What are rational numbers in Physics
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- Опубліковано 22 лип 2024
- Here's a question : what are rational numbers (ℚ) in physics?
A pointless question, indeed, but a beautiful one nonetheless...
Things to check out:
⊛ Technical notes on this video : sites.google.com/view/pointle...
⊛ 3b1b, Approximations of π : • Why do prime numbers m...
⊛ Steve Mould, Orbital resonance : • The Planets Are Weirdl...
⊛ Awesome website to play with continued fractions : tiusic.com/quasi_crystals.html (but it's about quasi-crystals, not orbital resonance)
⊛ A good video on KAM theory [not available yet, if you know one, please, leave a comment, I'll add it]
⊛ If you know about any other non-technical online content on a related subject, please leave a comment, I may add it to the list, [if it seems too technical, I might add it in the technical notes]
♥ Thank you to the 3b1b team and all of those who took part in organizing the summer of math exposition ♥
Musics : Moon And Star | Sandviken Stradivarius
by Wintergatan Build Tracks
These tracks can be downloaded for free at www.wintergatan.net
Free License to use these tracks in your video can be downloaded at www.wintergatan.net
Thank you to the Wintergatan team for making it so easy to access.
Sections:
00:00 ⇨ Intro
03:00 ⇨ Part 1: orbital resonance
06:00 ⇨ Part 2: continued fractions
09:00 ⇨ Part 3: continuity of rationality
12:12 ⇨ Part 4: the fake simulation
15:15 ⇨ Part 5: multidimensional rationality
17:35 ⇨ Recap
18:44 ⇨ End credits - Розваги
I do not want to discourage you to be interested in physics, as it is a very nice field. I really like it as well to the point that I'm currently doing my masters in it, so I understand your enthusiasm and don't want to be rude. But there are quite a lot of things really wrong in your video.
1. 1:57 The irrationals having measure zero on the reals does not mean, that all numbers are irrational. There obviously exist rational numbers.
2. 3:00 This explanation of orbital resonance misses the point of Bertrands theorem. The "stable" in the theorem has the meaning, that two points with initial conditions close to each other will stay close to each other indefinetely. This is the case in a 1/r newtonian potential. Other potentials will have points near each other diverging over time. This would be called chaotic in the case of exponential divergence, but there are also many cases with just linear divergence. The important thing is: There are all kinds of potentials with stable orbits in the sense of being "bound" between a lower and an upper limit. This means, that the planet will never escape the system.
One example for this is the planet mercury. Even in the absence of other planets, the orbit of mercury precesses, so isn't closed. This is because of the general relativistic correction proportional to 1/r^3 to the potential. Nonetheless, the orbit of mercury is energetically bound. It cannot escape or fall into the sun.
In actual fact, orbits with a small rational ratio of periods are particularly stable. There are many cases of orbital resonance making very stable orbits in the solar system, like Europa - Io - Ganymede or Neptune - Pluto. I would advise reading the wikipedia article on orbital resonance, as the reality completely contradicts your hypothesis.
3. Your explanation of continued fractions is quite nice. You are correct, that large numbers in the continued fraction expansion are good places for rational approximation. However calling phi "The most irrational number" is mostly meant as a joke. I believe that your ordering of "irrationalness" does not work at all. For example, according to it, Pi is more rational than 2. As Pi = 3 + 1/(...) and 2 = 2, and a_0 = 3 is bigger than b_0 = 2. This doesn't make any sense practically.
4. 13:28 Non-gaseous planets also have tidal effects.
5. You list of these orbital period ratios, you say explain the structure in the rings of Saturn. But in ratio to which other period?
In actual fact, most of the empty strips in saturns rings are due to a moon being in that specific region, gravitationally vacuuming up the dust. However some of the regions really are cleaned up by resonance.
love reading comments like these, learned a lot
"Physical phenomena are always continous" ( 2:30 ) - Are they though?
