21–30 of 44 posts
One interesting result implies that numbers like 3^(sqrt(3)) will be transcendental (ie no polynomial will evaluate them to 0). https://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theo...
One interesting result implies that numbers like 3^(sqrt(3)) will be transcendental (ie no polynomial will evaluate them to 0). https://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theo...
sin x = x Half the problems in EE become trivial once you learn this. Sometimes the universe does a bad job of complying with the approximation though.
sin x = x Half the problems in EE become trivial once you learn this. Sometimes the universe does a bad job of complying with the approximation though.
Are you familiar with the Taylor series? That's the first organ of the Taylor series, something like two decades ago I checked how accurate it goes past 20 organs: https://dotancohen.com/eng/taylor-sine.php
Happy to see someone else who watches Michael Penn videos.
YouTube has become a fantastic place for this long tail of content, in this particular case a bunch of interesting math problems and tricks presented on a blackboard. Or, even full classes, from a person focused on honing pedagogy.
3blue1brown is another amazing channel for math as well.
I have a feeling that this sort of content is the seeds of very great things for humanity. In the 20th century, ET Jaynes talks about how people never get credit in academia for creating simpler paths to greater understanding. But with YouTube, creators can both reach an audience and also find patrons to support them, or maybe even make a living off of YouTube directly with enough viewers.
Motivated students have such resources at their fingertips just from an internet connection, if they happen to get lucky enough to find the right resources.
sin x = x Half the problems in EE become trivial once you learn this. Sometimes the universe does a bad job of complying with the approximation though.
pi = 3.2 (that is an assignment statement) https://en.wikipedia.org/wiki/Indiana_pi_bill
Earlier quoted context omitted.
pi = 3.2 (that is an assignment statement) https://en.wikipedia.org/wiki/Indiana_pi_bill
My aero engineering friend from university winds me up every time I see him saying that pi = 22/7 - I finally stopped getting angry, checked and it’s pretty good! I’m still glad he didn’t decide to design planes after he graduated though!
Reminds me of a cool proof I saw recently that there are two numbers a and b such that a and b are both irrational, but a^b is rational: Take sqrt(2)^sqrt(2), which is either rational or not. If it's rational, we're done. If not, consider sqrt(2) ^ (sqrt(2) ^ sqrt(2)). Since (a^b)^c = a^bc, we get sqrt(2) ^ (sqrt(2))^2 = sqrt(2)^2 = 2, which is rational! It feels like a bit of a sleight of hand, since we don't actual…
I wonder what the easiest to prove example of a, b irrational with a^b rational is? The easiest I can think of offhand would be e^log(2). To prove that we need to prove that e is irrational and the log(2) is irrational. To prove log(2) is irrational one approach is to prove that e^r is irrational for rational r != 0, which would imply that if log(2) is rational then e^log(2) would be irrational. To prove that e^r is…
Reminds me of a cool proof I saw recently that there are two numbers a and b such that a and b are both irrational, but a^b is rational: Take sqrt(2)^sqrt(2), which is either rational or not. If it's rational, we're done. If not, consider sqrt(2) ^ (sqrt(2) ^ sqrt(2)). Since (a^b)^c = a^bc, we get sqrt(2) ^ (sqrt(2))^2 = sqrt(2)^2 = 2, which is rational! It feels like a bit of a sleight of hand, since we don't actual…
I wonder what the easiest to prove example of a, b irrational with a^b rational is? The easiest I can think of offhand would be e^log(2). To prove that we need to prove that e is irrational and the log(2) is irrational. To prove log(2) is irrational one approach is to prove that e^r is irrational for rational r != 0, which would imply that if log(2) is rational then e^log(2) would be irrational. To prove that e^r is…
You mean for rational r, don’t you?