The rotation problem and Hamilton's discovery of quaternions I

The rotation problem and Hamilton's discovery of quaternions I | Famous Math Problems 13a

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Күн бұрын

W. R. Hamilton in 1846 famously carved the basic multiplicative laws of the four dimensional algebra of quaternions onto a bridge in Dublin during a walk with his wife. This represented a great breakthrough on an important problem he had been wrestling with: how to algebraically represent rotations of 3 dimensional space using some kind of analog of complex numbers for rotations of the plane.
This is the first of three lectures on this development, and here we set the stage by introducing complex numbers and explaining some of their natural links with rotations of the plane. There is a lot of information in this lecture, so by all means take it slowly, and break it up by pausing and absorbing the ideas before going further. In particular the last slide (page 9) could easily be stared at for an hour or two.
Even old hands at complex analysis may find something novel here to stimulate their thinking, as I insist on a completely logical and rational approach to mathematics--no waffling with angles or ``transcendental notions/functions'' involving ``real numbers''. In fact such a pure algebraic approach is exactly what is needed to set the stage for a good understanding of quaternions.
In particular you will learn that the most fundamental fact about complex numbers is properly stated using the notion of quadrance, that turns are a viable substitute for angles, and that the rational parametrization of a circle is intimately linked to a quadratic map at the level of complex numbers. These ideas will prepare us for appreciating the rotation problem in three dimensions, which we tackle in the next lecture, and then the introduction of quaternions, which we explain in the following one.
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Пікірлер: 115

@gb590212h 11 жыл бұрын

Let me add that, despite the fact that some of my comments have a critical character, I have a high appreciation for your work. I have learned a lot of beautiful mathematics from your videos and your written work, and I thank you for it.

@smiley_1000 3 жыл бұрын

What an astoundingly beautiful introduction to rational trigonometry!

@njwildberger 11 жыл бұрын

Of course: the best resource (by far!) is my MathFoundations series of lectures.

@njwildberger 10 жыл бұрын

You are quite right: I didn't define division for complex numbers. Thanks for filling in that gap.

@AllanKobelansky 4 жыл бұрын

I appreciate your thoroughness and deliberateness. Well done.

@dhbaek9130 3 жыл бұрын

I appreciate your lectures sincetely! Those will change my life a brand new one! I can not forgot this forever!

@emanueol Жыл бұрын

easy to feel when someone truly sharing in empathic way any idea, thanks 🙏🌟🙂

@pythagorasaurusrex9853 7 жыл бұрын

Great lecture! Being a math teacher myself I know about the wonderful properties of complex numbers. But you introduced me a new sight on these numbers with "quadrance" and "reflections". New for me is the fact, that you don't use functions like cosine, sine and tangens in your lecture. Only using basic calculations with "plus" and "times". Love that. Thanks a lot. Looking forward to watch the other videos about rotation and quaternions.

@silencedidgood 11 жыл бұрын

You rock Professor Wilderberger....Hamiltonian numbers...the math of satellites and robots! I have been introducing my daughters (16 and 13) to these and your video will be one more piece of connecting them up to their future through our past

@7steelrainbow 8 жыл бұрын

Thank you for this great lecture. I was looking for a video to teach myself concepts of and mathematical approach to rotation in 3d space and found your video. It's been a quite long time since I learned complex numbers and basic operations with them in my school mathematics class. I forgot nearly all school-taught knowledge regarding them and I didn't ever know how complex numbers and their important concepts/theorems could be easily explained and visualized on x-y plane. Your explanation was so clear and kind that even a person who have been living very far from mathematics could understand this lecture very well. ( Before seeing this video, what I could think about complex numbers was only this definition of 'i' : i^2 = -1) If someone had told me that 'a+bi' has many interesting stories to be told when seen on x-y plane, having more values than just accompanying an imaginary number 'i' that satisfies i^2 = -1 as printed text , I would have liked those numbers rather than feeling awkward whenever I saw any number with a trailing 'i' in a textbook. :D Now, I'm going to watch the next video.!

@raingloom5334 9 жыл бұрын

This is probably the best "tutorial" I could find about quaternions. No oversimplification, but still understandable with some brain work.

@sijojosephdr 11 жыл бұрын

Thank you professor, for this marvellous lecture....

@allenlu9308 9 жыл бұрын

very interesting lecture style, i appreciated the extra history too!

