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Which combination?

love reading comments like these, learned a lot

I think I can respond to some of your replies here: Reply 2: The creator does put a video link of orbital resonance in the description by Steve Mould, which is an extremely good video on the topic. And in the video, he *exactly* explained the reason why orbital resonance happened: One key requirement is *elliptical orbit*, only then you can have the angular momentum restortion. Hence, the setup in this video is essentially different. Circular orbit can't have orbital resonance, and thus angular momentum restortion. But I do think the presentation of a Saturn ring is quite confusing and not a completely ideal case discussed here any more. Regarding the "divergence", its probably just a wording problem. If you read the blogs cited in the description written by the creator, he said: "The reason why I can say that it "diverges" is because I am looking for cases where the blue planet would keep a circular-like orbit. It's an additional constraint from which I can deduce the following : "if my planet did keep a circular-like orbit, I would be able to get it with perturbation theory; but I am not getting it, so the planet doesn't have one." It doesn't diverge as in "goes to infinity in a finite time" but it leaves the circular orbit. It "diverges from its original trajectory".I should have simply said that the orbit was unstable, but "diverging" is so much cooler. " The creator is too informal here... he used a quite confusing word. Reply 3: Regarding the most irrationality, it's actually a thing. You can look up Hurwitz's Theorem, www.wikiwand.com/en/Continued_fractions#A_property_of_the_golden_ratio_%CF%86. Golden ratio is one of the most difficult number to be approximate by rational number. This is also the reason why golden ratio appeared so often in plants or animal hair. You can look it up for Mathemagician video, I remembered that he once maked an excellent video explaining that. For the actaul ordering, I havent read into it, and I am not sure whether the "ordering" of irrationality is well-defined. However, I believe it may be workable by considering the upbound error when approximating, similar to idea meantioned by Hurwitz's Theorem. This definition is related to looking at next integer in the continued fraction. But again, I think the creator did not explain his idea well.

@@abeliangrape953 I'm still not convinced. On Reply 2: Assuming circular orbits works in many cases, but obviously not in this one, because the red planet definitely disturbs the circular orbit of the blue planet. Forcing it to stay circular leads to a wrong conclusion. After reading his article, I still think he's wrong. He says: "Perturbation theory works according to the following scheme: 1. Do the math 2. If it doesn't diverge, the solution will look like that 3. If it does diverge, the solution cannot be obtained through perturbation theory." The he goes on to do some calculations (which he does not show) and then divide by zero somewhere. He claims that this shows, that it's not possible to do perturbation calculations here, and the orbit doesn't stay on the circular path. This is not how perturbation theory works. Instead you perturb the easy equations scaling with some small parameter (like the red planets mass). Then you assume that the solution looks like the unperturbed solution with some error term. Then you derive a linear (although in this case second order) differential equation for the error term. If this implies exponential divergence for the error term, the system diverges, if it shows oscillation or decay of the correction, the solution is stable. In both cases, perturbation theory works and gives you an approximate solution for small deviations from the unperturbed solution. Dividing by zero somewhere doesn't show you that the system is instable, but that you are either in a singular point, or (more likely) that you made an error somewhere. Indeed I'm not convinced, that he even divides by zero at all. He writes: "Here, ω is actually the vector (ωr, ωb). And N is the vector (n₁,n₂) in ℕ² over which we sum. ωr and ωb being commensurable means that : ∃ n₁ n₂ ∈ ℕ : n₁.ωr=n₂.ωb so ∃ n₁ n₂ ∈ ℕ : n₁.ωr - n₂.ωb =0 i.e. ∃ N ∈ ℕ² : N.ω=0" The last line doesn't follow from the ones before, because the second line is the scalar product between (ωr, ωb) and (n₁,-n₂). But (n₁,-n₂) is not in ℕ². Indeed the scalar product of two vectors with strictly positive entries will never be Zero. Also, he is right to say, that a circular orbit is impossible in this system. But this is trivial, as there is not a central potential here. Nevertheless, it is perfectly possible, to have a slightly non-circular orbit in the same place. (given the perturbation is sufficiently small.) So this isn't able to explain the gaps in saturns rings. On reply 3: I'm aware, that the golden ratio is especially hard to approximate and often jokingly called the most irrational for that reason. I'm also pretty sure, that the ordering he defines is well-defined. My point is, that it totally fails in capturing anything about "irrationality", saying for example that pi is more rational than 2. This is just nonsensical.

