Euler’s identity proof for calculus 2 students!

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The Euler's Identity is considered to be the most beautiful equation in math, which says e^(i*pi)+1=0, t-shirt here: amzn.to/427Seae This video is for calculus 2 students after we cover the Taylor series. We will first do the classic proof of Euler's formula where we use the Taylor series of e^x, sin(x), cos(x), and complex numbers to draw the connection that e^(iz)=cos(z)+isin(z). Then we will plug in pi and get the equation e^(i*pi)=-1, which is the same as e^(i*pi)+1=0. Euler's formula is also the main building block of complex analysis.
Calculus 2 power series pdf 👉 bit.ly/calc2series
See how we can integrate e^(ax)*sin(bx) with Euler's formula and no integration by parts: 👉 • Integral of e^(ax)sin(...
0:00 Proving Euler's Formula e^(iθ)=cos(θ)+i*sin(θ)
4:58 Check out Brilliant
5:52 Proving Euler's Identity e^(i*pi)+1=0
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31 июл 2024

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Комментарии : 211

@blackpenredpen Год назад

Shop an Euler's Identity t-shirt: amzn.to/427Seae

@WestExplainsBest 2 года назад

As a teacher, the ability to speed up like that would be amazing.

@themibo899 2 года назад

*3 hours later* kid in the back who didn"t listen: yooooo how can you have the sqrt of -1 ??? I just checked on my calculator and it says "undefined" worse even: Everyone in the class: fuck he's right, we think you made a mistake sir

@anshumanagrawal346 2 года назад

@@themibo899 lol

@sudirosudiro4218 2 года назад

ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-3XjcbRYYyyQ.html proof of polynomial (x+a)^n

@timhaines3877 2 года назад

I had a physics professor who would just cartoonishly wave his arms and say "as any fool can see!" and then write down the last statement of the derivation. It was his way of not-so-subtly hinting that the physics is more important than the n-th degree of the math- he would show us resources that contained the full derivations later. That might not work in a math class, though.

@RobinHillyard 2 года назад

You forgot to mention that Euler’s identity also employs the four principal mathematical operators: addition, multiplication, exponentiation, and equality. ;)

@blackpenredpen 2 года назад

Oh yea! Thanks.

@slavisastojakovic3400 2 года назад

Whenever he says 'Ladies and gentlemen' I know magic is about to happen.

@blackpenredpen 2 года назад

😇

@plislegalineu3005 2 года назад

more like maθ

@eleSDSU 10 месяцев назад

@@plislegalineu3005 weird way to spell mathgic

@okenyodong1420 2 года назад

Amazing proof, thank you so much. I was struggling to wrap my head around Eulers identity for months.

@leziehr 2 года назад

i have been looking for this equation’s derivation for ages, thank you for this! this really is the most beautiful equation ever.

@blackpenredpen 2 года назад

😃

@General12th 2 года назад

If you're ever curious about where those series come from (or you don't feel like remembering them), just derive them yourself! Suppose cos(x) = a + bx + cx^2 + dx^4 + ... . Set x = 0, solve for a, then differentiate both sides with respect to x and solve for b. Differentiate yet again and keep going! Within four or five iterations, you should see a pattern emerge. You can do the same thing for any function, like sin(x) or e^x or ln(1+x). It's much easier than having to remember them.

@catnip202xch. 2 года назад

Yeah and it’s the Taylor series and I fucking hate it so bad it’s the bane of my existence lol

@General12th 2 года назад

@@catnip202xch. Deriving a Taylor series shouldn't be that hard. Do you know how to differentiate most functions? Can you set x = 0 and solve for coefficients one after another?

@Goku_is_my_idol 2 года назад

Taylor series just assumes every function can be expressed as a polynomial function of infinite order. Let f(x)= a0 + a1x + a2x²+.......+anxⁿ+....upto infinity f(0)=a0 f'(0)=a1 f"(0)=2a2 ....... And so on fⁿ(0)=n!an Then just multiply individual an values to xⁿ and thats the mauclaurin series. I should mention this is a specification of Taylor series. The original one's a bit more complicated to derive: f(a+h)=f(a) + hf'(a)+..... Remembering things is hard. Try to comprehend the underlying concept and you wont need memorization.