@boyastro100 10 жыл бұрын

Amazing lecture...I also like your NOVEL thinking about numbers. Thank you very much for your efforts.

@williamolenchenko5772 5 жыл бұрын

Thanks for the ultra-clear lecture and novel approach to the subject.

@njwildberger 5 жыл бұрын

You're welcome!

@pleisterman8660 8 жыл бұрын

Thank you very much for posting your videos. I really like your approach, and i am learning a lot.

@njwildberger 8 жыл бұрын

+Pleisterman Thanks. Do you know I have a Patreon site? It is a community of people that support what I do.

@pleisterman8660 8 жыл бұрын

+njwildberger Hello thank you for your reply, I was really busy so it took a little time. Although i have been watching some of your other videos. I learned math 30 years ago and angels have always baffled me. I even was on the verge of ending my education on this. In the past few years it has become a bit more clear. I will invest a little more time in understanding it and you helped me a lot in getting this in my head. And no, I don't know about your community but i am interested and have a look on the net...

@SimpleStory95 7 жыл бұрын

First I'd like to say that I greatly appreciate the value of theses video lectures and hope you continue to make them well into the future. I am not sure what I believe. I consider myself to be a very well versed amateur mathematician. I think that I believe in all types of mathematics that are consistent, even if they are not perfectly defined (not sure how to phrase that). That being said I find this approach INCREDIBLY intriguing. Whether or not you believe in irrational numbers, this "rational trigonometry" seems to have something to it. At the very least it has definitely opened my mind to really think about continuity, length and angles much more deeply (especially continuity). I already have 100 new questions about something I though I understood quite well. What do you lose when you replace the reals with the rationals? What do you gain? How might the differences between using the rationals verse the reals relate to the hyperreals? What other extensions to the reals could you use? What is the main requirement for continuity? Are many of the beautiful things I've seen in mathematics a result of continuity, or of the reals? And that's just some of the things I'm wondering about continuity! I'm not sure these are the questions you intended for, but I don't think that matters, its more important to me that new questions are being asked (at least for me). So thanks again

@njwildberger 7 жыл бұрын

Thanks for the comment. You will want to have a good look at the MathFoundations series of videos. Although we are approaching 200 of them, I would still recommend to start from the beginning. Hopefully it will be an eye opening experience. The story of modern mathematics with its dependence on " real numbers" is quite unfortunate: we have been fooling ourselves in a major way. The gap between our beliefs and computational reality however is getting so large and obvious that it is time to consider alternatives. My book on Rational Trigonometry can be downloaded for free at my Research Gate page, you could also find it at amazon.

@njwildberger 11 жыл бұрын

Yes there are in fact several good rational alternatives to the usual polar coordinates. One is described in my book, using quadrance and spread instead of distance and angle, and there you can see various applications of it to calculus questions as well as to describing classical curves. There is also another alternative which I call `rotor coordinates' which I will describe in this series at some point.

@njwildberger 10 жыл бұрын

My book is called 'Divine Proportions: Rational Trigonometry to Universal Geometry'. You can download the first chapter to get an idea of the main thrust of rational trigonometry at wildegg.com.

@hippohippi9273 2 жыл бұрын

Love your lecture! Thank you!

@njwildberger 11 жыл бұрын

That's right. The deficiencies with using angles becomes much more obvious in three dimensions; this explains somewhat the poverty of current practice in bringing the obvious richness of our 3 dimensional space to life in mathematics.

@aboyapart 11 жыл бұрын

Thanks for your response. I enjoy your work. It is clear to me that you love to share knowledge and I thank you for that. Whenever I can, I like to analyze things critically. If you could point me in the right direction to some article or video going deep about the problems of irrational numbers, it would be greatly appreciated.

@njwildberger 11 жыл бұрын

Yes the turn really wants to be defined to have values on the projective line, and in this case the projective version of the formula u1 + u2 + u3 = u1 u2 u3 holds.

@aniliitb10 9 жыл бұрын

really impressive lecture!!

@pouncebaratheon4178 8 жыл бұрын

Firstly, thank you very much for this and all your other math videos. They are very clear and fill a much needed niche of videos that otherwise have little to no presence on KZfaq. Still, to beat the dead horse: Do you not lose all sorts of useful properties by avoiding irrationals? As an example, do you have no problem with the fact that for the function y(x) = x^2-2, we have y(1)0, but no rational x such that y(x)= 0? I recognize the computational issues and the notion that working in the rational field can develop mathematical prowess, but the latter isn't terribly different from making students solve differential equations without referencing the complex numbers (it will make them good problem solvers but is also frequently an inefficient solution method), and the former is a product of our biology and technology rather than a question of existence. In particular I think the nature of a qubit as compared to a standard bit of information is of interest. And being that those are a part of the 'real world' in the sense that we've already made quantum computers, it's hard to denounce their existence.

