Eulers number is irrational - proof & song

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This song shows you, why e is irrational.

TheGermanFox

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"e is irrational and Euler proofed it. „
"Let‘s have a look at Fourier‘s proof.“

phosphor

Next video: Fermat's last theorem - proof & song (10h)

crymp

Lyrics:

*Chorus:*
Euler's number is irrational.
Euler proved it so there is no debate.
Euler's number is irrational,
and it's about 2.718

*Verse 1: LHS is an integer*
Leonhard Euler was the first one to prove it,
and he used the simple continued fraction expansion to do it.
But we will do it now in another way following a proof which was first introduced by Fourier.
And the series: '1 over n factorial' will be,
our starting point as the definition of e,
where n starts at zero and goes to infinity.
And now let us see, what happens if there would be,
and integer p, and a natural q, such that e equals p over q, okay!
And since we've had all natural numbers in our series, not just a few,
we must at some point find our q.
Let's subtract everything up to this point and if you, do now multiply by the factorial of q,
then right at the start, you might already see, q factorial over q must always be,
an integer since you can cancel out q, and cancelling out is what we our gonna do with the rest of these fractions because we can see,
the whole denominator will now always be contained inside the numerator so we can say,
this cancels out and leaves an integer, okay!
And we can now conclude, for the left hand side, it must always be an integer, Alright!

*Chorus*

*Verse 2: 0<RHS<1*
And if we now take a look at the right hand side,
we will again see factorials,
which we have to divide,
only this time the numerator factorial of q is contained in the denominator, which leaves you,
with one over a product, and there you can see,
this product just gets longer and longer,
and if we observe every single factor is bigger than q,
so every single factor must be at least 2,
then we know, if everytime we only write 2, the denominators will have to get smaller, so you know,
the whole fraction has to be _bigger_ right now but for this series,
I'm just gonna show you how,
you can visually see what this has to be,
just take a half and a quarter and an eighth and see,
how it gets closer and closer to 1,
so you know, this series converges to 1.
But didn't we show that our right hand side must be smaller, which is tough,
becuase we stay greater than 0 if we add positive stuff,
but there clearly is no integer between 0 and 1,
so this is the point at which we have won,
because this contradiction right here shows you why,
no matter which p and which q you would try,
it can never work, and therefore we see:
e can't be rational, Q.E.D.

*Chorus* ...281828459045235360287471!

Approximately!

Mystery_Biscuits

Did anyone ever think that for making videos like these you have to know Maths, Music, English and also be a fairly good poet? Wow, this guy's a really talented person.

nickpro

Richtig gut mit dem Schwibbogen im Hintergrund!

TTR

Amazing; the proof itself is unexpectedly simple and the presentation is brilliant. Subbed.

technoguyx

Yes! I’ve been wanting this for a while. I love your appearances in mathvengers and I love your quadratic formula song but I don’t speak German. Hopefully this means you’ll be doing more english stuff :)

fanyfan

Imagine Eminem covering this masterpiece

atkv

Already listened to this about 50 times, lmao, this is brilliant, btw. speed 1, 25 is also cool

michaldvorak

Duuude this is awesome! The song is catchy, the proof is quite elegant and your explanation is excellent. I love the “QED” part and the “aproximately” parts. And the best part is that we can see your excitement in the video! I hope you do more videos in english because I don’t know german haha.
I found you thanks to the Mathvangers video, it’s so good to see that there are many youtubers starting to make fun math videos!

AndresFirte

I wish I had a teacher like you..
Brilliant sir.
Love from Afghanistan

aliyankhan

Klasse! Der wievielte take war das? :D

juliusalbe

Yo we want to collab with you on another math song. We want you to make an appearance in the video, you can help with lyrics if you want. Respond if you’re interested :)

boblincent