His point wasnt that mathematically, all numbers are irrational. It was that a certain family or physicists would argue that all "measurable values" in physics would be irrational. i.e. you will never find a stick with a length of exactly 2 meters, it will be an irrational number close to 2 (according to this interpretation) This is converse to, for example, some of the greek mathematicians who believed that all numbers could be constructed with a compass and rule, so would believe that any stick you measure the length of would have a length belonging to the set of constructible numbers. The other example given in the video was that all our measurements have finite precision, so will only ever be rational numbers. It was more of a metaphysical/philosophical point, not a mathematical one.

Yeah I wonder what topology over the reals that would lead to… I mean, using that order to build a triangle inequality, and see what a norm respecting that inequality would be like…

The golden ratio is the MOST IRRATIONAL and so the LEAST RATIONAL. Sorry if I didn't make that clear enough, the "ir" in "irrational" tends to not be as easy to hear as I thought, I should have paid more attention to that.

Sorry, perhaps I misunderstood your message, but I do assure you that both Rational and Irrational exist. There is also a very well defined operation called division that can be applied on any of them. As for what you call "not being able to compute" it is true that is some meanings of computability (like "getting all the digits in a finite number of operations) some divisions are not computable. However, in the case of divisions "n/d" with "n" and "d" integers the exact value can be computed in any basis in a finite number of operations (due to the fact that their digit sequence will either be finite or periodic). And in any case, at no point is "computablity" required here. So yes, there are things to see here. As for the expression "naval gazing" (which I don't believe you intended to be a compliment) I do agree that (although their exist some) it is hard to see useful applications of this subject. Nevertheless, note that the name of this RUclips channel was not chosen at random😆. Here are links backing up what I said at the beginning of the message: en.wikipedia.org/wiki/Real_number en.wikipedia.org/wiki/Rational_number en.wikipedia.org/wiki/Division_(mathematics)#Of_real_numbers

@@pointlessquestions5774 What's the answer to 3 divided by 4?

@@noahbody9782 Their exists a single answer (for each structure you build to BE the rationals). However that answer has many names. The most famous name is "3/4" pronounced "three fourths" but you probably also know "6/8" "9/12" or "30/40". In the traditional way these all refer to the set A={ (3.k, 4.k), k∈ℕ*}. But in the context of real numbers, it may also be the set B={X∈ℚ^ℕ : limⱼ Xⱼ-A = {0} }. Note that the set B is often written using a "base notation". For example, in base 10, B is written "0.75", but in base 4 it is written "0.3" and in base 2 "0.11". Yet if I may, all of this was already explained in the Wikipedia articles I sent you. You will notice here, I did not define the expression "lim" (well, it is defined in the Wikipedia articles). I do encourage you to read them if you still don't see why 3/4 can be computed. If you don't feel like reading all of it, I suggest you check this simplified video (and the three flowing ones) it will only take a few hours. ruclips.net/video/dKtsjQtigag/видео.html&ab_channel=AnotherRoof If you have neither the time to read Wikipedia or to watch that video, I can only ask you to trust me : I know most of those rudimentary things, and I can tell you (from experience, knowledge or whatever) that they do exist, and that in most of the meanings of "compute" one CAN compute "three divided by four". Sorry if that sound a bit patronizing, I'm doing my best. Hope that answers all your questions.😁

@@noahbody9782 The (rational) number such that when multiplied by 4 gets you 3. Do you want a definition of 4 and 3 as well?

and obviously you are going to write it as .75 , you are working with a denary number system.