@samferrer 5 жыл бұрын

This is very interesting ... specially for integer optimization ...

@njwildberger 11 жыл бұрын

You should appreciate that the way we have introduced i here is completely algebraic and logical: we have pinned it down to completely explicit multiplication between pairs of (rational) numbers. This way we don't need to philosophize as to whether or not ``i really exists''. It would be nice if we could extend this algebraic approach to other ``irrationals'' but that turns out to be increasingly difficult and problematic as we proceed further.

@Pengochan 8 жыл бұрын

It's quite impressive how far you can get this way. I think it opens another view on things worth keeping in mind. OTOH real numbers and transcendental functions are much too useful to just throw away :) . Nevertheless i can see how this concept nicely fits with quaternion-rotations.

@adityadas.mr.cosmos357 5 жыл бұрын

Well explained ...hiked it.......👍👍👍👍

@samferrer 5 жыл бұрын

I agree with the "non irrational". The smallest unit - Planck constant, is the deepest we con go into a given dimension ... so, we perceive discretely ... hence in terms of rational numbers. Now, as a consequence, all that maters are the natural numbers, because they map the rational numbers ... and thus, we don't need the negative numbers either ... Z modulus does the job.

@DDlol01 8 жыл бұрын

I have to thank you so much! (and another youtuber who sent me here @yourworstfriend AKA Daniel Finlay) I finally under stood how quaternions work (not exactly, but I now have an idea) I have to keep watching part II and III^^ (and I probably will recommending you)

@nlight8769 10 жыл бұрын

Thank you very much for the lecture on this approach, because without any grade in anything, I've been able to understand complex numbers without having to dumbly accept statements that made no sense to me, even though I saw they worked thereafter. Though, I would like to suggest a very slight change (a detail) : after you've explained that (a,b)(c,d) = (ac-bd,ad+bc), you explain that (a,0)(c,d) = (ac,ad), ok, you replace b by 0, that's fine, but next you explain that (0,a)(c,d)=(-ad,ac), even though that is correct, this is slightly misleading just by the fact that where we usually saw "b", we see "a" instead... this is a detail, but saying that (0,b)(c,d) = (-bd,bc) keep things a bit more coherent in the "physical" distribution of the naming... Anyway, thanks again Professor

@njwildberger 11 жыл бұрын

Turns are actually naturally defined between lines, but are oriented, in the sense that the order of the lines matters. In other applications we can ask how to deal with directed lines, i.e. rays. For this the additional concept of half-turn is required. This is implicitly contained in this lecture, though I haven't spelt it out explicitly.

@unknownalien2117 4 жыл бұрын

Thank you so much

@njwildberger 11 жыл бұрын

True: but that equation already suggests how we can deal with division: since 1/z is the same as (z bar)/Q(z).

@njwildberger 11 жыл бұрын

Alternatives to the standard dogma are being developed. In particular, pretty well everything on this channel is designed to show how we can do mathematics concretely and correctly, without the usual waffle with infinite sets and real numbers. In particular the MathFoundations series is attempting to lay out, from first principles, the foundations of a truer mathematics that everyone can believe in.

@njwildberger 11 жыл бұрын

In addition, note that this formula for say u3 in terms of u1 and u2 is a lot like relativistic addition of velocities, in line with an earlier comment of yours on Bayes theorem and probabilities. Also have you written anything on the relation between spread and the lemniscate? That was a particularly interesting comment (sorry for not having responded to it earlier!) that ought to be expanded on.

@yasheesinha8181 4 жыл бұрын

This is so organized and overall a really nice video! Great work! I was wondering, however, that if you ditch angles, how would you go about defining something like, say, a complex logarithm? Or complex exponential's period for that instance. Is there still a way to go about it or do we just not use those ideas in this analysis?

@MrJosephArthur 11 жыл бұрын

Nice introduction to complex numbers connecting only with rationals. Questioning the non-rational numbers is a very interesting task in order to improve the rigor of mathematics, especially the mathematical analysis based essentially on the real number system. But, a reformulation of the analysis without considering the current real numbers might create other problems, maybe in applications due to the new way of interpreting the numbers. Though, it is much better the true objects, of course.

