" wow..." and started to laugh at myself.

this is beautiful.

[nash911]

Wow

[najeebmir123]

That was so intutional.U are awesome!!!

[volintine]

yup u r right

[ajwchin]

Isn't Euler pronounced "oiler"?

[spase667]

Yes it is.

[wkrepelin]

I love Euler's formula but it is a pretty well understood transform from the "reals" to th"complex" number spaces. I don't think the claim that it's not understood adds to the dramatic content. Why include it?

[khanacademy]

It is well understood as a tool and we understand its properties, but the relationship of these fundamental numbers shows some order to the Universe that I don't think is well understood by anyone (at least that I know).

[wkrepelin]

I see. I think I misinterpreted your statement (actually, at this point, I think it's rather obvious I did). In that sense though, how well understood is any of mathematics?

[CogitoErgoCogitoSum]

Indeed. The both of you are right. Eulers formula is magical. But how well understood is any of mathematics? I toast to that! The reason why Eulers formula appears magical is because its difficult to fully grasp (thats the definition of understanding). Simple expressions like 1+1=2 are more easy to understand. No less magical, just taken more for granted. Can you actually prove 1+1=2?

[elrefugiodelachota]

With the right axioms, you can.

[CogitoErgoCogitoSum]

what? Math is math. Its a natural extension of the same. I dont put arbitrary "only real" restrictions on my answers

[elrefugiodelachota]

but every numeric group (integers, rationals, reals) have it's own definition. In the case of the two first, they are taken as some propertie of the reals (the smaller inductive group with addition of the 0 and the opposites in integers for example), but then in reals you don't have definitions, you have axioms, and usings that axioms you prove 1+1=2

[CogitoErgoCogitoSum]

So? The properties may be proven differently but are otherwise the same. The same operations work, do they not? Are you arguing with me on how it was proven? Because I dont deny a difference in proof. If youre arguing that the operation isnt permitted then you are wrong. But I get the impression that you already kow that it works. So what are you talking to me for?

[elrefugiodelachota]

I think you don't understand me, probably because i can't write what i want to say, i'm not an english talker so going to the point, i was trying to say that 1+1=2 can be proved that's all, sorry for the missunderstanding

[cbot141]

why have so many of the polynomial approximations become "unavailable".

please reinstate these. thank you

[BreachedWall]

@khanacademy There are a lot of things we don't understand...yet in your previous video you described this particular relationship as evidence for God. Why? Maybe there is a hidden, not currently understood order - but why should this be a higher order? In the past, when man didn't understand the rainbow, he promptly connected it to God or some higher order. Surely you're just taking the same, naive viewpoint?

[V2PRC]

@khanacademy I agree with you, Sal. What we know is only a drop of water in the ocean of mysteries of the great universe. We'd better humble ourselves.

[rishkaal420]

@V2PRC Very true, it's just like our knowledge is just a mere quark in the midst of the multiverse:) We have all seen people who have used their intelligence for diabolical reasons or people who have attained mathematical enlightenment but have not shared to the world. But we have just seen one person who truly finds it entertaining and to spread joy and his mathematical enlightenment to the world. I am actually going to give a toast to Sal Khan on my graduation night!

[V2PRC]

@rishkaal420 Are you a mathematics major?

[rishkaal420]

@V2PRC oh no im not. I have just finished my high school in Singapore. I did the IB Diploma Program. What about you?

[anvilsalesman]

so, can this be used to find the natural log of a negative number???

[smartmusicfreak]

Yes, it can. With with definition of ln(a*b)=ln(a)+ln(b) and with the definition of ln(-1)=i*pi, ln(-x)=ln(-1*x)=ln(-1)+ln(x)=l­n(x)+i*pi

Therefore ln(-x)=ln(x)+i*pi.

[CogitoErgoCogitoSum]

Let me amend smartmusicfreak's answer.

ln(-x) = ln(x) + i*pi*(2n+1), where n is any integer (therefore 2n+1 is any odd number).

There are infinite answers to natural logs of negative values