What if we define 1/0 = ∞? | Möbius transformations visualized
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- čas přidán 31. 05. 2024
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Defining 1/0 = ∞ isn't actually that bad, and actually the natural definition if you are on the Riemann sphere - ∞ is just an ordinary point on the sphere! Here is the exposition on Möbius maps, which will explain why 1/0 = ∞ isn't actually something crazy. And this video will also briefly mention the applications of the Möbius map. As is the case for all videos in the series, this is from Tristan Needham's book "Visual Complex Analysis".
There will also be things like circular and spherical inversion, which are really neat tools in Euclidean geometry to help us establish lots of interesting results, this one included.
This video was sponsored by Brilliant.
Video chapters:
00:00 Intro
02:38 Chapter 1: The 2D perspective
08:43 Chapter 2: More about inversion
14:33 Chapter 3: The 3D perspective (1/z)
19:38 Chapter 4: The 3D perspective (general)
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SOURCES:
[That 2012 paper] Rigid motion 1-1 Möbius map: scholar.rose-hulman.edu/cgi/v...
Möbius transformations revealed: • Möbius Transformations...
Accompanying paper: www-users.cse.umn.edu/~arnold...
Unitary iff rotation: users.math.msu.edu/users/shap...
Möbius iff sphere: home.iitm.ac.in/jaikrishnan/M...
Rotation of Riemann sphere: people.reed.edu/~jerry/311/ro...
Circle-preserving implies Möbius: onlinelibrary.wiley.com/doi/e...
Problem of Apollonius video: • Problem of Apollonius ...
Power of a point: www.nagwa.com/en/explainers/7...
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MORE CONNECTIONS OF MÖBIUS MAPS:
Sir Roger Penrose lecture on the book with Rindler (Spinors and space-time): • Sir Roger Penrose on c...
The book: www.cambridge.org/core/books/...
Hyperbolic geometry: assets.cambridge.org/97811071...
Conformal mapping (fluid mechanics): math.berkeley.edu/~iliopoum/T...
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Music used:
Aakash Gandhi - Heavenly / Kiss the Sky / Lifting Dreams / White River
Asher Fulero - The Closing of Summer
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I know that this title is quite clickbait, but hopefully this is a good enough exposition on Möbius maps. There are lots of links in the description to help you learn more as well, because I have missed something there, like group properties, or its relationship with matrices (which give rise to the PSL(2,C) I mentioned in the end) [Actually I thought about “generalising" the idea of 1/0 = ∞ to rational functions and more general still, meromorphic functions, but this video is long enough.] Again, this is basically an animation of the book Visual complex analysis by Tristan Needham, so that the ideas can come alive. Hopefully you will like this video!
1/0 can be -infinity
or -infinity =+infinity
like -0=+0
1/-0=1/+0
@@user-bt1hf9cr5n In some sense, yes! The infinity here corresponds to infinity in all directions, so it can be approached via the positive real axis, in which case, it would intuitively be called +infinity, or the negative real axis, or -infinity; or the imaginary axis with positive imaginary part, so i*infinity and so on...
thanks for solving my problem.
For any electrical engineer wondering: the Smith chart is indeed a particular case of Möbius transformation, of course with the physical context (impedance, reflection coefficient, etc.) that makes it actually useful in the design of RF circuits.
I do not know that!
@@mathemaniac The Smith Chart maps the half-plane Re(z) ≥ 0 to a circle.
I've used them for years, but just recently learned that the Smith Chart is one example of a Möbius Transformation.
Whats more useful is that we can take reciprocals just as easily on the chart. Admittances are much easier to work with when u wanna do multiple stub matching
@@douglasstrother6584 Strictly speaking it also maps the negative resistance, or Re(z)1. MWO has this implemented for example.
Also, this means that you can think about the impedance matching as spinning and translating the Riemann sphere such that the load impedance is on the lowest point :D
@@Danilo_Djokic True. Where would we be without oscillators & amplifiers? It would be pretty quiet!
It's unfortunate that Mathematicians don't mention the Smith Chart as an Electrical Engineering example of a Möbius Transformation, and that Electrical Engineer Professors don't mention that the Smith Chart is a useful application of a broader class of transformations in the complex plane.
That moment when Mathemaniac möbbed every complex function in the room with infinities was truly an achievement of all times. Mörbios is definitely one of the transformations ever invented.
I laughed when Möbius said "It's Möbin time" and möbed all over the real number line.
Completely agree with the math education rant! I wish that balance of big-picture ideas and small details could be more common in lectures and textbooks
Yes exactly!
I wish....
But my professional expectation is that its going exactly in the opposite direction...
I think so too, that's what makes math fun! I mean... it's not wonder that most people absolutely hate math when all it's ever been to you is rote memorization and calculation, that IS pretty boring! But math can be super exciting and fascinating, and perhaps even FUN!
I think part of the problem is that on some level, the beginning kind of _has_ to be boring and simple. You need a solid foundation to be able to get to the interesting bits (or to even understand why they're interesting at all!). I don't know the best way to handle that, but I know that what we're doing definitely isn't it.
I genuinely _love_ math. I think it's one of the most fascinating things to learn about, and I bet you do too. I've have always loved math, and I owe that to a string of truly wonderful passionate math educators, but most people aren't that lucky. Which is exactly why this (and many others on YT) are vital in helping people to see what mathematics can be with a good teacher.
I really appreciate what they're doing here because I die a little bit inside every time somebody says they're scared of math. No! You shouldn't be afraid of math! Math should be afraid of YOU because you're going to solve the shit out of it!
where does he say that I can't find
Great video as always!
7:46 If c = 0, then neither a or d can be 0 since ad-bc is not 0. Then f(z) = (a/d) z + (b/d) is some combination of Enlargement/Shrinking, Rotation and Translation.
8:10 Yes we need all 5. But I don't know how to prove it.
11:24 At the center, the tangent lines to the image circles are parallel to the original lines respectively. Thus the angle formed by image circles remains the same.
13:30 Let a point P be d away from a circle C of radius r(relatively much larger than d) with center O, i.e. |OP| = r + d (the case of r - d is similar).
The image of P after the inversion is P'. Then |OP'| = r^2 / (r + d) = r - d + d^2 / (r + d) and PP' is orthogonal to the circle.
As r approaches infinity, the circle becomes a line L. |OP'| approaches d. Thus P and P' are both d away from L while on different sides of L. Also PP' is orthogonal to L. Thus P' is the reflection of P about L.
Thanks for attempting the exercises! For 8:10, we actually don't need all 5, and there is freedom to choose which to give up!
At the last part, when you say |OP'| = r - d + d^2/(r + d), that's already enough to conclude reflection as a limit: as r goes up, the fraction vanishes!
also, for the first question, there is actually no rotation necessarily. as f(z) = (a/d) z + (b/d), this is a scaling (what he calls Enlargement/Shrinking) by a factor of a/d followed by a translation by b/d, no rotation
@@mathemaniac I'd say that the 5 transformations are a bit loosely defined though. can a rotate about any point? what about a point at infinity? lots of stuff like that. if all these transformations can only be done at a fixed point, then you need all 5 of them.
This video was pretty dense, but in a good way (so will def need to watch it more than once to unpack everything). But WOW. This video was amazing, and my mind was particularly blown when you explained a stereographic projection in terms of a 3D inversion map. Thank you for this wonderful content!!
Thanks!
I thought this said Morbius transformations :(
I started learning complex analysis when you first announced this series in the 1st video and I have completed learning a month ago.
was it good
Sorry that each video just takes a very long time to make haha
Congratulations. Are you currently working toward completion of a degree?
You completed... learning? I'm sorry, but my processor does not comprehend that sequence of words... would you mind explaining?
Beep boop.
@@idontwantahandlethough I think he means copleted a subject or a course. Suspect a loss of detail in translation. To quote Tigger: "You have to read betwixt the lines."
From what I can tell, of the 5 types of transformation in a Möbius map, if we allow arbitrary axes or circles, then translations and rotations are just compositions of reflections (translation if the reflections are parallel and rotation otherwise). Meanwhile, scaling can be achieved by inverting across two concentric circles. _But wait,_ the video covered that reflections are really just inversions across circles passing through infinity. After all that, Möbius maps should all be capable of being boiled down to a series of inversions across multiple different circles.