@Muslim_011 2 года назад

Brilliant 👍

@eleSDSU 10 месяцев назад

Bold of you to believe I can do this for any function :3

@dr.rahulgupta7573 Год назад

Excellent presentation of the topics in a beautiful manner by an experienced teacher . Vow !

@jordimayorgisbert6490 2 года назад

Amazing !!! I don’t know that development of e^z using Taylor Series. Thank you !!!!

@user-yr8lm3sv4s 2 года назад

new inspiration on the formula,thank u

@JSSTyger 2 года назад

I agree that this is the most beautiful equation in math. I learned it in my fifth and final year at the university, taking a course i didnt even need. Needless to say, I am quite grateful I took that course.

@Mark-nm9sm 2 года назад

I am watching your 100 infinite series little by little and the difference in hair style is so nuts compared to now. Also the mic is now a pokeball amazing

@idrisShiningTimes 2 года назад

This equation id really beautiful, like no words to describe this masterpiece

@sahincpa1230 Год назад

Very very nice presention. I highly appreciated you sir.

@michapodlaszuk9025 2 года назад

Wow, it's really cool how easily I can show people that I'm done with the proof, thanks

@stevenlin2005 2 года назад

I’m a 10th grade student in Taiwan, and I chose pre-calculus as my elective course (it’s really PRE-calculus, we won’t even mention Epsilon-Delta definition of a limit this semester!) Though I can’t fully understand most of your calculus videos, I still find them interesting! And I wish to have a teacher like you ：）

@uchihasasuke7436 2 года назад

Make sure you know unit circle very well haha 加油同学

@valerium730 2 года назад

The second you wrote down "1-(1/2!)t^2" i immediately knew what was up, and it's beautiful.

@valerium730 2 года назад

like no joke i actually shouted "no way!" out loud the second i saw it

@meiljohnbueno3149 2 года назад

I really admire those people who really understand maths and its complexities. Wisdom is really a gift ❤️❤️

@eleSDSU 10 месяцев назад

You might be confusing knowledge with wisdom.

@octs609 3 месяца назад

@@eleSDSU No they are definitely wise.

@fred-ev7zd Месяц назад

THIS EQUATION IS SO FUN!!!

@idjles 2 года назад

Not only does it have the 5 most important numbers, it also has the 4 most important operations: equality, addition, multiplication and exponentiation.

@blackpenredpen 2 года назад

See how we can integration e^(ax)*sin(bx) with Euler's formula and no integration by parts: 👉 ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-E_ecLWj_rcM.html

@rosaorwinter910 2 года назад

I have a genuine question for you, if e^(iθ)=cos(θ)+i*sin(θ) then wouldn't that mean that if you plug in 2π you would get +1, meaning that i×2π=ln(1) so therefore i=0?

@mazroatussaadah617 2 года назад

O km momok

@abdelhadiech-chaibi2921 2 года назад

Can you calculate integral of exp(x) over t^2 +1

@velivelmu8530 2 года назад

another one I liked, in latex: f(x) = e^{-ix}(\cos x + i \sin x) \\ f^{\prime}(x) = e^{-i x}(i \cos x - \sin x) - i e^{-i x}(\cos x + i \sin x) \\ f^{\prime}(x) = e^{-i x}(i \cos x - \sin x) - e^{-i x}(i \cos x + i^2 \sin x) \equiv 0 \\ f^{\prime}(x) = 0 \;\;\; \forall \; x \in \mathbb{R}\Rightarrow f(x) \text{ is a constant} \\ f(0) = e^{0}(\cos 0 + i \sin 0) = 1 \cdot(1+0) = 1 \Rightarrow f(x) = 1 \;\;\; \forall \; x \in \mathbb{R} \\ \\ 1 = e^{-ix}(\cos x + i \sin x) \Rightarrow e^{ix}=\cos x + i \sin x \;\;\; \forall \; x \in \mathbb{R}

@AirshipToday 2 года назад

@@rosaorwinter910 No, because if you were to divide both sides by i aka multiply by -i you get 2π=0 and understand that ln(x) has infinite solutions

@RomanoPRODUCTION 2 года назад

It's beautiful ❤️

@hakimalaarossi7632 2 года назад

Thank you ❤❤❤

@amiralidorodian1932 5 дней назад

thanks so much

@iandmetick07 2 года назад

Hey Steve I love your videos, I would like you do Reimann hypothesis. What's it and what is the problem there and what's so difficult to solve it 😂

@thxmori 2 года назад

BPRP, I have been following u for o long time, can you describe Catalan’s constant that my teacher ask me to study it.😄

@154bharshbhanushali9 2 года назад

Hello Sir , Facing a lottt of difficulty in solving Definite Integration of Greatest Integer Function questions , Plz If possible then , Make a separate video for it 🔥🔥🔥🔥🔥🔥🔥🔥🔥

@juniorisrrael3184 2 года назад

Esto prueba que e^x es una función sin ceros en todo C.

