The Most Beautiful Proof

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I swear contradiction is in every proof of irrational numbers. I swear.

Math is the best thing that humanity has ever accomplished.

e is e-rational

Small inaccuracy at 3:15: 1/b doesn't have to be STRICTLY less than 1. It could be equal to 1. It make no difference in the proof (there already was a strict inequality in the chain). If 1/b=1 then b=1, that is, "e" would have to be an integer. It is well-known that 2<e<3: this can be proven in an elementary way. So either you shouldn't have said "strictly" at 3:15, or you should have mentioned that 2<e<3, making the denominator of the hypothetical rational number at least 2. Great video, though!

I've honestly never seen a proof that e is irrational before, and now I'm surprised that the proof is so simple.

3:23 technically you haven't ruled out b=1 at this stage so that last < should be <=. But you still get 0 < x < 1 thanks to the other two. That said, nicely done!

right, but how did Fourriere come up with the initial equation for x? it didn't just pop out of thin air thanks to Fourriere's geniality, no. there was a thought process behind that, that led him to deliberately choose precisely this definition for x. The motivation was to analyse the difference between "e" itself (as a sum of all terms of its Taylor/Maclaurin series) and the partial sum of the same series, up to b-th term. then you scale up the difference by multiplying it with b!. and that's what should have been said explicitly in the video, imo: Why are you doing that? why are you multiplying by "b!"? It is to make both, the fraction a/b and the partial sum, integers. The partial sum is an integer bcs you're summing for n=0..b, so b > than all n, and b!=1*2*3*..*b, so b! is divisible by every integer smaller than b => every term of the partial sum is an integer. - so that's why he deliberately chose x to be specifically THAT formula. bcs it makes it easy for him to prove that x is an integer. the second part, x < 1, comes from the fact that factorials grow so quickly and i actually like how the video treats that part.

Beautiful. I had to pause on all the summation manipulations before understanding them, and I'm going to have to watch a few more times to get the rest. (I'm about 70% on board with the inequality at 2:53.) Thanks for the concise, quality explanation.

Okay, now prove it's transcendental. (I'll wait).

wdym? e=3=pi=sqrt(g)

So if you assume e is rational, you can prove there is an integer greater than 0 and less than 1. And you can prove other things like 2 = 6, or Abraham Lincoln was a carrot.

I was a math tutor for 5 years and ive gone about 2 years without actively tutoring the subject or learning it. Gotta say, its an attractive subject but some of this definitely went over my head. I need to sit down and do this by hand to understand it better

This is proof is by Joseph Fourier, and for me it is one of the proofs that I find not too difficult to follow, as compared to proving pi for example. Please do more videos on more famous proofs!

Sir can you make videos on conic sections including ellipse , parabola and hyperbola including its applications and also it's book for self study. Please sir.

Brit, he math guy.

3:00 I thought a^1 + a^2 + a^3 + ... = -1+ (1/(1-a)), because the summation is supposed to start at k=0 to use the expression 1/(1-a). For example, 1/2 + 1/4 + 1/8 + ... = -1 + (1/(1 - 0.5)) = -1 + 2 = 1. This seems like a big problem in your steps, but if that's the case, then we should have an easy route to the proof: 0 < -1 + 1/b, thus 1 < 1/b, thus b < 1, however, no such integer was chosen for b as b was supposed to be 1, 2, 3, ..., or etc.

I am in love with this proof. Do you have something similar for the number "pi"?

I absolutely ADORE this proof. Can’t believe I could follow, and even spent time making sure of parts where you go too fast for me. However, the real genius is in picking x. Where does that come from, what prompts it? Honestly, when I try to understand that, I feel very stupid again. Any insights? Thank you.

Why are those numbers alarmed? Are they stupid?