Oh. oh my. _Oh my god._ @8:34, when you say "however, the transformation looks a bit too complicated to understand - is there an easier way?" and at that exact moment the background projection hits the point where the 2d image distorts to become a 3d object (or rather, a shape on a 2d plane that our brain immediately interprets as 3 dimensional), and then totally unfolds into your standard 2-dimensional plane... holy crap my dude. If that was intentional it was truly a genius choice. Amazing work :) If that was your video editor, please give them a raise. If it was you, please give yourself a raise. Or at least a cookie. Chocolate chip maybe?
(seriously though, did you purposefully make your words and the background animation sync up like that? I _need_ to know!)
No, that was not intentional, because the rendering of the animation was simply to smoothly transition from one Möbius map to the identity map. It just turns out that in the middle it might look like it is a 3D object.
2:05, You almost got me there.. A certified Vsause reference
Yooooo morbius transformation
No way, it's morbin' time
Can you make a video about conformal mapping? I'm struggling with it and I find your videos really enlightening!
Thanks for the compliment! There might be a mention of it for the next video about complex differentiation.
This is awesome! Thanks for making it! The only suggestion I have us at 10:40 or so. The green and blue lines are a little hard to distinguish on some screens. It might be wise to use two colors that are further apart next time.
Yes, noted. I usually do colour code things in my videos, like when I draw objects, I usually colour the first one yellow, then the second one cyan, and then green and so on. I might consider changing those colors because they do look similar!
@@mathemaniac perhaps just a deeper blue than cyan
Love your videos! I would love if you could make a video on pythagorean spinors and quantum mechanics. There are a couple of papers relating them to apollonian gaskets. You can find them if you google them. I find the connection fascinating and I think that given your expertise, I bet you would be able to flesh out some really unique.
I will check them out!
Absolutely blew my mind when you mentioned stereographic projection
Glad to hear that!
Thanks for the brilliant video!
(That was 25 minutes?! Didn't feel that long)
Brilliant is the definition of a well targeted sponsored video. Than you two =)
Thank you so much for this amazing video!
Glad you liked it!
New here, just wanted to say, your use of squares to silently show your work is intuitive.
Thanks!
Love your rant on math education, I agree!
Please help me with this part 21:44
Why is not just the composition of the two rigid motions?
This video is amazing. Can you do a more in depth one about Penrose's Twistor identity at the end? His hand-drawn projector slides are nice but I bet you could make something so much more striking with this modern animation.
Arguably Great video !!! Ali!!!
Thanks!
In most cases we want to keep "1/0" as undefined. But the point of the video is to show you that, under a suitable context, we may assign a symbol "infty" to the expression "1/0". At times, "infty" is just a symbol, but sometimes (like in this video), "infty" is in fact a "number".
Perhaps the phrase "infty is in fact a number" unsettles people. Indeed it is a slight abuse of words, but it can be made rigorous by the following two steps: 1. extend your number system to include a new point denoted by "infty". eg in this video we have complex_union_{infty} 2. the (extended) number system is a set, and by a "number" we simply mean that it is an element of this set.
The caveat is that step 1 requires a suitable context. Therefore the take-away is: If people tell you that "1/0 = ...", ask them "Under what context?". This video answers: let me give you one suitable context through two ingredients, the Riemann sphere and the stereographic projection.
That's a nice way to put it.
@@mathemaniac Keep up the good work =)
@@strikeemblem2886 Well, I disagree that this is sufficient to say that ♾ "is a number." Despite the analytic applications, you cannot do arithmetic with ♾ consistently, which is a defining feature of what we almost always mean when we talk about a number system. But I agree with the rest of your comment.
@@angelmendez-rivera351 valid and great point, you insist that a number system has to be a field so as to do arithmetic. Then you would also have a problem with things like quarternions and ordinal numbers. Here, it is simply meant that we extend the binary operations + x, possibly dropping any algebraic structure in the process.
.
There is a nice discussion on stackex titled "Is infinity a number?", and I pretty much agree with the top answer.
@@strikeemblem2886 *valid and great point, you insist that a number system has to be a field so as to do arithmetic.*
I did not say it has to be a field. Notice that the natural numbers form a semiring, yet they are still what most people would consider a number system, and you can do arithmetic with them. Addition and multiplication are well-defined. The integers form a ring, but not a field. These are still algebraic structures, and you can do arithmetic with them. The projective complex plane, or Riemann sphere, while interesting in terms of its applications in analysis and its topological structure, it lacks sufficient algebraic structure. Sure, technically, - and / are well-defined, but + and · are not.
*Then you would also have a problem with things like quarternions and ordinal numbers. Here, it is simply meant that we extend the binary operations + x, possibly dropping any algebraic structure in the process.*
In what sense are we extending the operations? I know that, by notational convention, a·♾ = a for nonzero a, for example. But since things such as ♾ + ♾ and 0·♾ are not defined, I am not sure in what sense this forms an algebraic structure. I suppose that, if one considers functions from a proper subset of (C+{♾})^2 instead, then one gets a unital semigrouid for the addition and multiplication. But at that point, I am fairly certain most people would not feel comfortable calling this a number system, and I think this goes for many mathematicians as well. Now, I understand this is largely a matter of semantics. There really is no formal definition of the word "number" that most mathematicians have agreed to, and the boundary between just an algebraic structure and a number system can get a bit vague and fuzzy. And for the record, I am not dismissing the utility or the interest or partial structures, or what have you. But to the extent that these are drastically different from the algebraic structures that we typically associate with number systems, to the extent that I think most people would not recognize them as such, I am not convinced that grouping them with number systems is the most conceptually healthy intuition to instill on people, hence my objection.
I mean, I know none of this actually matters when you have a degree in mathematics and you actually know what you are doing. Whether you call ♾ a number or not does not matter to a person that understands the underlying theory: calling it a number will not bring about any silly mistakes by someone educated on said theory and actively researching it. But what I am concerned about here is educating laypeople who, at most, have a bit of undergraduate level education in mathematics, and are just looking at Internet resources to strengthen their understanding or are just doing it for fun. For these people, whether you classify something as a number or not does matter, and tends to evoke a pretty consistent image in their heads. When they ask "is ♾ a number?," what they really mean is, "can I add, subtract, multiply, or possibly even divide ♾ with all other numbers in a given number system in a way that is consistent?" And since that is the image people have, I think that is the operational definition of "number" with which we should approach the question, especially in light of the fact that many people who make videos on CZcams always say "♾ is a useful concept, but not a number; you cannot do arithmetic with it or compute with it." Since it is apparent that, at least intuitively, number systems have to be a total algebraic structure of some sort, I find it extremely unhelpful to take the liberal approach used in higher mathematics of treating "number" as synonymous with "literally any mathematical object" for this specific context of discussion. It does not answer any questions, and it does not give people the deeper understanding they need about how they should think of ♾. Among higher level mathematical discussions, that usage if the word number is perfectly fine, but I find it unfriendly and obtuse to do so in the context of answering the question "is ♾ a number?".
Great video! I was wondering about this for a long time and it's a great addition to the math videos on CZcams. What happens if we set the degrees in a circle from 360 to a larger highly composite number with more divisors?
There is no way this is true. The Sumerians definitely knew of numbers much larger than 360 that were highly composite. They wrote about such large numbers frequently. 360 is just the most highly composite number that is also closest to the number of days in a year, according to a solar calendar, so that 1 degree of orbit approximately matches one day of the year.
Also, your question is misconceived. There is no reason why we should think of uniformly partitioning the circle into any number of segments to devise a unit of rotation. Mathematically, this is unnatural. This is not an issue with 360 being arbitrary choice for the number of degrees in a circle, but an issue with trying to define a unit of measurement for angles in terms of fractions of a rotation to start with.