@user-tt7wu2zh7h 2 года назад

The beauty of math in one episode

@carultch 2 года назад

Are you proud to see mathematics making your alphabet famous?

@AznJsn82091 2 года назад

I will get a tattoo of that in the future!

@bhavydugar6665 2 года назад

What I like best is that he is one of the few Americans to correctly pronounce Euler’s name is

@GaryFerrao 2 года назад

OMG the sudden close up of your face for the sponsor message was so creepy lol 😅😂

@randomlyselected4550 2 года назад

आशीर्वाद from India ✋🏻

@lekiwi_4145 2 года назад

Thx

@meesbaker Год назад

Would it be a proper derivation to say the space in the circle with r=1 equals pi, so integral from 0-1 of sqrt(1-x^2) = pi/4. Then complex substitution gives pi/4 = i/2*ln(-i) which can be transformed into the euler formula e^ipi = -1

@sanjeevkr.1497 2 года назад

Love you sir

@huizylove 2 года назад

Do you have a cal2 review? I need it your the best!

@blackpenredpen 2 года назад

I got you. 100 calc 2 problems. ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-Kwyk_mteyNc.html

@huizylove 2 года назад

@@blackpenredpen Thank you! 😊

@dimastus 2 года назад

Well, it is good and valid proof/derivation, but I think, that using infinite series in that case is like shooting at fly with a bazooka. Why not to use limit definition of e^x and geometric meaning of complex number operations?

@volodymyrgandzhuk361 2 года назад

It would be too boring

@DawidEstishort 2 года назад

Doesn't the geometric interpretation come from the fact that you can represent e^iz as cosz+isinz? I mean without it we could represent a complex number as a point in coordinate system, but we wouldn't know where e^iPi is.

@dimastus 2 года назад

@@DawidEstishort no, geometric interpretation can be derived from trigonometry. Any complex number is defined by real and imaginary parts or x and y coordinates, like vectors. But you can go to polar coordinates r and t. Then real part x = r*cos t , imaginary part y = r*sin t. So we have z = x + iy = r(cos t + i sin t). If you try to multiply two complex numbers with coordinates x1 y1 and x2 y2 (or r1 t1 and r2 t2) in cartesian you will get a mess, but in polar, after some trigonometric exercise, you will get a new number with r = r1*r2 and t= t1 + t2. (Lengths are multiplied and angles are added). With this as a starting point, you can use the limit definition. e^it=lim (1+it/n)^n So you need to multiply the number 1+it/n by itself n times and tend n to inf. Now recall, lengths are multiplied so the resulting length is lim sqrt(1+t^2/n^2)^n = 1, and angles are added so lim (t/n)*n =t. And so e^it=1*(cos t + i sin t). Hope, I've convinced you.

@DawidEstishort 2 года назад

That makes sense. I didn't notice limits work nicely with lengths of vectors here. Although I think in this case series approach is more elagant. Using this approach he just had to use the definitions of functions as a series and everything else was shown in the video. I think if he wanted to use lim approach in the video he would need to spend too much time with trigonometric calculations to show where everything comes from (like showing why multiplicating complex numbers in polar coordinates works how it works). Cheers

@dimastus 2 года назад

@@DawidEstishort indeed, moreover, this video is targeted to calc 2 students. So, yes, I guess series approach is better when you study them. I just really like proofs with geometric origin, even if they are longer or not intuitive. It's like some kind of art or sport for me.

