[HN Gopher] A joke in approximating numbers raised to irrational...
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A joke in approximating numbers raised to irrational powers
Author : nomemory
Score : 102 points
Date : 2024-11-18 16:41 UTC (2 days ago)
(HTM) web link (www.andreinc.net)
(TXT) w3m dump (www.andreinc.net)
| xdavidliu wrote:
| for a second I thought 404 was the joke. Tried thinking hard for
| maybe 10 seconds to figure out why it was the joke, but then
| realized it was not.
| nomemory wrote:
| An unfortunate mistake...
| NameError wrote:
| 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 actually
| have to know whether sqrt(2)^sqrt(2) is rational for the proof to
| work.
| tzs wrote:
| 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 irrational for irrational r it suffices to prove
| that e^n is irrational for all positive integers n.
|
| We'd also get the e is irrational out of that by taking n = 1,
| and that would complete our proof that e^log(2) is an example
| of irrational a, b with a^b rational.
|
| So, all we need now is a proof that e^n is irrational for
| integers n > 0.
|
| The techniques used in Niven's simple proof that pi is
| irrational, which was discussed here [1], can be generalized to
| e^n. You can find that proof in Niven's book "Irrational
| Numbers" or in Aigner & Ziegler's "Proofs from THE BOOK".
|
| That can also be proved by proving that e is transcendental.
| Normally proofs that specific numbers are transcendental (other
| than numbers specifically constructed to be transcendental) are
| fairly advanced but for e you can do it with first year
| undergraduate calculus. There's a chapter in Spivak's
| "Calculus" that does it, and there's a proof in the
| aforementioned "Irrational Numbers".
|
| [1] https://news.ycombinator.com/item?id=41178560
| cruegge wrote:
| I think a = sqrt(2), b = log(9)/log(2) with a^b = 3 is
| easier. To show that b is irrational, assume b = n/m for
| integer n, m. Then 9^m = 2^n, which can't be the case since
| the lhs is odd and the rhs is even.
| xanderlewis wrote:
| > To prove that e^r is irrational for irrational r
|
| You mean for _rational_ r, don't you?
| tzs wrote:
| Yup!
| tyilo wrote:
| Also see https://math.andrej.com/2009/12/28/constructive-gem-
| irration... for a similar proof using 2^(log_2 3)
| brianush1 wrote:
| pretty sure you have a typo, should be "If not, consider
| (sqrt(2) ^ sqrt(2)) ^ sqrt(2)."
| seanhunter wrote:
| 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.
| nightpool wrote:
| Kinda weird that we don't get a graph for the final "solution"? I
| was looking forward to seeing how it compared to the other plots!
| nomemory wrote:
| Good idea. I think I've rushed into publishing this. Anyway,
| I've reposted the link, and I will do the graph you've
| mentioned.
| parsimo2010 wrote:
| After following the correct link from @nomemory in the comments,
| this is good for a bit of a chuckle once you see the formula. If
| you can evaluate the formula you probably have a calculator or
| computer on hand and could compute the original value to double
| precision (I'm not even sure that the approximation would compute
| faster, but I didn't benchmark it).
|
| But even though the approximation has no value in a real world
| application, the description of getting to the approximation is
| really good. I've never heard of Pade approximations before, and
| I liked the lead in from small angle approximations and Taylor
| series. I'd say this post is accessible to (and can be
| appreciated by) advanced undergraduates in engineering or math or
| comp sci.
| nomemory wrote:
| https://news.ycombinator.com/item?id=42174105
|
| New submission here.
| sevensor wrote:
| 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.
| pkoird wrote:
| I am not sure I understand. Sin(x) approaches x only when x
| approaches 0. When else does the universe does a bad job with
| this approximation?
| philipov wrote:
| sin(x)=x in the same way that c=p=1 when doing cosmology.
| bubblyworld wrote:
| At least you can often recover the constants after the fact
| with dimensional analysis in cosmology =P
| mr_mitm wrote:
| 1=c=G=hbar and sometimes =k is not even a joke, that's
| just natural units. Pi=e=1 however ... is only half a
| joke, because cosmologists are often only interested in
| orders of magnitudes, and even those are sometimes
| approximated.
| adgjlsfhk1 wrote:
| the joke is that sometimes the universe is bad at making sure
| x always approaches 0.
| dotancohen wrote:
| 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
| sevensor wrote:
| Oh yeah, for sure. And if you like a good time, compare the
| Taylor series at x=0 for sin(x) to that for exp(jx).
| xelxebar wrote:
| > 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.
| seanhunter wrote:
| 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.
| brookst wrote:
| Any sources to Bach that up?
| selimthegrim wrote:
| I wonder what Grassmann did
| xg15 wrote:
| Was thinking of organ pipes and imagining it as the first
| tune. Might also be fitting.
| m463 wrote:
| pi = 3.2
|
| (that is an assignment statement)
|
| https://en.wikipedia.org/wiki/Indiana_pi_bill
| wmwmwm wrote:
| 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!
| defrost wrote:
| That was a near miss for the industry, real aerospace nerds
| use 355/113 ...
| nomemory wrote:
| Fun fact, in the book Life Of Pi, the kid stays exactly 227
| days on the boat with the tiger.
| setopt wrote:
| Since e^(2pi) = 1, we can also conclude that e^(2pifx) =
| 1^(fx). This makes Complex Fourier Transforms quite trivial.
| enugu wrote:
| 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...
| immibis wrote:
| No polynomial with rational coefficients. Of course x-y
| evaluates to 0 when x=y, even if y is a transcendental number.
| wging wrote:
| 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.
| enugu wrote:
| Yes, I should have mentioned polynomials with rational
| coefficients(or indeed any algebraic numbers as coefficients
| due to transitivity of being algebraic).
| jbmsf wrote:
| Happy to see someone else who watches Michael Penn videos.
| epistasis wrote:
| 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 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.
| lanstin wrote:
| 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.
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