What is most natural is to think of angles that are congruent, and place them in the same equivalence class. We want them to be congruent under direct isometries, not indirect ones. The class of angles in Euclidean space can be partitioned into these equivalence classes, which can be totally ordered, such that they are order isomorphic to the interval (-p, p] for any real p > 0. Since each real number corresponds uniquely to a class of directly congruent angles, this real number can then be thought of as the measure of the angle. p is the measure of a straight angle, and so represents a half rotation. p is arbitrary, and a choice of p amounts to a choice of scale factor. p = 180 is equivalent to using degrees to measure angles, but this is by no means a natural choice, in light of what we are working with. There is one obvious choice that is natural, p = 1, but there is a most natural choice that is not obvious, and this choice comes when thinking of a circle whose center is the vertex of an angle, and thinking about the relationship between angles and arclengths. The choice is given by p = π. The reason arclengths are more natural, and more fundamental, is because of how they generalize the notion of angle to the other conic sections, and even to other metric spaces, where arclengths can be defined, but not angles.
This is awesome! Thank you :)
Thanks!
Beautiful video and explanation. If I only may suggest a tiny clarification, the Riemann sphere is a model of the EXTENDED complex plane
Which is the complex projective line P1(C), not to be mistaken for the real projective plane P2(R)…
I toyed with this idea many years ago, I defined 1/0=inf, but I also defined 1/-0=-inf, I haven't really comprehended what that really implicates to this day but I drew some nice circles and parabolas and philosophised on axies so that was fun
This hardly relates to the video, but as a layman with limited math experience, and a programmer, I personally have found tons upon tons of cases where defining division by zero as being equal to zero has been extremely useful to me, even if it is ultimately invalid by some definitions.
It has been incredibly useful in game dev to treat division by zero as equal to zero, for lots and lots of vector math, trig, etc. It's been so useful, I've often even gone so far as to create code that'll handle division by zero for vectors and numbers this way, as well as certain trig outputs, and it often produces consistent and accurate results.
When you don't define division by zero you admit that a literally infinite number of operations don't map to *any* value in the set of real numbers. I don't like that at all, it doesn't make much sense to me. Contrary to imaginary numbers where you get a 1:1 mapping for your inputs to outputs, in this case you get a mapping of your inputs to a single output. However, I would argue that for 0 this is actually valid, because if you multiply anything by zero.
This isn't really proof or even evidence but a fun piece of intuition, you can make the statement that the average of the sum of the values of 1/x and 1/-x will always be zero where it is defined so we can extrapolate that it should also be zero where the divisor is zero. 1/0 and 1/-0 are the same (-0 = 0) and the only single number a which sums to 0 like this is 0. This is nice because it means scale or proportion is irrelevant. This is as far as I know pretty consistent and works correctly in lots of formulas and equations, and in my opinion, it's a better definition than undefined because of that.
where else is it found useful?
The statement at 9:00, I wish all math authors followed this!
Congrats!
Inversion can be interpreted as reflection on a circle. Compositions of such reflections generate the whole conformal group.
Compared to the affine group emerging from usual line reflection. The conformal version adds dilatation and contractions. They come from the composition of inversions on circle sharing the the same center.
Yes! (I think you haven't mentioned rotation, but this should be easy: successive reflections in two different lines) So only inversion would work!
@@mathemaniac Exactly.
Since lines are flattened circles passing through ∞, the reflections, translations (reflections with 2 parallel lines) and rotations (reflections with 2 concurrent lines) can be recovered from them.
I could barely keep up with this video. I'm gonna have to rewatch this a bunch of times to get all of it🤯
Edit: I loved it
An in depth video. Back in 2014 @Numberphile had a demonstration of inversion by hand. The video was titled _Epic Circles_ . Keep publishing good mathematics videos!
Thanks!
I'm not sure that video exists anymore, but here is a newer video from a year ago from his channel czcams.com/video/bJOuzqu3MUQ/video.html
@@floydnelson92 Thanks! Love this video.
for the problem at 7:49, here is my solution:
if ad-bc=0, then ad=bc, therefore, bc-ad=0.
since bc-ad=0, the problem shown below turns into 0/(z + the initial translation by d/c and the enlargement/shrinking by c^2), which reduces down to 0.
now, the equation simply is f(z) = a/c + 0/xz, which still eventually reduces to a/c.
therefore, if ad-bc=0, then all complex inputs would be reduced to a single output, i.e. a circle with radius 0.
at least i think i got this right. correct me if i'm wrong!
8:15 off the top of my head I remember that every isometry is a composition of at least 3 reflections, meaning that if you can chose any lines to reflect upon, you don't need rotations or translations
What about stretching and squishing? Must it be there, or can it be a combination of the inversion and reflection?
excellent content really mind blowing
Thanks!
Really high quality
Thanks!
أذكر عندما كنتُ صغيراً وأفكر في حل مشكلة القسمة على الصفر اعتقدتُ أننا بحاجة لاستبدال خط الأعداد بدائرة الأعداد، حيث يكون الصفر في أسفل الدائرة والنقاط على اليمين تمثل القيم الموجبة وعلى اليسار تمثل القيم السالبة، وتكون النقطة في أعلى الدائرة هي النقطة التي تمثل ±∞ في وقت واحد وهذه النقطة تمثل إحداثية ناتج القسمة على الصفر، وهكذا تم التغلب على فكرة الاقتراب من +∞ من اليمين ومن -∞ من اليسار، لكن بهذا المفهوم فإنّ جميع الأعداد التي ستقسم على الصفر ستعطي النتيجة نفسها وتمثل بهذه النقطة
bro Ive been saying this for years now
You don't need complex numbers for this argument. You could see that inversion on the real line is the same as rotation on the stereographically projected circle. Then 1/0=∞ since 0 and ∞ swap places under this rotation. Also I would like to learn what the relationship between rigid motions of spheres and PSL(2) is (because we know that they both correspond to mobius maps). Also I love the video : )
Yes, that's right - since this is a series on complex analysis, this has to be an argument on complex numbers...
I think the relationship between rigid motion of spheres and PSL(2, C) is the most direct when we link them to Möbius maps, but there might be some more direct ones that I don't know of.
Your videos are great at making hard mathematical concept look easy. Thank you and keep up the good work!
Thanks so much!
The circle of inversion part brought back flashbacks to Numberphile's Epic Circles video.
Do asymptotic representations of functions survive the introduction of such operation?
No, not quite.
Congratulations on getting a sponsor!
Thank you!!
This makes me totally want to do relativistics in terms of the Riemann sphere and then have a look how black holes look from that perspective! That should shine a light on the idea that our universe is in fact inside the black hole of another universe! Unfortunately I have no time, so I hope someone else will do this...
i'm not sure I understood anything but as an engineer, 1/0=infinity works just fine anyway
Are you okay with 0*infinity=1 ?
yes, intuitively, division-by-very-small numbers, ALMOST overflows into multiplication-by-very-large-numbers, but for this to make sense, you need a few constrains, that are not always useful.
No it does not make generally sense to equate the 2. Whenever you divide (by a very small value (that approaches 0)), this is a clear tell-tale, that your model/axioms approach a much simpler special-case == near-asymptotic-case, that often does not exist in a special case. The division-by-0 is an emergent property of your generalization, and it points out a constraining-range of a generalization.
Where ever you multiply in a function, your inverse of that function introduces a division (unless the function is its own inverse or something like FFT) , and your possible-parameters may or may not enable a division by 0, and you then have to filter out a boundary, to evade the precision loss, that results from dividing by too small numbers, by using something like SmoothMinimum(a,b)
within that low-precision-area (where you divide by almost-0) you then use a simpler special-case function, and outside of it, you must use the generalization, that added the division-by-0 case.
im pretty sure all transformations can be described by 3 reflections. i got super into projective geometry while studying perspective projection in computer graphics, and i remember that fact (along with many other concepts like point/line duality) sticking out as being astounding
thank you very muh for this video.
Glad you like it!
I've always thought that you could define translation in terms of rotation. By using a circle with infinite radius (a line), a rotation along that circle is effectively a translation along that line
This is something I haven't thought of, although you can't really define the angle of rotation that way... But among those 5 transformations, both translation and rotation can be reduced to some combination of the others.