@rogo7330 2 года назад

2:07 that pause was funny, sorry)

@blackpenredpen 2 года назад

Lol

@bogdyee 2 года назад

There was a problem once that I found but I could not understand the answer and why it works. The problem sounded like this. Given a board of M rows an N columns find the number of ways to tile an MxN rectangle with 1x2 (they can be rotated) dominoes. (something related to aztec treasure problem but on a rectangular board) and the formula that I found on a forum is like (I tested it with a program and it does indeed give the right answer for several M and N test cases). 2^(M*N/2) * ( cos^2(m*pi/(M+1)) + cos^2(n*pi/(N+1)) ) ^ (1/4) over all m,n (integers) in the range 0

@berdly3367 2 года назад

E^(i*pi) + 1 = 0 literally reminded me of quadratic

@herbie_the_hillbillie_goat 2 года назад

The rabbit hole goes deeper. Euler's identity contains the five fundamental mathematical constants, three fundamental operations (addition, multiplication, exponentiation) and equality. Of course, this invariably brings up the tau debate and the fact that e^iτ=1.

@rasseboiii3505 2 года назад

Is there any way possible to solve the equation x^^x=4? The answer is 2 obviously, but can we confirm thats the only one? Talking about tetrarion, not just regular exponentiation

@crochou8173 2 года назад

hi ive been wondering at what value A will A^x=logA(x) have a single solution maybe make a video out of it? thanks

@andrewkarsten5268 2 года назад

Is this log base A of x or log of Ax? Log base 10 or e if that’s the case?

@JESUSisLORDDd344 2 года назад

would there be a time when u do more induction

@naeemalqise8978 2 года назад

A technician takes X hours to visit the stores in a couple of streets in one shift, and when he finishes his tour, another technician revisits the same stores again, needing other X hours to finish the second shift, and so on. For example, if the first technician started his shift at 1:00pm, he finishes it at 7:00pm and the second technician starts his shift at 7:00pm and finishes it at 1:00am, and so on. The supervisors used to make a quick meeting with all technicians twice a day at 1:00 am and 1:00 pm, so they need to finish their tours exactly 1:00. Identify the mathematical notation for the number of shifts should be made by the technicians in order to achieve this, write the name of this mathematical value, and find it for two tours, one with X=7 and another with X=11 I need the solution please

@sr.tarsaimsingh9294 2 года назад

Everybody knows they are watching RU-vid, But I know I am watching a Legend. 😁

@neowise9264 2 года назад

6:28 Yes senpai

@bhavydugar6665 2 года назад

Is this the only way to prove as we can use the analytical way of splitting the graph of e^ix into the real and imaginary part and obtain cos(x) as the real part and sin(x) as the imaginary part and then obtain this identity

@3manthing 2 года назад

No. I know of a proof via calculus. If you consider a function f(t)=e^(it)/(cos(t)+i sin(t)). f is a nice function, since cos(t) and sin(t) can not be both zero, for any t, so our function is devoid of poles and discontinouties. Taking a derivative f'(t)=(i e^(it)(cos(t)+i sin(t))-e^(it)(-sin(t)+i cos(t)))/(cos(t)+i sin(t))^2 ; f'(t)=(i e^(it)cos(t)-e^(it)sin(t))+e^(it)sin(t)-i e^(it)cos(t))/(cos(t)+i sin(t))^2 f'(t)=(i ( e^(it)cos(t)-e^(it)cos(t)) - (e^(it)sin(t)-e^(it)sin(t)))/(cos(t)+i sin(t))^2 f'(t)=(i ( 0 ) - ( 0 ))/(cos(t)+i sin(t))^2 f'(t)=(0)/(cos(t)+i sin(t))^2=0 This implies f is a constant for all t (f(t)≡A, A is some constant). But if f is constant for every t, we can evaluate it at any t, to see what A is. Since f(0)= e^(0 i)/(cos(0)+i sin(0)) and i 0=0 i=0,e^(0)=1, cos(0)=1, sin(0)=0 so f'(0)=1/(1+i 0)=1/1=1. So f is constantly 1, so e^(it)/(cos(t)+i sin(t))=1, or e^(it)=cos(t)+i sin(t). ▧

@mikethegamedev 2 года назад

Good vid

@28sarthakdwivedi44 2 года назад

Sir can u visit India once ? There are so many here to appreciate your work especially me.....

@vaughanwilliamson173 2 года назад

...and when you sketch e^(i pi) on the complex number plane with the unit circle drawn, e^(i pi) + 1 = 0 is just so obvious!

@mrocto329 2 года назад

buuut you have to prove that r(cos x + isin x) = e^ix which comes from taylor series.

@brouquier7172 2 года назад

If you use tau (equal to 2*pi) instead of pi then you get e^(i*tau) = 1, which to be honest looks even better!