This is such a great angle of looking at the "rational and irrational numbers share the same number line" topic. I love how you found a physical example to really give weight to this pointless question.
Thanks!
I think one cannot make a nicer comment ☺️.
I'm glad you enjoyed the video.
Awesome video! The engineers will still tell you that pi = 3 after watching it though
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
Congratulations on this. In my opinion:
- Comedy: 9/10
- Topic interest: 10/10
- Sound and music: 9/10
- Animations: 8/10
- Structure of the presentation: 6/10
I find it somewhat confusing in some parts. Is not always clear where you are going and sometimes it feels like some subtopics are concluded prematurely before adressing the next thing. As a physicist this was wonderfull! but I feel that it will be quite difficult to follow for anyone that hasn't completed a physics or maths degree. And it shouldn't be like that since it really is something that intuition can be built in. Overall I consider this an excellent educational video, and for me in particular, quite a fascinating one! But if you want some contructive criticism, that's all I can come up with. If you want to get to a larger audience you need a sligtly better narrative structure next time (looking at open ends to close them or at least to acknowledge that they will be left opened, so the audience can see the mental map behind the explanation). With that in mind I think your videos will be exceptionally good, so kudos to you. I'm happy that I found this channel in its infancy.
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.
Numbers were defined by the ancient Greeks (in Euclid's elements*) to be a name that describes the measure of a ratio of magnitudes.
Only ratios of commensurable magnitudes have a measure, which means √2, pi, and e are not numbers but rather constants, which are the attempted or failed measures of an incompenserable ratio.
Magnitude is the concept of size, dimension, or extent. E.g. length, area, volume, time, force...
Ratio is a comparison of two homogeneous magnitudes. E.g. my height : your height.
Two magnitudes are commensurable if they share a common divisor magnitude. E.g. -- and --- with common divisor -.
Two magnitudes are incommensurable if they share no common divisor. E.g. a square's diagonal and a square's side.
The number 2/3 describes the measure of the ratio -- : --- by telling us how many times the common divisor - measures the antecedent (first) and consequent (second) part of the ratio. 2 = measure (-- : -) and 3 = measure(--- : -) therfore 2/3 = measure(--- : -).
√2 does not describe the measure of the ratio square diagonal : square side since it's incommensurable, which is why √2 can never be written out in a fully descriptive way and is not a number.
(*) Euclid didn't state very well, and translation funkiness.
Dude make more vids, this was great
Really interesting video man, I hope I see more of your stuff in the future
this was a very... unique video to me, because i sort of already knew everything? but i still learned a lot!
i doubt many people would have the same perception, but it was heartwarming, even, to see things i learned on my classical mechanics and orbital mechanics classes, recognise them, understand where and how you got those ideas, and... still be delighted but the whole video
i only wish i had found your channel sooner, because im definitely subscribing
The interesting thing i took away frrom your video, was about how error in measurements really made it so you couldn't know if anything in physics was really rational or not. Since error gives a range and there are an infinite number of rational and irrational numbers left in the possible range. Yes according to mathematics irrational numbers have a higher cardinality than rational numbers, but the thing being measured being more likely to be irrational doesn't guarrantee that it is irrational.
Great video, and told in a very funny way!! Thank you!!
I haven't got the timw for the full video, but the idea here is more or less one of the cornerstones of analytic number theory, particularly the circle method. I rather enjoy how there is a simple indicator function for irrationality which is _sorta_ continuous.
I love it! That was such a fun ride. I more or less understood it...
Subscribed at 481
You’re gonna make it big
You might enjoy this: en.wikipedia.org/wiki/Sol%C3%A8r%27s_theorem
It is an old theorem on quantum foundations. Despite the initial assumptions not mentioning the continuum, it derives that the Rationals are not suitable for quantum mechanics. We need to use something equivalent to a real, complex, or quaternionic Hilbert space.
😍 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.
