In this follow-up video to his “e to the i pi for dummies” video the Mathologer sets out to properly explain the coolest features of the famous number e and the exponential function e^x. Find out WHY e is irrational, how you go about calculating the first 1,000,000 digits of e, WHY the exponential function e^x is its own derivative, etc.
Here are links to the videos that I refer to in this video:
e to the pi i for dummies: (this is the video I summarise at the beginning)
Indeterminate: the hidden power of 0 divided by 0: (about derivatives, among other things)
Math in the Simpsons: e to the i pi: (this is the video that I refer to at the very end)
This week’s t-shirt I made myself. Check out this wiki page about this pretty identity
Thank you very much to my friend Marty Ross for proofwatching drafts of this video and helping me to get the words “just right” and to Danil Dmitriev the official Mathologer translator for Russian for his subtitles.
Enjoy!
source
1/{ e ^ (π/2) } = ( i ) ^ i
bruv she homer on my simpson till im mathing it
Thanks a lot! very well explained and now it's clear to me why the derivative of e^x is e^x same as why (e^nx)' = ne^nx!) that's absolutely great and astonishing.
Homer Simpson 😊😊😊😊😊
178482301^(1/19) is sick close to e. There seems to be an integer sequence {1, 3, 8, 19, …} (I lack the computing power and/or know-how to go further) such that e to this power is closer to an integer than any lower value (3, 20, 2981, and the aforementioned 178482301). (I've excluded 0 for obvious reasons.) Haven't checked OEIS. But this means that 3^1, 20^(1/3), 2981^(1/8) and 178482301^(1/19) make closer and closer approximations to e, and these are the local minimum error cases.
I typed e on google this video also homer simpson lol
The symbol ”Ü-mlaut” sponsoring the ”Simpsons”-episode must be a tribute to Burkard’s German-ness 🇩🇪😃😅.
The only confusing aspect for me is the usage of log notation for natural logarithm, instead of ln. Otherwise great video, thanks!
I saw your more difficult video discussing Eulers e first. So I was happy to also come across your "Dummies" video Thank you.
You probably didn't even intend this but I have a deeper insight into hyperbolic geometry because of watching this video.
8:40 I think you mean: ”The error, we get, by chopping off, at the Nth term, is LESS, THAN 1/N!.”; seeing, as we arrived at 1/N!, by simplifying the bracketed expression; which, in this case, made the bracketed expression bigger; and, indeed, the expression:
”error < 1/N!”, on screen, is correct. 🤔
At 16:50, Homer would have applied Simpson's rule to calculate the area under the curve 🙂 🙂 🙂
I never understood logarithms before but now at least I have an inkling about how they are used. It seems to me that smaller and smaller measurements of slope between points on a graph function is the basis of calculating the derivative. Why not just say that the derivative is the exact value of that slope?
The Euler identity comes from the Euler formula which is obviously wrong because one side is the exp function ranging to infinity but the other side is finite (sum of 2 trig functions each ranging from -1 to 1). So, both Euler 's inventions are wrong.
I enjoyed the explanation….it's very clear without looking complicated….I recall the quote from Einstein that if you can't explain it simply, you don't understand it well enough….you did it…thank you…
I found this to be an informative , well presented an complete addition to my own line of studies an though process. Thank you.
e=n^n/(n-1)^n for n –> infinity My math formula 😀
thank you, best explanation because it actually explains why, which i think is the most important to understanding
It worked well
Explanations are clear, thank you
Very good will watch again just for review.
I’m a dummy, does this mean that pi = ln(-1)/i?
You are genius!!!
hey so what's (∞/∞!)? could we define that as an approximate ratio, for practical computation purposes? if you call it zero then that'd basically contradict the small-angle approximation method, right?
Lost me in under 5 minutes.
Very understandable.
informative video about e and pi these are some other videos https://youtu.be/21_ezQTkxK0 https://youtu.be/dbHLxn5D1cU https://youtu.be/Xu-xAi1VSQE
7:47 If you instead replace all the factors in the denominators with 8, then the error would be less than 1/(7×7!)=0.0000283447… which is a much closer upper bound of the true error which in this case would be 0.0000278602…
I wonder how much closer you can get to the true error while still "simplifying" the right hand side to a small-ish closed form expression in terms of N, where N is the number of terms you take from the series, i.e. in this case N=8.
Very confusing video😂
I cant replicate what you have shown after watching this video but I did understand everything you did.
Great video. I will definitely watch it again
e can be expressed in a fraction, look: 1+(1÷(x!))
Another great video from mathologer! Except for one part (at least for me), which is the proof, that e must be irrational: Just slightly change 19/7 to 18/7, and the unequality holds (18/7 minus the chunk in the brackets is a fraction less than 1/7!). It's easy to see, that this is true for a/b in general, as long as there are no further restrictions to the integers a and b, and as long as we are not talking about the absolute value of the result (Mathologer did not).
Am I wrong? If yes, I'd appreciate a short explanation, why. In that case, pls. keep it 'simple & stupid', because I'm not a mathematician, and English is not my first language…