The "angle" of rotation is in fact simply the distance being translated. You can also go the other way and argue that standard angles of rotation are instead distances in spherical geometry. My reasoning for defining the angle as such is from the relation between complex numbers, hyperbolic numbers, and dual numbers. With complex numbers, e^iφ moves φ units along the unit sphere, while hyperbolic numbers (aka split-complex numbers) e^jφ moves φ units along the unit hyperbola, which is sometimes called the "hyperbolic angle." Dual numbers behave similarly, but in flat geometry, so e^𝛆φ moves along the straight line at x=1 by φ units.
Translations in certain systems behave somewhat like dual numbers, which is why the comparison is relevant.
You know what got me is studying point set topology. It's why I recognize the symbols and it teaches how to create proofs ultimately leading to proofs for algebra. And I did 1/0 on my chromebook. It correctly displays the infinity symbol
can you point me to any good topology resources you've found online? The wikipedia pages on the topic are.... _dense,_ to put it mildly, and I'm having a hard time knowing where to look. Any help would be greatly appreciated (regardless of the density of the reading material! Normally that doesn't bother me, I'd say I actually prefer it, but the topology pages on wikipedia are truly something else lol)
edit: or not online! I'll take anything I can get my hands on at this point
@@idontwantahandlethough Zach Star's channel might have some videos on books on the topic.
The infinity itself is an undefined concept. There is axiom of infinity, induction and lim, but no infinity itself. Especially not as a scalar.Usually the symbol is shorthand for "any positive number" or "any negative number". If you consider infinity to be a number larger than any other numbers, then dividing 1/infinity, 2/infinity etc would give distinct infinitesimals. It also preserves sign (0 is unsigned, and loses such information). Same way, 2*infinity would be a different kind of infinity from 3*infinity They did that in earlier calculus, before introducing lim. In the end you an collapse them to normal 0 and infinity.
A very, very nice video :) Thank you very much. A question - is this right, that the inversion reverses orientation? 1/z is a holomorphic map and holomorphic maps are true to orientation and conformal. You must reflected your curves on the real axis, then it is true to orientation, isn't it? Best wishes from germany
So for the **geometric inversion** (or inversion with respect to a circle), it does reverse orientations (hence anticonformal), and of course, a simple reflection across the real line also reverses orientation. Both operations preserve the magnitude of the angles.
Since 1/z is a combination of both operations, it preserves both orientation and the magnitude of the angles, and hence yes, it matches the prediction that holomorphic functions are conformal (where the derivative is not 0, but 1/z does not have derivative 0 anywhere on the normal complex plane anyway).
thank you
Isn't translation just a combination of rotations?
And inversion is a combination of scaling & reflection?
If I'm right, the 5 tranformations can be simplified to just scaling (grow/shrink), rotating, & reflecting
Can you elaborate more on why inversion is a combination of scaling and reflection? (I am referring to reflection only across a line here)
@@mathemaniac I guess I was picturing your animation at 3:57 where tracing the circle counter clockwise traces the inverted circle clockwise. But now that I think about it, we don't need points to be unique (map to specific new points) so reflection does nothing & inversion is scaling + translation (which is just a combination of reflections)
@@AlexanderQ689 In particular, since we know all the actions preserve circle, we can just translate C1's center to C2's center, and scale the diameter till they match
Although circles do map to circles, inversion doesn't do the scaling and translation *uniformly*, i.e. a circle with radius r does not always map to a circle with radius kr, where k is a constant. It depends on the position of the circle.
7:30 note that differentiating with respect to z we get |M|/(cz+d)²
When Robert Siliciano wrote "Constructing Mobius Transformations with Spheres" he was still an undergraduate.
22:41 I didn't find a paper that demonstrated it, but the video "Möbius Transformations Revealed [HD]" (which you also have linked to) demonstrates it, is created by people different that the one who wrote the paper, and was copyrighted in 2008.
I don't think the video demonstrated it - the point here is Mobius map can be *uniquely* represented by a rigid motion of the Riemann sphere, but the video only shows that a Mobius map *can* be (but not necessarily uniquely) represented by a rigid motion. The paper I mentioned explicitly wants to show the uniqueness, as shown by their introduction: "The main result shows that for any given Mobius Transformation and so-called admissible sphere there is exactly one rigid motion of the sphere with which the transformation can be constructed."
@@mathemaniacAh, okay, that may be right. So maybe the connection had already been known for a long time, only that the one-to-one correspondence was first shown in that paper?
Yeah - but that is quite a "basic" result in the sense that it doesn't involve advanced mathematical tools, which is why I was surprised it was only shown so recently.
is there any relationship between mobius maps and matrices?
YES! I have said in the pinned comments that this is something I have missed - just that the video is too long so I didn't include this part on matrices...
I don't see any pinned comment?
Yes, every Möbius transformation az+b/cz +e is uniquely described by the matrix (a,b)(c,d). And if you concatenate two Möbius transformations you multiply their matrices.
Möbius transformations (meromorphic bijections of the Riemann sphere) form a group under composition which is isomorphic to the projective general linear group PGL(2,C). Each transformation corresponds to an invertible 2x2 matrix (see Caspar's comment) uniquely up to multiplication by a nonzero complex constant.
@@Caspar__ Uniquely up to multiplication by a nonzero complex constant. Also e vs d is a typo :)
congratulations for the 69k subscribers!
Nice
Very entertaining video.
Thanks!
hope you got a mil one-day.
That's way too ambitious, but thanks for your appreciation!
Are the moebius maps another name for affine transformations from linear algebra or is there some subtle difference?
You need all the transformations, I think, and you can prove that with matrices.
No - affine transformations and Möbius transformations are really different: for a start, affine transformations maps lines to lines, but Möbius transformations and map lines to circles. Affine transformations might not be conformal, but Möbius transformations must be.
You don't need all transformations, and you can do everything with just one of them!
Don't bother with Brilliant if you are on the level you watch this for fun - 2x year subscriber
Vote up, nice video, thanks for sharing :)
Thanks for the compliment!
¡Gracias!
Can you tell me, why is infinity so important for the complex analysis so that it is meaningful infinity representing as a point on the sphere? I mean why is the extended complex plane so important? The only reason I can introduce is, that you can whole functions (functions that are holomorphic in the complete plane) classify after the kind of isolated singularity in infinity. Do you know more reasons?
I do not know too much, but I think it is way easier to represent Möbius maps if we just include infinity, as demonstrated in the video. (Excluding infinity will make Möbius maps lose the group properties)
But I suspect the main reason has something to do with geometry and topology. It makes the complex plane compact, and the description of meromorphic functions(i.e. holomorphic functions from C to Riemann sphere) more convenient.
Its importance is less about complex analysis, and more about projective geometry and algebraic topology.
How long does it take you to edit a video like this one? Thank you
No exact figure, but easily 70+ hours.
Bosons with spin 2 are represented by a complete revolution of the Mobius strip, which is central to understanding how mass, energy, space, time emerge from a singularity, the two poles of the Reimann Sphere.
That's just a way of representing the fact that the spin group is a double cover of the special orthogonal group though. That's not directly related to mobius transformations (although they also form a semisimple Lie group)
"By abuse of notation...". Wow. This is candid, even for a mathematician.
It's not ridiculous and makes sense even in 2D. The inversion works with non-unit circles (of non-zero radius) too, producing the same infinite 'circle' regardless of the radius of the original circle. Proving once again anything divided by zero is not just undefined, but more precisely, infinity.
I believe that there has been a historical mix of things going into the education of math in schools. Some of the jargon is complex because it builds on other complex ideas and has to be very precise so that people can communicate exactness correctly. But also much of it was made overly difficult so as to bar it from being taught to the commoners. Sometimes very simple ideas are given very large names that intimidate people who do not know what the word means. I believe that modern society should rename some of the jargon to be more indicative to the lay people what is meant by the idea of the jargon. I think the jargon should be about as "simple" as the idea that it means to express. In many cases, this is the case, but in many cases it is not.
I don't think math jargon is created with the intent to intimidate, just so that is can be reffered to later, for the sake of compactness
Then you need to learn. There are places where women aren't allowed an education. The same goes for caste society in India, some people are barred.
This isn't exactly what was stated, but why the hell were bibles written in latin for such a long time when only the all-male clergy spoke latin?
Can you do a video on homeomorphism and diffeomorphism?.