@danodet 11 жыл бұрын

I'm glad you appreciated my comment relating the lemniscate and the spread. With your permission, I might tried to add this to the wikipedia page about spread.

@nicholastan3027 9 жыл бұрын

WWWWWWWWWOWWWWW this video is awesome

@hier0phant336 8 жыл бұрын

thank you

@njwildberger 10 жыл бұрын

You're welcome!

@njwildberger 11 жыл бұрын

I have an article `Set theory: Should you believe?" that you might like to look at, along with all the MathFoundations videos, of course!

@njwildberger 11 жыл бұрын

15 is a fine age to start thinking seriously about mathematics. Just make sure you have fun!

10 жыл бұрын

thanks, good job!

@deemotion 6 жыл бұрын

Very Good !

@phpnepal 10 жыл бұрын

Thank you sir! This is exactly what i was looking for, for my game engine.

@lidorshimoni5470 2 жыл бұрын

how this approach goes on your engine?

@phpnepal 2 жыл бұрын

@@lidorshimoni5470 sorry it has been a while. i think i used this for fps engine.

@STIVESification 8 жыл бұрын

Thank you

@aboyapart 11 жыл бұрын

Thank you very much, Professor! The article's title got me excited because I have asked that question to myself. Definitely want to look at it, where can I find it?

@pickeyberry4060 Жыл бұрын

Video Content 00:00 Introduction 06:24 A(rational) primer on complex numbers 27:47 While quadrance is a rational analog of length,what is a rational analog of angle?

@alibalkan4202 3 жыл бұрын

Thank you very much! The way you explain is just great.. if a fathead like me understands what you teach everybody will

@theoremus 3 жыл бұрын

Thank you Prof. Wildberger. You have just stimulated me to think about slope or turn in the case of a Cavalieri shift condition. I will have to investigate this further.

@yousifucv 10 жыл бұрын

Hello WildBerger, thank you for your videos. I watched this video and there are some things I don't get, and I hope you can clarify. 1. At 44:30 you mention the fact that a line through the origin does not necessarily intersect the unit circle, how would that be the case? 2. At 46:40 you say that the line that goes from O to z^2 meets the unit circle when you divide it by its quadrance. Then you proceed to divide it by Q(z), not Q(z^2), how come? 3. It's not clear to me how you would get that point on the unit circle by dividing by either Q(z) or Q(z^2). Is it a similar idea to how you can normalize a vector by dividing by it's modulus?

@hansleet 9 жыл бұрын

Here's how I understood it (please correct me): 1. He said it depends on the point's parameters. Some points would require the use of an irrational square root which he tries to avoid. On the other hand the point from 2. and 3. always exists in the rational sense. 2. This confused me aswell, but when you think of the analagous vector normalization, you divide the vector by its length. Here you divide z^2 by one side of its quadrance ( i.e. its 'hypotynuse-area' ), again you skip the square root because you already know Q(z) ( the area of Q(z^2) would be Q(z)*Q(z), but we just need one side of the square ).

@pbierre 9 жыл бұрын

To say that a line thru the origin might not intersect the unit circle at 2 points is appealing to the notions that • infinite precision is possible for rational numbers, but not the irrationals • a 2D line that appeals to irrational numerics to represent it is not as mathematically airtight as a 2D line that can be defined with infinite precision using rational numerics. Example: the 2D line at 30.0 degrees tilt requires irrational numbers to describe it using any representation based on [ dx dy ]. • since the rays emanating out from the origin are "peppered" into the two categories having rational vs. irrational numerics to stand behind them, then you would conclude that the unit circle is also peppered with point values can be similarly categorized. The viewpoint treats lines and intersection points having irrational numerics as non-existent. I'm stating the argument, not arguing for it. There is another more mainstream view of numerics that says that irrational numbers don't pose any practicall barrier to accuracy and precision. If you have the time and the need, you can spend it pursuing as many decimal digits as you want for the value of square-root(2). There are an inexhaustible supply of decimal digits defining this quantity, so imprecision can be moved off the table as an issue no matter what practical level of precision is required. People in scientific computing understand (at least those with training) the pitfalls of finite precision math operations, and routinely avoid any negative consequences. The world we live in would be greatly impoverished to dismiss irrational numbers as untrustworthy, and no major disasters can be traced back to the admixture of irrationals and rationals in the practice of mathematical software apps.

