A proof that e is irrational - Numberphile

Proof by contradiction always feels like ending a story with "and then they woke up and it was all a dream"

"R has to be positive because it's just a sum of positive terms." Now that's rich coming from you lol

"e is approximately 3" Smells like ENGINEER in here

More professors and teachers should be like Ed. When you don't really know something at the moment just say "I don't really know".

"e^x is the only function that differentiated it gets back to itself" Zero function: angry analytical noises

"e" who shall not be named...

This feels like an oldschool numberphile video :0

Why e is useful: When you are solving differential equations (which wind up describing an awful lot of things when you look carefully) you get lots of situations where a rate of change is related to the value of a thing. (ex: The rate of bunny births/deaths is related to how many bunnies there are.) When you find a solution, or even an approximation, for these sorts of things, e pops up all over the place. Compound interest, radioactive decay, population modeling, temperature change over time - all involve e.

14:28 "We know that R has to be positive, because its a sum of positive terms". The irony of this being said by the same guy who did the "1+2+3+4+... = -1/12" video.

All mathematicians shoukd write "ta-dah!" At the end of their proofs instead of QED.

The smiliest astrophysicist in the planet is back!

Love that it was pretty close to completely rigorous and had minimal hand waving

Camera man: Why is it important? Mathematician: Wrong question!

Btw if anyone’s curious about how he got that series expansion e^x = 1 + x + 1/2x^2 + 1/3! x^3 ... , a really easy way to verify that this makes sense is to use the property that e^x = d/dx e^x. If you take a derivative of each of the terms in the infinite series, they all kind of “shuffle” down. 1 -> 0 so it disappears, x -> 1, 1/2x^2 -> x, etc! (One of the reasons I think this expansion is so neat is it’s another visual way to see why e^(i pi) + 1 = 0

4:53 The moment you understand the choice of thumbnail

3:26 I am surprised he didn't say the obvious reason: that property lets us solve a whole bunch of differential equations that model physical and non-physical dynamics.

Prof Ed's voice is just so calming. I'm pretty sure I transcended into some dimension of e just listening to this video.

The talk about the derivatives was a bit of a tangent...

3:14 Brady is a brilliant interviewer. I love how he's able to ask "normal human" questions and how those are the ones that experts trip over and make them think. I bet Brady could have asked all kinds of very in depth detail questions about some obscure technicality and Prof Copeland would have had a quick answer but a simple "Why is that useful?" is not a thing that he has thought about :-D

Man I love Ed. It’s a pleasant surprise every time he shows up