It might not be a full video, but this is something that I can consider.
Alright 👍. Looking forward to watch your videos on them.
19:03 very nice wok.
Wow the three R's ! Ty
Haha 😂 loved that Vsauce reference at 2:05!
2:06 LOL at the Vsauce reference
division of anything by zero is left undefined in general because of the lie everyone's accepted that fractions are numbers. of course it makes more sense to define it some specific way in specific cases, because the reason it's left undefined is a result of there being multiple such cases which contradict each other. this results in division by zero not behaving like a function (surprise! probably because it's the slope of a vertical line, which by definition is not a function... what a shocker!). but in a world where people expect every division problem with defined numerator and denominator to yield a number... this is a hell of a paradox.
the solution to the paradox is to realize that fractions are always vectors, and actually never numbers (by which I mean scalars), and as such it's usually extremely important to understand the relationship between the units of the components even more than computing the ratio between the components. for instance, atan(2/0) is obviously pi/2, but if you attempt to compute 2/0 before looking at the function that's being applied over it you end up stuck. and even taking limits won't necessarily help, as the result can yield either positive or negative infinity, and atan(-2/0), which is what it will appear to be if you try to take the limit from the left, is 3*pi/2, which is obviously not correct.
moral of the story, math is not about computing values, so rushing to compute things before looking at what you're doing is a foolish move. as such, 1/0 being best handled as infinity in some cases while remaining undefined in general makes sense, and honestly this kind of thing should be shown to students far, far sooner than it is. because the mentality that we need to teach the basics first, and computation is basic... is completely wrong.
A question. The inversion is a one to one mapping - one point maps onto one other point.. But all the points at infinity map onto the one single point of the origin. So that's for me actually a strong argument for the indefinitness of 1/0 in the context of mapping operations.
In the projective complex line, which is a way of making this rigorous, there is only one point at infinity.
totally agree!
Nice
In GLSL, 1.0/0.0 is Inf but 0.0/0.0 is NaN
In fact iirc any number divided by zero (except for zero) is Infinity
I looked glsl up and i was almost offended at how obvious your statement is! Thanks for the insight!
Someone please answer 11:22 why
"1/0 = ∞ makes more sense than you think" Jokes on you, I thought it already made perfect sense xD
There is a much graceful definition for an algebra that allows division by zero. Actually the periodic table of elements presents such an algebra.
I don't know what you mean. How does that algebra work and what does it have to do with the periodic table?
I have to agree with Alexander here. What are you talking about?
Ima need to rewatch this when it’s not 4:30am 🙃
Wow , it's live
It's Möbin' time.
It confused me at the beginning but when it went 3D, I was like "Ohhh"... I finally get what's going on.
That's exactly what I was hoping!
Thank you..With all my respect I wish you have completed the series yourself.
czcams.com/video/X83vac2uTUs/video.html
on the 3:47 point in the linked video
Does the inversion mentioned in this video have something to do with the duality between points and lines mentioned in the linked video?
I've learned a lot from this video.
Most important thing being that I'm stupid :D
Interesting
I'm put in mind of a graffito from Caltech:
(unfortunately, I can't reproduce it perfectly using my phone's keypad)
1/infty = 0
(rotate 90⁰ anticlockwise)
-18 = 0
(add 8)
-10 = 8
(rotate 90⁰ clockwise)
1/0 = infty
If I walk far enough on a mobius map, will I get to the other side?
I've been claiming this for a while, but I still get the old retort: "But that violates one of the Axioms: Division by 0 is Undefined"
Here's my claim and we'll start off with the expression y = x.
y = x is an expression and a statement. It is not exactly an equation as there are no direct operations that perform a specified transformation. Yet it is still an equation as it does express either a direct assignment or an algebraic equality, not a boolean equality comparison. We can easily state that f(x) = y and substitute the change of variable y into functional notation. This then gives us f(x) = x. And now we have an actual function. Our very first algebraic function. There are many unseen properties from this basic innocent looking function so let's cover some of the more important ones. First and foremost, this is the identity function (what goes in comes out unchanged). Second, it is linear. Third it is the foundation to all other functions either it being Algebraic covering all other polynomials, exponentials and logarithms, geometry and trigonometry and even calculus including vectors and their applied transformations.
We can quickly build a small table to see the linear relationship of this function that is also associative. You can swap (x,f(x)) with (f(x), x) and still produce the same exact results.
x | f(x)
----- | -----
-3 | -3
-2 | -2
-1 | -1
0 | 0
1 | 1
2 | 2
3 | 3
And we can use a 2D Cartesian Coordinate system with an X and Y axis that are orthogonal, perpendicular to each other that are either 90 degrees or PI/2 radians apart. This will become important a little later. This gives us the diagonal line that goes through the origin and increases towards infinity in the +X+Y directions or the 1st Quadrant, and goes towards -infinity in the -X-Y directions in the 3rd quadrant.
Why is this line so important? It is a bisector. It has perfect symmetry and it also implies rotation from the basic principle of something being equal or equivalent to another object. I didn't say itself as this would imply x = x and this is not the case! Context here is everything!
Now in order to understand why this is so important we have to refer to the Slope-Intercept form of the line y = mx + b where b is the y-intercept and m is the slope defined as rise over run. The slope can be determined by any two points on a line from the following equation (y2-y1)/(x2-x1). It is the ratio of the differences of rates of change in y and x. This can also be expressed as either deltaY / deltaX or simply dy/dx.
Within the original expression y = x we can see that b intercepts the y-axis at the point (0,0). We can see that it's slope has a value of 1 as it is constant and unit length of change in both directions. So we can easily take y = 1*x + 0 and get y = x as the multiplication of 1 and addition of 0 do not change the quantity. Most don't make this connection, but multiplication of 1 and addition of 0 when it comes to transformations are exactly the same, not in that they are the same operation just that they will always yield the same result. Most might understand this through basic arithmetic from elementary school, but how many consider this in the terms or within the context of transformations? I'm sure some do, but not all. This will become apparently important as we step through this process.
So, how does this have anything to do with all of the other branches of mathematics? Since we have a slope of 1 and it passes through the origin (0,0) and we know that it bisects the XY plane in the 1st and 3rd quadrants we can easily see that it generates a 45 degree or PI/4 radian angle between the line y=x and the +x-axis. Once we have a generated angle of rotation we can also express any geometry.
Since slope is defined as dy/dx or rise/over run, we can take any point on this line within the 1st quadrant, draw a vertical line coming up from the +x-axis at the x-value and this will be our (x2,y2) coordinate where (x1,y1) will be the origin (0,0). This vertical line we can label it as dy. The distance from the origin (0,0) to (x,0) we can label as dx. Now we have a Right Triangle in standard position that has two legs that equal x with a hypotenuse of x*sqrt(2). And thus we now no longer have just integers or fractions, but also irrationals and this is just the first one. Since we have irrationals or radicals we also have quadratics and once we have quadratics we also have complex numbers!
Since we have a defined right triangle we also have the 6 Trigonometric Functions. Of the 6 we are only interested with 3 of them within this context as they are the Sine, Cosine and Tangent. From a Right Triangle Definition Sine t = opp/hyp, cos t = adj/hyp, tan t = opp/adj.
And from these definitions we can easily see that theta t is the angle next to the origin (0,0) between the line y=mx+b and the +x-axis. Since we now have theta a measure of angle we also have rotation. Before I move on we can also briefly express a few properties of triangles as well as their direct relationship to lines or linear equations. We know that all 3 interior angles of a 2D Euclidean Planar Triangle adds up to equal 180 degrees or PI radians and we also know that the angle of a straight line is also 180 degrees or PI radians which is a very important relationship that will come later.
Now back to the relationship of a linear equations and the trigonometric functions. From the definitions of the trig functions we can see that our dy = opposite and our dx = adj sides. With this we can easily construct our slope formula in terms of a trigonometric expression. y = mx+b can be written as y = (sin(t)/cos(t))*x + b which is also equivalent to y = tan(t)*x + b.
We can test this with two points on this line and I'll keep it simple with the points (3,3) & (5,5).