I'm not sure I fully understand what you are saying but you might be interested to know the following things: -You are talking about multiples of charge of the electron (so I feel I can assume you ave heard of Quantum Mechanics). Consider to quantum states of the same object (let's say ↑ and ↓ for the example). If you now put your system in a symmetric superposition the state is (↓± ↑) / √2. The "√2" arising from the norm constraint of QM. But if you consider a more special superposition that consist of 3 times ↑ and 4 times ↓ you get (3↑±4↓)/5. Now "√2" is irrational, but 5 isn't. So irrationals are currently unavoidable in modern physics. (This example is actual based on the Pythagorean triples, so you can, of course, find similar things in GR, but they'll be more complicated to write). -Speaking of GR, indeed orbits are NOT stable. But know that Newtonian physics is simply the low filed approximation of GR, I wouldn't call them "not relevant" (especially knowing that orbital resonance can create instability of higher magnitude than the second order term of GR). If you don't feel convinced by my argument (which, I agree, is kinda circular), perhaps you might be interested to know that the phenomenon of instability of rational orbit is in fact much more general, and can be extended to any phase space of any (non-quantum) physics. Here's a link, if you want to check it out : en.wikipedia.org/wiki/Kolmogorov%E2%80%93Arnold%E2%80%93Moser_theorem Of course, when I say "can be extended to any phase space" it doesn't mean that this makes orbits stable in GR, just that you can see "different patterns" emerge for rational and irrational values. -I think you are thinking of Johannes Kepler (not Galileo). Note that, my claim is based on Newton's equations (which I wouldn't call kooky) and (on a historical note) that in the time of Kepler, the measurements of the orbits of planets (and their imprecisions) allowed for this theory. Note that he (actually Tycho Brahe, his boss) made more precise measurements to verify this hypothesis and proved himself wrong (and found what are now known as Kepler's laws).

@@pointlessquestions5774 I love how so many people are like "rational numbers don't exist !" when they're everywhere :'//

😍 That was so interesting, thanks. I don't fully get it yet, so I'll have to read more about it, but thanks for introducing me to it.

wow that sounds super interesting, I mean, that means a lot- There is always the question "yeah but what if it's just tiny discret intervals", proving it cannot work - at least for quantum system's phase spaces - is a huge deal !

Woah, wait, that page doesn’t seem to mention an assumption that the division ring in question be characteristic zero, but the conclusion implies characteristic zero. So… I guess that rules out QM over finite characteristic? At least, for infinite dimensional vector spaces.

I don't think so (because the first value compared is the integer part, when it exists). But you are correct, I kinda lied when I said that the golden ratio was the most irrational number.😂 And it has to do with the integer part : the most irrational number would be the continued fraction with 1s everywhere EXCEPT for the integer part, that would have to be 0. But (and this is why I allowed myself to lie on this point) that number would be Φ-1. And it turns out that Φ-1=1/Φ. So the TRUE most irrational number is the inverse of the golden ratio. The golden ratio is only the second most irrational number. But since in my example (jupiter) I was looking at the fraction of period T₁/T₂ it still works fine knowing that T₂/T₁ will be the most irrational one. Physically the reason why it doesn't matter whether you look at the 1rst or 2nd most irrational number is because since there are an uncountable number of real numbers, it corresponds to an infinitesimal difference.

@@pointlessquestions5774 Can we agree that in the continued fraction expansion for 10, the a_0 coefficient is 10. And for phi, the a_0 coefficient is 1. Therefore, by your definition of the total order, 10 > phi. So maybe you mean the "greater" means "more rational"? But then that doesn't work with your claim for zero, as the most rational. That's why I don't understand how you are using this total ordering to say some numbers are "more rational" than other. Maybe I'm misunderstanding something you are saying here, but I cannot figure out where.

@@purplepenguin8452 Yes, you are perfectly correct. The "greater-like" symbol does mean "more rational". I don't think you are misunderstanding much : you are just missing the part about the length : if a number has a shorter continued fraction than another, it is more rational. (see n_x and n_y at 8:22) And in the particular case of 0 (I say it at 8:45 but I agree it was hard to spot) it has "no coefficients" hence it being the most rational number. If you are ready to read a bit, I have re-written our complete discussion in Latek to properly (and rigorously) answer like I should have done from the start : www.overleaf.com/read/mmkzswjvfvks Hope this makes things better ! Thanks for being so interested in the video. Feel free to ask any other question.

@@pointlessquestions5774 Thank you for the long form reply. That makes more sense now with the length (not sure how I missed that). This gives me an idea. It feels a bit weird that transcendental numbers aren't somehow "more irrational" than algebraic numbers. One could make a new definition that compares first the length if finite, then if the sequence ends in repeating one could compare based on some combination of (length before repeating part, length of repeating part), and then finally if there is no repeating in the infinite sequence one could use your comparison of the first different element. This way we'd get a clean separation: any rational > any algebraic number > any transcendental number. (This order wouldn't work so well for your purposes, but it is interesting to me that it is possible.) Anyway, thanks for the explanations!