@PwerGuido 2 года назад

It does not

@brouquier7172 2 года назад

@@PwerGuido I think you'll find that it does.

@rajatchaudhary6246 2 года назад

Sir make a group in which you solve our query

@onyxa3144 2 года назад

I'm going for a math minor in college and I can say I'm definetly using this video to impress my family and friends

@wigfranz 8 месяцев назад

We know that e^(iø) = cos ø + i. sin ø and when we substitue ø = 2π, we get e^(i.2π) = 1 Say, take "ln" to both side, we get ln e^(i. 2π) = ln 1 = 0 i. 2π = 0 How could it be? Can you explain this case, why i. 2π can be 0?

@gamerff5454 7 месяцев назад

Taking the natural logarithm (\(\ln\)) of both sides in \(e^{i2\pi} = 1\) yields \(i2\pi = 0\), which is not true. The issue here is that the natural logarithm is a multivalued function for complex numbers, and in this case, it leads to an incorrect result. Thats what chatgpt says

@muhammadmahmoodalam7221 Год назад

What is this man? @blackpenredpen Should we consider pie 3.14159.... or 180 degree angle? Then e to the power i0 will be equal to 1. And put something else if you want a different result of your choice.

@_Domo_ 2 года назад

Well this is epic

@johndoyle2347 10 месяцев назад

Yes, the recursive form of Euler's Identity should be considered the most beautiful and important of lone equations, but the solution to the Basel problem and its reciprocal to define the moments in time where black holes get crushed into dark matter as part of a Big Bounce event, well...

@skltfz4997 Год назад

is e^z just taylor series proof for a=0 case? or if we proof a = 0 is true then it’s true for every a (in the case a= pi)?

@jagatiello6900 2 года назад

7:22 Ladies and gentlemen: Euler's Identity!

@Mono_Autophobic 2 года назад

I have a music video of this equation too

@sonalinegi9651 2 года назад

It also can be solved as following 🙃😅 e^i@ =cos@+isin@ (euler's identity) So, @=π (here) e^iπ= cosπ+isinπ , (sinπ=0 & cosπ=-1) Hence, e^iπ= -1 Now, e^iπ+1= -1+1=0 hence proved.

@user-kw2xs5rs2f 9 месяцев назад

thats only for one value and thats in fact not a proof, by that logic x^2 = 2x if we only consider x=0 and x=2

@NonTwinBrothers 2 года назад

This one has captions let's goooo

@dingle37 2 года назад

I’m an algebra I student trying to understand as best I can... 😅

@janmamu8721 2 года назад

E^(i•tau)

@avunz125 2 года назад

That's a nice validation. but how did we discovered that? Did someone just randomly tried to taylor expand e^ix? Probably not. How to invent math, that is the question.

@terzonxmrphy4119 2 года назад

this man planning to become an assailant sage or something?DAM!!!

@ImTheRisingPhoenix 2 года назад

@blackpenredpen Hello, after watching the video I made a little google search about the topic and people seem to say that the identity is "Euler's proof of God" and they are saying it because the equation is made from all the basic constants from every field of mathematics, could you elborate on that or is it just because the beauty in that math Euler said that in order for this to be true God must be real?

@maxv7323 2 года назад

I don't think most calc 2 students can show that the derivative of e^z is e^z given that the understanding of the derivative that we learn kinda relies on us using the reals (I think)

@General12th 2 года назад

Doesn't the limit definition of the derivative apply in the complex world too? I don't see why not.

@andrewkarsten5268 2 года назад

@@General12th you have to be very careful how you do that. In many cases yes, but you need complex analysis to actually show why it’s reasonable and the answers we get. You cannot just assume that things play nice like that, you first need to prove it before you use it.

@andrewkarsten5268 2 года назад

@Max V you’re right, the definitions taught up to calc 2 and 3 are dependent on using real number inputs. You need complex analysis to extend the definitions into the complex world before you can start taking derivatives of functions with complex inputs.

@krischan67 2 года назад

You may move -1 to the left side, I do not :p

@ruffifuffler8711 2 года назад

The faceted dimensional transition operator annihilates itself during a transcendental time stretch to remain within the singularity, and create the count, and 1st ordinal..