Linear: y = (5-3)/(5-3)*x + 0 = x
Trigonometric: y = (sin(45)/cos(45))*x + 0 = ( (sqrt(2)/2) / (sqrt(2)/2) ) * x = (~0.7071.../~0.7071...)*x = 1*x = x
y = (tan(45))*x + 0 = 1*x = x
As you can see these are the same exact expressions except the first is in direct terms of lines and the later are in terms of the trig functions and the angle theta between the line y = mx+b and +x-axis.
How does this pertain to 1/0 = +/-infinity or the concept that division by 0 Is Not Undefined?
(continued...)
(...continued)
We have to examine and see what happens when we begin to rotate y=mx+b about the origin (0,0) and look at the properties and the relationships between the angle, the slope, the 3 vectors of the right triangle as well as the interior area of the triangle as well as the relationship between these 3 trig functions.
We can first claim that m = tan(t) and from here on out I'll be referring to this relationship as dy/dx = sin(t)/cos(t) since dy = sin(t) and dx = cos(t). I didn't mention it earlier but when I stated that the hypotenuse was x*sqrt(2) from the points (0,0) and (x,y) this was derived from the Pythagorean Theorem, more on this later.
When we look at slopes or the ratios of opp and adj in regards to theta that is in the range of 0 < t < 45 degrees or PI/4 radians we can easily see that our ratio or slope will be in the range of 0 < m < 1.
When we look at the slopes or the ratios of opp and adj in regards to theta that is in the range of 1 < t < 90 degrees or PI/2 radians we can easily see that our ratio or slope will be in the range of 1 < m < +infinity.
We have to look closely at what is happening within both the sine and cosine functions as we approach 90 degrees or PI/2 radians. More than that we also have to take into consideration the properties of these functions, the relationship to each other, their ranges and domains, as well as what is happening to the area of the generated triangle.
The easy part is that both the sine and cosine functions are continuous oscillatory, rotational, periodical, transcendental wave functions. They also have the same range and domain. Their simple domains are the set of All Reals (*they can actually take in complex numbers as well but that is beyond the scope of this topic) and their ranges are [-1,1]. Another important property or relationship between these functions and their graphs is that they are the same exact wave function except that they are linear transformations of each other, yes horizontal linear translations to be exact. They are translated 90 degrees or PI/2 radians along the x-axis from each other. The final important property about them is in regards to their limits. Within Pre-Calc or Calc I, you'll learn that these functions are continuous across their entire domain as both their right and left handed limits exist for all points on their graphs. This never changes!
As for the area of the triangle that will become apparent when we examine their simplified limits when we look at them being expressed in terms of the tangent function.
When we have horizontal line that is parallel to the x-axis we say we have a slope of 0. We can use the points (2,2), (4,2) to derive the following.
From the slope equation: m = (2-2)/(4-2) = 0/2 = 0.
From the trig expression: sin(0)/cos(0) = 0/1 = 0.
Length of dy = 0, length of dx = 2, Area of Triangle = 0, Interior Angle of Triangle? Still 180 degrees as the triangle is now a horizontal straight line!
Now how about when theta equals 90 degrees or PI/2 radians where the slope is a vertical line parallel to the y=axis? We can use the points (2,2), (2,4) to derive the following.
From the slope equation: m = (4-2)/(2-2) = 2/0 = ?
From the trig expression: sin(90)/cos(90) = 1/0 = ?
Length of dy = 2, length of dx = 0, Area of Triangle = 0, Interior Angle of Triangle? Still 180 degrees as the triangle is now a vertical straight line!
If we take the simplified trig fractions for 0 degrees and 90 degrees being 0/1 and 1/0 we can see that they are reciprocals of each other, we can also see that the lengths of the opposite and adjacent side of the right triangle or dy and dx flipped from 0 and 2 to 2 and 0 respectively. Yet the area of the triangle in both cases is 0 and the interior angle still remains 100 degrees.
So how is it that horizontal slope can be defined as 0 since 0/1 = 0 makes sense, yet people struggle with vertical slope 1/0 being undefined?
(continued...)
(continued...)
Consider a real life example with motion or translations in various planes. Imagine walking down an alley between two city buildings on each side of you. The road is flat or level and you have 0 incline and no slope. Then you start to walk up hill, now you have positive slope, then it starts to level off again and back to no slope, then you walk down hill and you have negative slope, it levels off once again back to no slope. Finally you come to a third adjacent building to the other two making it a dead end. There is no turning around or going back. You look at the wall of the building and you see runged ladder. You begin to climb up the rungs of the ladder. What's your slope? Is it really undefined?
When you walked on the flat ground you had 0 slope as you only had translation in the xz-plane and you had +/- slope that slanted through the xz plane into the xy and zy planes. Yet when we have no motion in the xz plane and constant motion in the y plane all of a sudden it's undefined because of "division by 0"? Nah, I don't think so. It is most definitely defined. The result might be Ambiguous as it doesn't fit the model of a function where you have 1 input to 1 output as that is a 1 to 1 ratio, and with 0/n that is a null to many ratio which is okay, but when we have a many to null ratio everyone wants to throw their hands up, giving up and demand that its undefined. That's quite unintelligible to me.
You still have constant vertical motion!
So when we look at sin(90)/cos(90) = tan(90). We are taught to accept that tan(90) is undefined because of the discrepancy within its limits at 90 degrees where the vertical asymptote is. I'll express, ascertain and confirm how it's not Undefined and that it is Ambiguous. There is a difference between the two their definitions and the Context in which they are applied.
Sin(90)/Cos(90) each limit in both the numerator and denominator exist and are independently equal yet through trig substitution tan(90) posses a problem? Yeah, I'm not buying that one!
Let's continue a bit. So far we have expressed just about every aspect of mathematics from linear equations, to trig functions, some polynomials, some geometry (once you have a triangle, you can generate every other shape), well once you have a line, you can draw every other shape!
We even mentioned the Pythagorean Theorem. We talked about angles and rotations, but there was one thing we left out. The Circle. So let's make this full circle!
Here I'm going to use the very first arithmetic problem that 90% of the population is taught from a very young age and that is 1+1 = 2.
At first glance you might not recognize it, but this very first arithmetic operation that now actually includes an expressed operation that wasn't inferred or derived from y=x directly but just from the basic concept of counting itself lies within it the Unit Circle located at the point (1,0).
The general equation for a given circle is (x-h)^2 + (y-k)^2 = r^2 where (h,k) is the center point, (x,y) is any point on its circumference and r is the magnitude or length of its radius. We know that the unit circle has a radius of 1. If we look at the expression 1+1=2 and consider each digit value not as just a typical scalar quantity or amount as that will not be sufficient. We have to look at them in terms of being vectors on a plane.
The first 1 is a unit vector from the points (0,0) to (1,0). The + operator is a linear transformation. Yes it is a horizontal translation, but it also implies rotation, it's just not apparent at first sight. The act of translating the vector defined from (1,0) - (0,0) to its new position defined by (2,0) - (1,0) gives us two vectors with the point (1,0) being the center point between them, the point of reflection, symmetry and rotation. The = is either assignment or equality and expresses that the combination of these two are equivalent to the vector defined by (2,0) - (0,0). And as we can see we know that a Circle with a Radius of 1 also has a Diameter of 2.
We can take this information and apply it to the equation of the circle. (x-1)^2 + (y-0)^2 = 1^2 and this can be simplified to: (x-1)^2 + y^2 = 1 == y^2 = 1 - (x-1)^2 and this is one of the reasons why the Trigonometric functions have Pythagorean Identities. In fact if we look at the equation to the unit circle at the origin, it will become more apparent.
(x-0)^2 + (y-0)^2 = 1^2 = x^2 + y^2 = 1
Doesn't x^2 + y^2 = 1 look familiar? Since 1 is a perfect square we can see an exact relationship between the Pythagorean Theorem and the Unit Circle located at (0,0)
A^2 + B^2 = C^2
x^2 + y^2 = 1^2
What does this have to do with division by 0 and tan(90) not being undefined?
Well we have a perfect circle, full rotation of 360 degrees or 2PI radians. If division by 0 was undefined... then how is it that your tires can do full revolutions? How is it that time or a clock with hands on the walk don't stop rotating as long as they have power to run?