@hlogoma 9 жыл бұрын

Pierre Bierre I agree. Not to take away from Professor Wildberger's generosity in providing all of these videos for free and thereby exposing an amateur in mathematics such as myself to a wider arena of mathematics, I do think that in the interest of balance one should not restrict oneself to just this viewpoint. To that end, David Deutsch provides I believe, a more balanced, scientific and closer argued presentation on the question of infinity and irrational numbers.

@Anonymous-by5jp 2 жыл бұрын

@@hansleet Alex, this is off topic but are you any kin to the "father of modern analysis"?

@paul1964uk 11 жыл бұрын

A great lecture. I was going to query how we could expect to see turns of more than one-half expressed uniquely at that rate if we map to 'double' of each turn on the circle. But I see that using the vertical line Re(Z)=1 is enough provided the turns are made to be oriented (and therefore to go beyond a half turn we see a change of sign?)

@njwildberger 11 жыл бұрын

Thanks. Please let us know how your teenage daughters react to all this lovely stuff! I'm not confident my teenage daughter has quite the same level of enthusiasm.. :)

@njwildberger 11 жыл бұрын

Infinitesimals are a dubious example to support your statement; although they are commonly used informally by applied mathematicians and engineers, they are not officially part of the standard analysis set-up (although non-standard analysts have tried to introduce them, largely unsuccessfully).

@njwildberger 11 жыл бұрын

The map is constant on all multiples of z, but this is not really a disadvantage. It allows us to associate rotations to lines through the origin in a general field. Not sure about the geometric interpretation of the turns formula.

@gb590212h 11 жыл бұрын

Thank you for the reference to Zeilberger, I'll look up his work. My problem now is not convincing myself of the problems with the standard theory of real numbers and infinite sets. Rather, my problem is, how do these objections translate *concretely* into my mathematics? It seems I have the choice between A) a standard theory that has some serious flaws, but is familiar to everyone who has studied math, and B) a *non-existent alternative theory*...

@eltodesukane 11 жыл бұрын

u1 + u2 + u3 = u1 * u2 * u3 correspond to Tan A + Tan B + Tan C = Tan A Tan B Tan C for the angles of a triangle (with A+B+C=Pi).

@njwildberger 11 жыл бұрын

Charade is a bit strong, don't you think? There is a video in the MathFoundations series called the Magic and Mystery of ``pi''. There I discuss different approaches to what this is. I end that lecture in an indefinite way: claiming that ``pi" is probably a lot more complicated an object than just a regular number like 2/3. One day someone might tell us what exactly this is, in the meantime we can use it in an applied sense; a certain number of decimals, having various approximate properties.

@njwildberger 11 жыл бұрын

I don't really use pi in my book until I get to the last, applied, chapters. It certainly does not figure in the main set-up of rational trigonometry.

@pieinth3sky 7 жыл бұрын

I wish you left a link to this video under WildTrig15. It's kind of hard to realize that this video with quaternions in the title and quaternions in the first 7 minutes is sort of continuation of WIldTrig15. I'm also pretty sure that all results here would work in quadratic extensions of rationals or even in quadratic closure of rationals. I also managed to pull out an approximation of pi out of this, but can find the way to get the approximation of e. Phi(z) * phi (w) = phi(zw) clearly tells that there is an exponentiation with some base b around, but I can't figure out how to pull out the algorithm to show that this base b = e.

@anontrolo 3 жыл бұрын

Hey I think I found a neat proof of 33:10. From definitions, it looks like u(z,w) = u(wz*), where z* is the complex conjugate. u(zv,wv) = u(wv(zv)*) = u(Q(v)wz*) = u(wz*), as multiplication by Q(v) will not change the slope of any vector and essentially acts as multiplication by a scalar.

@njwildberger 11 жыл бұрын

Not all my viewers are able to watch all the videos; after all, there are now quite a few. So the occasionally viewer who just watches way one of the FMP series might be interested to know that I consider `irrational numbers' highly suspect (to put it mildly!) That might motivate them to investigate more deeply in the MF series, or elsewhere. But thanks anyway for your suggestions, I do accept that your position is a reasonable one, just one that I cannot follow for my own reasons.

@njwildberger 11 жыл бұрын

Conservative fields can be defined over the rationals.