@@purplepenguin8452 You are raising a very interesting point! I also wondered about the problem transcendental numbers and there continued fractions. The thing is, it seems to be a complete mess. Some transcendental numbers have a very "regular" continued fraction (check e, it's almost periodic) while others (like π) are just a mess. I believe that the reason why it is painful to extract is because algebraic numbers are both THE MOST RATIONAL ONES (fractions are algebraic) and THE MOST IRRATIONAL ONES (𝛟 is 1+√5/2 after all). But to be honest my actual knowledge on the topic is close to non-existent 😅, so I'm not going to be of any help, sorry. If you do find interesting stuff on the subject, please, to send it, I'd love to know more.

I'm glad you've found such a trivial solution to the problem. I wonder why I didn't see that. Jokes aside : I'm aware that orbits are ellipses (in fact, circles are ellipses), and I'm aware that taking that into consideration changes things which is why (at 19:43) I pointed out the APPARENT contradiction. If you want to understand the broader picture, consider checking this blogpost : sites.google.com/view/pointlessquestions/bonus-content/what-are-%E2%84%9A-in-physics It goes deeper into the problem (the section about the phase space contains your answer).

@@pointlessquestions5774 I didn't ask a question, but circular orbits are inherently unstable. At the first closest approach between the planets, planet to planet attraction would deform the orbit from circular. There is no time to average out the the inner planet's force effectively to be a ring even when the periods' ratio is irrational.

"At the first closest approach between the planets, planet to planet attraction would deform the orbit from circular" Indeed. There are only two cases where circular orbits can exist (1/r² and r constant central forces, it's Bertrand's theorem). However, this doesn't prove that "circular orbits are inherently unstable" Stability doesn't mean that the orbit will remain perfectly circular. Stability means that the orbit will remain in a close neighborhood of the circular orbit. Close being defined by a polynomial of the mass ratio (an ε, if you know what I mean). This infinitesimal is always true at the beginning (when starting with the correct speed and orientation) but need not remain true at all time. Hence the study of long times with perturbation theory. Now, speaking about time, when you say "There is no time to average out the the inner planet's force effectively to be a ring even when the periods' ratio is irrational" Indeed, "there is no time to average out" the forces get averaged out (again, see the blogpost : sites.google.com/view/pointlessquestions/bonus-content/what-are-%E2%84%9A-in-physics) Now, if you still believe that my whole video is wrong (I mean, my explanations might be very bad but I am talking about the fact that rationality has a destructive influence on the stability of orbits, in the case of a perturbation), I suggest you now refer to the theorem which I am trying to describe : en.wikipedia.org/wiki/Kolmogorov%E2%80%93Arnold%E2%80%93Moser_theorem Here's it's wikipidea page. In it, you will find the link to the first (imperfect) proof of this theorem (ie Arnold, Weinstein, Vogtmann. Mathematical Methods of Classical Mechanics, 2nd ed., Appendix 8: Theory of perturbations of conditionally periodic motion, and Kolmogorov's theorem. Springer 1997) And also links to better proofs and generalizations. If I misunderstood the theorem, please do tell me how. If not, please do understand that I do not know the complete proof and can therefore not keep arguing with you. I did what I could (of course, feel free to ask me questions if some sentences in the blogpost sites.google.com/view/pointlessquestions/bonus-content/what-are-%E2%84%9A-in-physics feel unclear, I'd be happy to correct that).

@@pointlessquestions5774 All central forces can have circular orbits; you only need F(r)=mv^2/r for a circular orbit. Only F=kr and F=k/r^2 necessarily have closed elliptical orbits. The converse of your statement that the outer planet is too light to affect the inner planet's orbit is that the inner planet is massive enough to alter the outer planet's orbit far more than epsilon; a minor perturbation cannot be applied. There is a middle ground where both planets are perturbed, but that leads to stable elliptical orbits with rational period ratio. Another possibility is that the inner planet is very close to the star and the outer planet is very far from the star so that the ratio of periods is a few tens. In this case the force from the inner planet is a small perturbation on the force from the star and both orbits can wander over "blurry circular bands" when the period ratio is irrational. Actually, having a denominator > 20 or so is "irrational enough" in the real world where other sources of gravity occasionally contribute.