@ruffifuffler8711 2 года назад

The count exists before the 1st ordinal, but has no presence until being recognized by the 1st ordinal, which then: a. bootstraps the count, b. then defines the 5 aspects of the presence of the singularity, c. conducts the euler matriculation of presence to exhaust the count.

@aymanad3437 2 года назад

What are you talking about???

@ruffifuffler8711 2 года назад

@@aymanad3437 the equation has depth antecedents b4 face value appreciation, and is not an identity.

@ernestgrigoras7658 2 года назад

I know this might be offtopic...but is there any solution for x^(x+1)=(x+1)^x ?

@ronaldjensen2948 2 года назад

Notice 2^3 < 3^2 but 3^4 > 4^3. Since both functions are continuous between x=2 and x=3 they must intersect in that range. Therefore, there is (at least) one solution.

@manangandhi3259 4 месяца назад

Here is a queston for you cos2x/1×3 + cos4x/3×5 + cos6x/5×7......infinity

@ewthmatth 4 месяца назад

e^i*tau=i^4 * *runs away* *

@dhrubajyotiray1214 Год назад

isn't this maclern's series

@BilalAhmed-on4kd 8 месяцев назад

Why radians and not degrees

@sakibhasanredwan3468 2 года назад

I want to purchase brilliant premium but i don't have even any card to pay and also i don't have money.You might be surprised but it is normal in my country like Bangladesh.But i hope ,when i would earn, i will purchase brilliant premium version

@xevira 2 года назад

not just irrational... _transcendental._

@miradota347 2 года назад

Ladies and gentlemen, we got em.

@irrafer8663 2 года назад

After 8 time of calculus 1 this is drug for me

@francescodeleonardis7621 2 года назад

I remember that shirt, buy white version, from a Simpsons episode.

@GUTY1729 2 года назад

HERMOSO

@lesarXD 2 года назад

kayra kibar chill dude edit: he deleted most of the comments

@varun3282 2 года назад

Then, e^ipie must be =-1 right.

@RukkuChinnavera 2 года назад

2:12 he paused I laughed

@parinvivek4289 Год назад

1:10 is it just me who got jump scare?

@michaelmapple8201 2 года назад

Am I wrong but wasn't there something that needed to be considered when infinate sum's are rearanged like that? Just curious why it is possible in this case while you can also get pure nonsense out of it othertimes.

@johndoyle2347 10 месяцев назад

i = the square root of e to the i times pi power. This describes stable particles. Teach correctly, sir.

@Peter_1986 2 года назад

"Everybody knows..." is highly context-dependent. You pretty much have to actively choose to study a bunch of optional math courses in high school to even so much as get introduced to imaginary numbers.

@kehana2908 2 года назад

hold on what i learned that like 9th grade math

@MadaraUchihaSecondRikudo 2 года назад

I think what's totally mind boggling about this is that e and pi are supposedly two completely unrelated numbers *found in nature*. e occurs naturally in anything that has to do with exponential growth, and pi occurs naturally in all circles. We did not invent or define those numbers. We observed them. And yet, we've proven that those two numbers are intrinsically linked through the imaginary world.

@user-rz7ek1wx1i 2 года назад

So 1 = 1! more beautiful

@carultch 2 года назад

0! is defined as 1. 1! is therefore 0! * 1, which also equals 1. It is pretty much the base cases of factorials that 0 and 1 factorial, are both equal to 1. Factorial is defined recursively, so it has to reduce to a base case to have any meaning.

@alexmoga8654 2 года назад

Why is this vid unlisted?

@officiallyaninja 2 года назад

how did you find this video 2 days early?

@reiterschwein3828 2 года назад

lol

@alexmoga8654 2 года назад

@@officiallyaninja it was linked on his home youtube page before it was posted lmao

@debajyotidas1 2 года назад

“Everybody knows that i^2=1”. That moment when nobody in my grade knows what I is...

@cubicghostgugajunqueira609 2 года назад

I 10 year end like your vídeos i Brasil.

@yami_the_witch 2 года назад

Not gonna lie, relating it to geometry is much easier. Just imagining it in the context of the complex plane makes it hilariously simple.

@purim_sakamoto 2 года назад

Calc2というのは高校2年生でやるのかい？ほうほう

@Rocky4719 2 года назад

My friends, next time your significant other asks how beautiful you think they are, just tell them they’re as beautiful as Euler’s identity. And then link this video to explain to them what it is