@njwildberger 11 жыл бұрын

There are different levels of understanding. First one should realize that simpler methods can be used to avoid irrational numbers in many cases. A second level is to understand that the current theory of irrationals is logically dubious, almost to the point of being fraudulent. An implication of the latter is that calculus needs to be developed much more carefully. It is still possible. Finally mathematics is far from dogma, which follows us wherever we go. See my MF series for more!

@peterkiedron8949 2 жыл бұрын

@njwildberger 11 жыл бұрын

Okay, I see your point. But getting people to reconsider their positions re ``real numbers'' is one of my important obligations. For example, perhaps you would not be so likely to rethink things if I had not stated my objections strongly, and for me, honestly.

@gb590212h 11 жыл бұрын

At around 36:30 you begin commenting on the question of the additivity of angles (which does not hold for turns or spreads). You say that this additivity comes at a very heavy price, and that this basically comes from the fact that we are "forcing linearity on the circular structure." That's not entirely fair, because quadrances too lack the additivity of their "traditional" counterparts, distances, and it would be a stretch to say that the heavy price we pay for the additivity of distances...

@njwildberger 10 жыл бұрын

Check out my paper `Set Theory: Should you believe?' available at my website under Views.

@christophergame7977 5 күн бұрын

I can now try to point my question. Let me vaguely define a 'sheer rotation' as a 'rotation of a three dimensional object that preserves nothing but the object itself'. Is that enough to try to say what I mean? My question is 'does a quaternionic rotation preserve anything other than the object itself?' Or 'is a quaternionic rotation a sheer rotation?' Can a quaternionic rotation be expressed by exactly three numbers? I am asking why are four numbers used for a three dimensional rotation? Are three numbers not enough to rid us of gimball lock? What does the fourth number in a quaternionic rotation tell us?

@njwildberger 11 жыл бұрын

How about spelling out somewhere what the exact statement is, and give a proof? That is a good starting point, before we start posting on wikipedia.

@Vinaykumar-ob8pt 8 жыл бұрын

Why do you call the result given for multiplication of two complex numbers to be a definiton? (within 11:26) It is just a result obtained by multiplication right. Edit: I see you did mention it to be a result obtained by simple multiplication, considering i=-1. But, even then, I don't understand why you said in the first place, the result to be definition.

@harshverma6425 8 жыл бұрын

Thats because he doesn't believe in i , which is sqrt(-1), like he doesn't believe in sqrt(2).

@njwildberger 8 жыл бұрын

Actually there is something of a difference between the way we usually think about sqrt(2) and sqrt(-1). The point here is that defining multiplication of ordered pairs of rationals gives complex numbers without invoking the existence of a separate entity called i=sqrt(-1); it arises naturally.

@cetjberg 6 жыл бұрын

Dear Professor Wildberger, I can work out the correct form of the turn from your (a1, b2) to (a2 b2), but only by reverting to the Euler Formulae. Do you have your method written out somewhere that I can read? I, perhaps like you, have always disliked those formulae for the sins, etc, of the sums of angles -- I never remember them, always have to derive them. I have a more serious request. Please consider this. What if your distance a is inherently irrational. What if you choose a triangle with sides, (1,1) and then rotate the coordinates so that a2 lies on the hypotenuse of the first triangle. Will not a2 be inherently irrational? Patience please, I'm a retired engineer and professor, and I will be 85 soon (I hope.) In closing let me say your lectures are superbly well organized and presented. Thank you. All Best, Charles A Berg

@njwildberger 6 жыл бұрын

Many questions about the form of turns etc come down to using the rational parametrization of the unit circle. I would suggest you try that. As for your question about what happens to the hypotenuse of a right (1,1) triangle if you rotate it: why not try it and see? Make yourself an isosceles right triangle with equal sides equal to 1 out of cardboard. Now rotate the triangle so the hypotenuse is along your ruler. Now measure the hypotenuse's length. Tell us what you get. But you might say: what about doing that mathematically, not physically? The point is that then we are working in a quite different framework. We no longer can just assume that words such as "length" and rotate" have intuitively obvious meanings. We must struggle to define those, and be prepared that they may not have well-defined meanings!

@cetjberg 6 жыл бұрын

Thank you.