1D fractals!
woah, algo failed this guy 😭
Your mother, here. Your dinner's getting cold. You've finished lying to those poor people, so come down and eat!!!
@2:26: Prof. Planck would like to have a word with you.
why is this so underrated?
Well how do you prove that e and pi combination is irrational? That's not proven in math yet
9:01, Is the golden ratio the maximum of your >~ operation (most rational number) or the minimum (least rational)?
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.
i've only just started the video, so my bad if this is addressed, but i'm at the part where he says "some use theoretical arguments to say all numbers are irrational" with a wikipedia excerpt about the lebesgue measure
the rationals having lebesgue measure 0 is NOT the same as "all numbers are irrational." this is an analogy: if you have a (mathematically ideal) dartboard, there are an (uncountably) infinite amount of points on the dartboard. now, if you have a (mathematically ideal) dart with an infinitely sharp point, and you throw it at the dartboard. the dart is going to hit one, and only one, point in the endless sea of infinite points, so the probability of the dart hitting that point is 0%. the probability of the dart hitting ANY particular point is 0%, but the dart is definitely going to hit some point. so it'd be ridiculous to say that there cannot be any points on the dartboard because the probability of hitting any particular one is 0%. just as the rational numbers having lebesgue measure 0 doesn't imply that all numbers are irrational, the probability of hitting any particular point on a dartboard being 0% doesn't imply there are no points on the dartboard. it just means there are just uncountably many other points around the one you hit (uncountably many irrationals around the rationals).
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.
You should have submitted it for #SoME2
Don't worry, I have. ;-)
The way you defined your total order, isn't 10 more "irrational" than the golden ratio?
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.
In your planetary sistem if the ratio of the periods is not a simple fraction, the orbit is unstable but over a very long time scale, greater than the life of the star.
Intéger?
so much comedy..
What an interesting pointless question you presented. How would you objectively determine how irrational a number is? Like pi is almost rational while the golden ratio is very irrational. There's a lot of numbers in the middle. I came across a similar motivated construction in my SoME entry tracing flower petals. ua-cam.com/video/-PslG1YMMNM/v-deo.html
I'd be curious to learn more about how to quantify a numbers "irrational-ness". If you've explained it or someone else has please share. Also, I was hoping for your 3 body simulation with irrational areas stable and rational areas unstable visualized. Maybe you can show that in your next video. Although the trolling picture of Saturn was comical. Thanks for the thought provoking entertainment!
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 ~
Just in case you don't have time to check the description, here the blog post I am referring to:
ua-cam.com/users/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
Actual orbits are ellipses instead of circles, so none of this applies. Rational ratios of periods are the norm and are stable.
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.
I like the way set theory defines numbers as how many elements are in a set by recursively putting an empty set in an empty set so it's no longer empty. It's like saying 0+0+0 is 3. Everything made of nothing. Welcome to the universe.
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"
This is actually super interesting! Do you have any resources for this? I loved how you can define a total order on the reals to see how irrational a number is
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…
I don't see rational numbers having any significance in physics. Just integer multiples of things, for instance integer multiples of the charge of an electron or as the case may be, one third of the mass of an electron. There is definitely no way that rational numbers are relevant to orbits, because that all fails when general relativity is involved, as now there are no stable orbits at all and everything spirals in as it radiates gravitational radiation. Sorry but this reminds me of galileo or whichever early scientist thought that the planets in the solar system had some significant mapping to the Platonic solids (when they thought there were 6 of them total). In other words a totally kooky idea that really really doesn't work.
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 :'//
There is no such thing as rational number. We can't divide one number A into another B (in mathematics or physics) unless A is a factor of B. Hence for something like 3/4 = 0.75 = 75/100 = 3/4. It's an operation we cannot compute. Nothing to see here. Mathematics, naval gazing taken to the extreme.
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 UA-cam 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.
ua-cam.com/video/dKtsjQtigag/v-deo.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.