You see, the universe doesn't see division by 0 as being undefined. To explain this in the best way possible, one has to use their imagination, open up their mind and think about it for a moment without jumping to conclusions that they've been embedded to believe before they could even factorize. When the tangent function reaches 90 degrees and division by 0 becomes apparent within the denominator of the fraction what's really happening here is that there is a Phase Shift between the Sine and Cosine Functions. The phase shifts are when theta = 0, +/-90, +/-180, +/-270, +/-360 and all of their integer multiples.
To fully understand what is happening during the vertical phase shifts (not the horizontal) which are +/-90 and +/-270 we have to take the same approach when people started asking about what is sqrt(-1) to fully understand this.
The trig functions can take complex numbers as arguments. We would have to go into the complex plane to see what's really happening. And you would have to take modulus arithmetic of complex numbers and angles into account. The Angle is 90 which is real, the fraction is 1/0 which is ambiguous within the real number planes XYZ but if we extend into XYZ & ijk complex counterparts... It becomes transcendental. The slope blows up towards +/-infinity when these vertical phase shifts take place. The Right Triangle has an area of 0, and yet it maintains an interior summation of angles at 180 degrees. It still has 3 vertices, it's just that two of its legs have equal value, and its third leg, the adject or dx leg has a length of 0.
It's a bit complex, yes and the result is Ambiguous because we can put in a single value and get an infinite list or set back, but it is surely not Undefined. It does have a Definition! It's called Vertical Slope! It is perpendicular to Horizontal slope that is defined as 0/1 based on the properties of the trigonometric functions. You can also use the dot product that is related to the cosine to see these relationships, you can check the laws of sines and cosines to see these relationships...
I don't agree with the terminology that division by 0 or tan(90) is undefined. I can state that it is Ambiguous with Vertical Slope!
Here's the main difference between undefined and ambiguous within the context of computer programming using the C/C++ language(s) as reference.
// Case A: Undefined:
void foo();
int main() {
foo(); // Compiler Error: Undefined function foo() called in main.
return 0;
}
// Here foo is declared but it lacks a definition.
// Case 2: Ambiguous:
int foo() { return 5; }
bool foo{ return true; }
int main() {
foo(); // Compiler Error: Ambiguous call to foo(): Did you mean (int)foo() or (bool)foo()?
return 0;
}
Being Ambiguous does not imply being Undefined. This nomenclature of division by 0 and tan(90+PI*N) for all integers N needs to be reconsidered and ought to be listed as Ambiguous or Vertical Slope or Purely Vertical Inclination that Tends Towards +/- Infinity!
Why? If we take the limit of the tangent function at 90 degrees from both sides independently they do both exist. However, they are opposite not in their magnitude but in their sign or their direction. The direction vectors flip away from each since the sine and cosine functional parts are at their intersecting phase shifts. What is this phase shift? If you graph the pure sine and cosine functions at look at the x axis where the tangent has a vertical asymptote, this is where the two curves are past their intersection to each other, the sine is at its peak or highest value and the cosine is 1/2 way through its cycle intersecting the x-axis. The Cosine is not changing its direction as this momenta, but it is changing its sign going from + to negative. Then when it reaches its minimum and changes its direction it's still negative but when it intersects the x-axis it is maintain its current direction but its about to change its sign. If we observe the sine in the same manner The sign is at its maximum and is about to change its direction but not its sign at 90 degrees, then the sine will change its sign after it intersects the x-axis when the cosine is at its minimum. The cosine wave will transition from + to - before the sine function will. They are parallel waves that have moments of intersections. They intersect at PI/4 + 2PI*N where N is an Integer. You can call these inflection points. And this property is also why they are Derivatives and Integrals of Each other except the fact that they pose different signs. This is due to the fact that the cosine is an even function and the sine is an odd one.
And I think I'll finish with this with that last pun since one equals one giving you y = x and one plus one is so much complex fun!
@@skilz8098 *I didn't say itself as this would imply x = x and this is not the case!*
Are you seriously insinuating x = x is false? Why?
*Within the original expression y = x, we can see that b intercepts the y-axis axis at the point (0, 0).*
No. We can see that b = 0, and the line given by the equation y = x intercepts the y-axis at (0, 0). b is just a numerical quantity. It does not intercept anything.
*Most don't make this connection, but multiplication of 1 and addition of 0 when it comes to transformations are exactly the same, not in that they are the same operation, just that they will always yield the same result.*
Of course. 0 is the identity element of the additive group, and 1 is the identity element of the multiplicative monoid. That is literally how they are defined in ring theory, after all. This means that there is monoid homomorphism f between the additive group of a ring, which is a monoid, and the multiplicative monoid of a ring, with f(0) = 1, and f(x + y) = f(x)·f(y).
*And thus, we now no longer have just integers, or fractions, but also irrationals and this is just the first one.*
No, this is incorrect. The problem is that you are treating geometry as a way to generate "a new number system" that leads to the real numbers, but this is not possible, because in order to get to Euclidean space, you must _start_ with the complex numbers. Euclidean space is a vector space over R, of dimension 2, with a Euclidean inner product. The real numbers are more fundamental. Geometry is defined in terms of real numbers, not the other way around. The real numbers are a totally ordered field that is Dedekind complete, and sqrt(2) is defined as sup({q in Q : q·q < 2}), where Q is the field of rational numbers as a subfield of the real numbers.
*Since we have irrationals or radicals we also have quadratics, and once we have quadratics, we also have complex numbers.*
Hold on. Quadratic polynomials are perfectly well-defined for the integers, without radicals or irrational numbers. Also, you can get the Gaussian integers without going through the complex numbers.
*Since we have a defined right triangle we also have the 6 trigonometric functions.*
This is problematic. You must define the measure of an angle in general terms before you can make sense of a trigonometric ratios. Using a single triangle will not do.
*When we look at the slopes or ratios of opp and adj in regards to theta that is in the range of 1 < t < 90 degrees or π/2 radians...*
This is so poorly phrased, it is unintelligible. What unit is the lower bound in? Stick to one unit, and stop referring to both.
*we can easily see that our ratio or slope will be in the range of 1 < m < ♾.*
No, because ♾ is not a quantity that is defined here, and the slope of a verticle line is not defined either. A vertical line cannot be parametrized in terms of the equation y = m·x + b. So the ratio is not defined when the angle is 90°.
*The easy part is that both the sine and cosine functions are continuous oscillatory, rotational, periodical, transcendental wave functions.*
You have a tendency to use technical jargon without having any clue as to what it means. There is no such a thing as a "rotational" function, this is nonsense. Also, oscillatory and "wave" mean the same thing, and transcendental is implied by the periodicity, since algebraic functions over the real numbers cannot be periodic. And yes, they are continuous, but more importantly, they are analytic.
*They are translated 90 degrees, or π/2 radians, along the x-axis, from each other.*
To be more accurate, cos(x) = sin(x + π/2). The sine and cosine functions are inherently defined in terms of radians, so mentioning degrees is a conceptual mistake here.
*Interior angle of triangle? Still 180 degrees as the angle is now a horizontal straight line!*
No. An angle of 180° degrees is composed of two rays sharing a common vertex and extending in _opposite_ directions. Here, the line forms an angle with the x-axis in quadrant 1 of 0° degrees, since the ray extending in the positive direction is extending in a direction _equal to_ the x-axis. This is an scenario where tan(0) = 0, not tan(π) = 0. π is not even an allowed value for t, because the way you have set-up the relationship between slope and angle, t is constrained to the interval (-π/2, π/2). This covers all possible slopes.
*If we take the simplified trig fractions for 0 degrees and 90 degrees being 0/1 and 1/0, we can see that they are reciprocals of each other.*
No, they are not reciprocals of each other, because 0·1/0 = 1 is false. What it means for two quantities u and v to be reciprocals is that u·v = 1. Also, as explained earlier, there is no trigonometric ratio for an angle of 90° in this context. The vertical line and x-line do not even project onto each other, so they do not form a triangle. There is no triangle with two angles of 90°.