@cetjberg 6 жыл бұрын

I thanked you for your reply before. I know what you mean by the cardboard triangle, of course. And as for your reference to the "mathematical" construction of the hypotenuse, I am reluctant just to accept that some of our concepts are not meaningful. Here's the way I see matters: You have correctly brought out the historical evolution of the mathematical concepts of area and Pythagoras' theorem. What I see in your excellent discussion of this matter is the evolution of a new and very useful level of abstraction. So that abstraction has certain points in it where the meaning of symbols are undefined. Well, I'll use the abstract level at the points where the symbols are well defined; I'll avoid the undefined areas. For example, if I am dealing with Euler's equation, exp(i*Pi) + 1 = 0, I will not try to put in the exact value of Pi, I will use a 7 or 9 or 15 digit approximation of that value -- obtained from a Leibniz series. I will obtain a rational number (the approximation that I choose) and proceed. I hope this is a better answer to your comment than my earlier one. I believe we must use the various levels of mathematical abstraction carefully, but use them nonetheless. Otherwise, we will always be defining concepts, and never making any material progress. As Before, All Best, Charles A Berg

@wdobni 5 жыл бұрын

in this new way of thinking are we still allowed to say that there are an infinite number of possible lines through the origin? or, rationally, there are only a finite but unknowable number of possible lines through the origin.

@njwildberger 5 жыл бұрын

In this new way of thinking we want to just stick to saying true and correct things. It is not necessary that there be a certain "number of lines through the origin". Number is something that applies to specified collections: one gathers together objects of a certain kind, and then can ask how many there are. But asking for the "number" of an unspecified collection of objects is not ambiguous. The reason that "lines through the origin" does not give a specified collection is because these things are indexed by ever increasingly large equations, and we are unclear when those will spill over whatever data storage we have.

@gregsamerson4255 10 жыл бұрын

Wait, if i squared is k squared and if k squared is j squared, then wouldn't i times j times k be i cubed, k cubed, or j cubed?

@njwildberger 10 жыл бұрын

Remember that in this algebra, the commutative law does not in general hold.

@gregsamerson4255 10 жыл бұрын

njwildberger Ah, I'm also confused as to what the difference between (0,1) and (0,-1) would be, since squaring either gives you (-1,0).

@njwildberger 10 жыл бұрын

Hi might say surprising, I would say suspicious.

@njwildberger 11 жыл бұрын

I don't see the logic of your argument here. Just because my remark doesn't apply as strongly when applied to a different situation doesn't diminish its validity for the domain I was addressing.

@njwildberger 11 жыл бұрын

On the contrary, I can show you a 4-vector very explicitly: here is one: v=(1,4,5,-3). I don't have to philosophize about whether or not `four dimensional physical space exists'. As long as we can write down explicit concrete objects, with coherent operations etc, then we don't need to insist on any one interpretation. Thus you should be skeptical of the usual gobbly-de-gook reasoning which insists that `irrational numbers'' exist without properly exhibiting one.

@TheReligiousAtheists 4 жыл бұрын

I don't quite get what you mean, here. A simple and rather trivial demonstration is drawing a circle with radius 1 unit and asking what the length of it's circumference is (measured in those same units). Would *you* consider that a proper demonstration? If not, why?

@mohammadbinmahbub9160 3 жыл бұрын

rationally speaking: if i=j=k==sqrt(-1), then how is ijk==(-1) ????

@jolez_4869 3 жыл бұрын

The ijk property is just a definition like i=sqrt(-1).

@streamapp 9 жыл бұрын

So you know people have proven that certain numbers of are irrational, right? Like pi, for example. The proofs aren't that hard, so why do you insist that irrational numbers don't exist?

@njwildberger 9 жыл бұрын

David Jackson One problem with your statement that it is not clear what `certain numbers' refers to. Do you have a prior theory of `numbers'? If so, what is the definition of this term?

@streamapp 9 жыл бұрын

njwildberger How about pi?

@streamapp 9 жыл бұрын

njwildberger I mean I see that you mean symbols aren't numbers, but irrationals are phenomena that shouldn't be dimissed as "not numbers" because they have significance...And they can be expressed as ratios, just often relations of physical space...like pi

@njwildberger 7 жыл бұрын

Please see my MathFoundations videos, and/or my paper: Set Theory: Should you believe?

@njwildberger 11 жыл бұрын

Unfortunately the orthodoxy is omni-present in modern mathematics. There are, at present, only a few dissenters---but of course that is going to change! Doran Zeilberger is another prominent skeptic, you could look for his opinions. Another thing to do is to actually look carefully at existing so-called introductions to `real numbers'. Have a look at some calculus textbooks, and see what kind of wishful hand waving occurs when this subject is being introduced!