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A joke in approximating numbers raised to irrational powers

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Re: A joke in approximating numbers raised to irrational powers

#31

Post by brianush1 »

NameError wrote: post #4

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…

pretty sure you have a typo, should be "If not, consider (sqrt(2) ^ sqrt(2)) ^ sqrt(2)."

Re: A joke in approximating numbers raised to irrational powers

#32

Post by nomemory »

wmwmwm wrote: post #27
m463 wrote: post #20

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!

Fun fact, in the book Life Of Pi, the kid stays exactly 227 days on the boat with the tiger.

Re: A joke in approximating numbers raised to irrational powers

#33

Post by xelxebar »

dotancohen wrote: post #15
sevensor wrote: post #12

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

> That's the first organ of the Taylor series

Guessing that "organ" is a typo for "order", but somehow I kind of like envisioning Taylor series as living organisms, with terms being individual organelles.

Thanks for the smile in the morning.

Re: A joke in approximating numbers raised to irrational powers

#34

Post by tzs »

xanderlewis wrote: post #30
tzs wrote: post #11

Earlier quoted context omitted.

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…

> To prove that e^r is irrational for irrational r You mean for rational r, don’t you?

Yup!

Re: A joke in approximating numbers raised to irrational powers

#35

Post by lanstin »

epistasis wrote: post #26
jbmsf wrote: post #24

Happy to see someone else who watches Michael Penn videos.

I came here to say the same thing! 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 th…

math does every 30 or 50 years simplify stuff. It's hard for the originators to do it, they get so familiar they are able to get weird intuitions that make the difficult tractable. I listened to some of the simple groups people talking about it, and they just had crazy detailed knowledge about all sorts of group properties and prime properties and so on. Totally inscrutable without devoting your life to it.

Re: A joke in approximating numbers raised to irrational powers

#36

Post by seanhunter »

xelxebar wrote: post #33
dotancohen wrote: post #15

Earlier quoted context omitted.

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

> That's the first organ of the Taylor series Guessing that "organ" is a typo for "order", but somehow I kind of like envisioning Taylor series as living organisms, with terms being individual organelles. Thanks for the smile in the morning.

No Taylor liked to communicate the series musically on various organs. IT got expensive and that’s why noone ever goes beyond the first two or three terms.

Re: A joke in approximating numbers raised to irrational powers

#37

Post by enugu »

wging wrote: post #22
enugu wrote: post #18

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...

Small but important correction: no polynomial with integer coefficients (equivalently, rational coefficients). p(x) = (x - 3^(sqrt(3))) is a perfectly fine polynomial with real coefficients.

Yes, I should have mentioned polynomials with rational coefficients(or indeed any algebraic numbers as coefficients due to transitivity of being algebraic).

Re: A joke in approximating numbers raised to irrational powers

#38

Post by seanhunter »

NameError wrote: post #4

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…

Well the proof I would use is let a = e and b = i(pi).

e^(i theta) = cos theta + i sin theta (Euler's identity) thus e^(i pi) = cos pi + i sin pi = -1 + i(0) = -1

We know that e and i pi are irrational (in fact i pi isn't even a real) and -1 is rational.

Therefore there exist two numbers a and b such that both a and b are irrational but a^b is rational.

In fact log of just about anything is irrational so e^(log x) works as well for just about all rational x, but Euler's identity is cool so I wanted to use that.

Re: A joke in approximating numbers raised to irrational powers

#39

Post by brookst »

seanhunter wrote: post #36
xelxebar wrote: post #33

Earlier quoted context omitted.

> That's the first organ of the Taylor series Guessing that "organ" is a typo for "order", but somehow I kind of like envisioning Taylor series as living organisms, with terms being individual organelles. Thanks for the smile in the morning.

No Taylor liked to communicate the series musically on various organs. IT got expensive and that’s why noone ever goes beyond the first two or three terms.

Any sources to Bach that up?

Re: A joke in approximating numbers raised to irrational powers

#40

Post by selimthegrim »

seanhunter wrote: post #36
xelxebar wrote: post #33

Earlier quoted context omitted.

> That's the first organ of the Taylor series Guessing that "organ" is a typo for "order", but somehow I kind of like envisioning Taylor series as living organisms, with terms being individual organelles. Thanks for the smile in the morning.

No Taylor liked to communicate the series musically on various organs. IT got expensive and that’s why noone ever goes beyond the first two or three terms.

I wonder what Grassmann did
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