*So how is it that horizontal slope can be defined as 0, since 0/1 = 0 makes sense, yet people struggle with vertical slope 1/0 being undefined?*
What you are failing to understand is that this notion of slope is an artifact of the coordinate system. If I rotate my coordinate system by 45°, then both the horizontal line and the vertical line have well-defined slopes: -1 and 1, respectively. If I rotate it 90° instead, then the horizontal line has undefined slope. Slope is not a geometric quantity, and it is solely an artifical construct, born from the choice of coordinates. If I choose polar coordinates instead, then I no longer have to worry about certain lines having undefined slope, and all lines can be described by an angle of declination. None of this changes the fact that 0 has no reciprocal.
@@skilz8098 *Yo begin to climb up the rungs of the ladder. What's your slope? Is it really undefined?*
Yes. Again, slope is not a geometric quantity. It is not a property that lines have, not in mathematical anstraction, and not in the physical world. It is an artifact of coordinate systems. Here, you have a hidden the fact that you chosen a coordinate system implicitly: you have chosen for the x-axis to be the axis tangent to the geoid in the direction that you are facing, and the y-direction to be perpendicular to the line. And yes, when using this coordinate system, your slope when climbing the latter is undefined.
*Yet when we have no motion in the xz-plane and constant motion in the y-plane, all of a sudden it's undefined because "division by 0"? Nah, I don't think so.*
It is most certainly undefined, and this is a consequence of your choice of coordinates. To be clear, it has nothing to do with whether your motion is constant or not, or how many directions you are moving in. If I rotate the axis, my physical motion is the same, but now I am moving in all three directions, and my slope is defined. Again, slope is not a physical quantity. You can petulantly insist like a child that it is physical, but it is not: it is literally defined in terms of the coordinates you choose, and the coordinates are an artificial construct, not part of the actual geometry. Also, there is no such a thing as the y-plane. You should be talking about the y-axis instead.
*The result might be ambiguous as it doesn't fit of a function where you have 1 input to 1 output, as that is a 1 to 1 ratio, and with 0/n that is a null to many ratio, which is okay, but when we have a many-to-null ratio, everyone wants to throw up their hands, giving up and demamd that it's undefined. That's quite unintelligible to me.*
It is unintelligible to you because you clearly do not understand any of the mathematical concepts you are talking about: you do not understand how coordinate systems work, how functions are defined, how geometric properties are coordinate-independent, or the fact that division by 0 being undefined has literally absolutely nothing to do with anything you have said so far. You could not even distinguish an axis from a plane, calling it the y-plane instead of the y-axis. Not understanding mathematical concepts beyond the very basics does not give you an excuse to think expert mathematicians who have studied the subject for decades are unreasonable and unintelligible. Maybe, you should actually study and understand the subject first, and only later develop an opinion.
*You still have constant vertical motion!*
I know! But slope is not a measurement of constancy of motion. It is a measurement of coordinate change projected specifically to the plane tangent to the motion. You cannot project motion that is perpendicular to a plane onto that plane, at least not within Euclidean geometry, which is where you have defined your Cartesian coordinate chart. See? Maybe if you actually understood what slope is, instead of being baffled at people, there would be no issues.
*I'll express, ascertain and confirm how it's not undefined and it is ambiguous.*
"Undefined" and "ambiguous" mean the same thing in the English language and in mathematics.
*There is a difference between the two...*
No, there is not.
*sin(90)/cos(90) each limit in both the numerator and denominator exist and are independently equal....*
Are you stupid? How are they equal? sin(π/2) = 1, while cos(π/2) = 0. And they are independent. Remember that cos(x) = sin(x + π/2).
*...yet through trig substitution tan(90) posses a problem?*
You are so ignorant. Forget sin(π/2). Can you tell me what 1/cos(π/2) is? Yes, cos(π/2) is defined. 1/cos(π/2) is not. The limit expression lim 1/cos(x) (x -> π/2) does not exist. I can prove that it does not exist. And yet lim cos(x) (x -> π/2) does exist.
*So far we have expressed just about every aspect of mathematics from linear equations, to trig functions, to some polynomials, some geometry,...*
Are you kidding?! Every aspect of mathematics?! No! Where on Earth did you mention category theory, topological spaces, simplex theory, differential algebra, Galois theory, graph theory, googology, computability theory, combinatorial game theory, functional analysis, measure theory, type theory, order theory, lattice theory, group theory, set theory, complex analysis, non-Euclidean geometry, tensor analysis, or anything else? You are so arrogant that you think you have covered every aspect of mathematics, yet you have not touched even one percent of one percent of it all.
*The + operator is a linear transformation.*
No, it must definitely is not a linear transformation. A linear transformation is defined as a homomorphism between two vector spaces. You really have no clue of what you are talking about, do you? You could not bother to just take a quick look at Wikipedia?
*The act of translating the vector defined from (1, 0) - (0, 0) to its new position defined by (2, 0) - (1, 0),...*
That is not how vectors work. A vector in Euclidean space is not a line segment from one point to another. A vector in Euclidean space is a point in Euclidean space. (3, 5) is a vector. In physics, we would represent it by 3 *i* + 5 *j.* (0, 1) is a vector. (1, 0) is a vector. (0, 0) is a vector, called the zero vector. You are confusing a vector space with an affine space.
*Well, we have a perfect circle, full rotation of 360° or 2π radians. If division by 0 was undefined,... how is that your tires can do full revolutions? How is it that time or a clock with hands on the walk don't stop as long as they have power to run?*
Do you seriously not understand how rotations work? Doing a rotation does not require dividing by 0 at all, at any point.
*You see, the universe doesn't see division by 0 as undefined.*
It absolutely does, which is why in physics, you never divide by 0, in the equations that describe the universe.
*To fully understand what is happening during the vertical phase shifts, which are +/-90 and +/-270...*
Okay, so you really have no clue of what you are talking about. If you did, you would know the shifts, or whatever they are called, also occur at +/-450°, +/-630°, +/-810, etc. I cannot believe you are trying to lecture people with college level education in mathematics and think you know better, all while being this ignorant.
*...we have to take the same approach when people started asking about what is sqrt(-1) to fully understand this.*
No. That approach has been tried, and it does not work. It can be proven it does not work. There is no logical contradiction that emerges from considering an algebraically closed field where i^2 = -1. Was it counterintuitive? Yes, but not logically impossible. But 0 having an inverse is logically impossible: it is in the same category of impossibilities as a non-circular circle, a married single, a bluish yellow, sqrt(2) being rational, a solution to the inequality x < x, or a solution to the equation |x| = -1. These things are not merely counterintuitive. They are actually impossible, and it can be proven that they are.
*The right triangle has an area of 0, yet it still maintains an interior summation of angles at 180 degrees.*
No, it does not, because it is not even a triangle anymore, as it does not have 3 distinct angles.
*It still has 3 vertices.*
No, it has two.
*The adj leg has a length of 0.*
A line segment cannot have a length of 0. That is literally part of the definition of a line segment. By definition, a line segment is connects two _distinct_ points. A "line segment of length 0" is literally just a point.
*I don't agree with the terminology that division by 0 or tan(90) is undefined*
Because you do not understand it.
*Being ambiguous not mean undefined.*
Within CC+, it does not. Mathematics does not care. The English language does not care. And some programming languages are inferior to others.
@@angelmendez-rivera351 The y-plane is the vertical plane that is perpendicular to the horizontal xz plane. The y-plane consists of both the yx and yz portions. It's straight up. The horizontal plane is aligned with the horizon and in the real world is typically based off of sea level. If you ever drive through the mountains they will have signs with the gradient of the mountain or its steepness. This is slope. The slope is a physical quantity with respect to two vectors that generates an angle, especially when one of those vectors is fixed to the horizon.
Is this video related Geometric Algebra ? I saw lots of similarities.
Conformal Geometric Algebra is precisely the algebra that describes these angle-preserving transformations. It rarely gets much attention since people tend to focus more on pure Euclidean Geometric Algebra, Projective Geometric Algebra, or the Spacetime Algebra depending on their field.
I am really hoping that Mathemaniac makes a video about Geometric Algebra at some point.
its like there is 1 of something that will be given to 0 persons, it can't be given to 0 persons because that just doesn't work like that, so that 1 of something just stays here and never gets given, so technically its infinity as long its still will be given to 0 persons