Fourier's Proof that e is Irrational

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We prove that e is irrational by examining its series expansion and asking for a little bit of help from our friend Joseph Fourier.

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I warn you that the proof is much harder and requires the knowledge of Calculus I (and maybe also Calculus II).

LetsSolveMathProblems

This is the best proof of the irrationality of e I've seen that's understandable at a precalculus level. Well done.

nestorv

An integer between 0 and 1...those are rare indeed

uchihamadara

Very elegant proof. Easy to understand. Thanks for posting! You've earned yourself a subscriber!

chessandmathguy

Take a bow Mr Fourier. That is tres elegant!

adandap

I really appreciate your work, it's just so great the way that you show us how to solve these problems, and enlighting us with so cool maths. Thank you so much, a warm hug from Colombia ^^ .

thetheandrein

It was really beautiful and elegant.
Thank you for sharing it and thanks to the Great Joseph Fourier for establishing such a nice and easy-to-understand proof.

mohammadmoradi

Thank you for making your videos quick, easy, and clear.

haydnenthusiast

The way you explained this proof was very clear and concise. Best explanation of this proof by far.

WaynPain

That was beautiful. I was very excited to be able to see the idea behind the rest of the proof after you reached the point of distributing b factorial. Watched this while taking a break from an assignment, excellent use of time. Can't wait to watch more of your videos.

StNick

This proof is so elegant! e is my favorite number!

alkankondo

I found another method to prove x is smaller than 1. I just used the smallest possible values of b.
For b = 0 (impossible since e=a/0): we would get x=e-1.
For b = 1 (impossible since e=a and e isn't an integer): we would get x=e-2.
For b >= 1 (possible): x would be smaller than e-2 and 0<e-2<1

thomasq

Thanks man, that was a pleasure to watch!

saitaro

thank you. my professor did this problem in class and explained it poorly and be couldn't answer our questions so looks how to prove this and you helped me. the problem was he didn't explain or expected us to know was that this a geometric series and we can apply that formula. terminology is soooo important and him not saying the right things left us sooo confused. thank you 😊

axelvaldovinos

This is an astonishingly elegant proof. I actually feel a little emotional knowing how intelligent human beings can be. Just wow. I can't wait to study math at a higher level!!!

monikaherath