Thanks for all the critic. Let me give you more information : Sound & Music : no glory here, I borrowed a friend's expensive mic and did not compose the music. Musics : Moon And Star | Sandviken Stradivarius by Wintergatan Build Tracks These tracks can be downloaded for free at www.wintergatan.net Animation : well, I think you are being too generous here, I made them with PowerPoint (I did not have time to learn manim) Structure : I'm not quite sure if you dissociate the narrative from the structure, but you are correct, this is probably the worst aspect of my work. To answer the particular point you have raised (unclear bits and never really concluding), I was simply not able to do better. Since you are a physicist, you probably missed it, but the concepts I am dealing with (chaos, continuity, density...) are concepts that cannot be understood without a very rigorous (and thus long) math lesson about them. I chose to talk about "imprecision" to avoid talking about open sets; I chose to talk about "diverging" to avoid the word "chaos" etc... If you want less fuzzyness, I recommend that you check the following blogpost I wrote (with readers like you in mind) sites.google.com/view/pointlessquestions/bonus-content/what-are-%E2%84%9A-in-physics It makes everything so much clearer (but requires actual math). Last but not least : English is not my native language... I guess it has consequences of some sort... Thank you for your very constructive critic (although, as you can see, I was already aware of most of it). If you are interested in helping me seriously improve, I'm open to co-writing stuff. On the blogpost's website you can find the ideas I have for future pointless questions : sites.google.com/view/pointlessquestions/future-projects I am just a lonely physics student, so I would accept any kind of help.

Wow, big comment, let me answer step by step -First, thank you very much -Nice video, although since you only consider rational numbers, it's no longer really in my expertise... -Now for the big part: when you say "quantify" what do you mean. Do you mean "if I have two numbers, I can say which one is the most irrational" in which case I did give the answer in the video. (Pause on the definition of x ~<~ y and read it carefully, you'll get there, it's the lexicographic order). But if you mean "can we get a measurement of how rational a number is?" Than the answer is NO. (I mean yes, but not the way you expect it). No in the sense that there is not "A WAY" to do that, but many different ones. This is due to the precision problem (or the fractal nature of rational numbers). A MEASURE could be (for instance) the 15th digit in the continued fraction. Another could be the 235th. Or you could take the sum of all coefficients multipled by powers of 0.1 (if you want a nice way to do that, check out the question mark function). But these methods are all different, because each gives a different weight to each degree of precision. And if you want a measure that has infinite precision, then it simply the boolean function "this number is a rational one". This is also why I didn't talk much about the simulations (I have done a few) the simulation's results depends on the time scale I choose and the precision I choose WHICH IS NOT NEGLEGABLE in a chaotic sytem. Hence my not going into any details. I have (technically) answeres all of your questions but if you are interested in these, I recommend that you check the blog post I made that goes deeper into thoses details. I think it'll make things clearer for you because your question requires the understanding of the concept of chaos, phase space, and density wich I am not going to explain in a RUclips comment 😅 Hope my answer satisfies you. If not we can start a long conversation on questions.pointless@gmail.com Thanks for watching (and reading)

Just in case you don't have time to check the description, here the blog post I am referring to: ruclips.net/user/redirect?event=video_description&redir_token=QUFFLUhqblhwa1N1UEx1UVlIYUZWRVlyWnFpbXJTNm9QZ3xBQ3Jtc0tsQzBtaTkwc1ZSMTBZU2NreUFnU2dGT2Q4SExabjBFSHZ5a2taeTZmV21ZNVFybkZJVzRleUg3c2wtTXZHcmx1Rmp6TGN6SGhmMnBhNW9raHkwc0tMTkIxRENNZjZ5VFc2U1pnNUg1ZjhUbTNrdzlFUQ&q=https%3A%2F%2Fsites.google.com%2Fview%2Fpointlessquestions%2Fbonus-content%2Fwhat-are-%25E2%2584%259A-in-physics&v=4FfQSBaTyjw

Thanks! I think one cannot make a nicer comment ☺️. I'm glad you enjoyed the video.

No it doesn't say 0+0+0=3, it says 3={0,1,2}={0,{0},{0,{0}}}

@@antoniusnies-komponistpian2172 yea, recursive or nested, that's what I meant by that.

Numbers come from geometry, not set theory. Go read the elements. The ancient Greeks had this figured out long time.

Also, you can't get something from nothing for the same reason that no line can have any area.

@@bananamanjunior7575 numbers have been conceptualized many different ways depending on theory and culture, amazingly they still seem to work together. There is no simple ”this is it & how it is"

Don't worry, I have. ;-)

Thank you so much. As for engineers, they now know the justification for that : " 7 ≈ ∞ " (ouch, it's even worst)

​@pointlessquestions5774 come on, you just didnt have the guts to finish that pi approximation and get the 22/7 engineer love so much