Sounds like this series will lead up to the generalized Stokes Theorem, which explains why, in any and every case, the boundary of a region contains information about the interior of the region. Looking forward to that!
@@BlueGiant69202 I don't know enough about geometric calculus yet at this moment. For now, I just lumped it in with differential geometry, but I will probably regret that at some point ;-) If you have any good (= understandable) references, do let me know.
The 3rd video in the complex numbers series will give a (simple!) example of a Lie group. Nothing too advanced yet of course, but it should provide you with a strong first intuition.
@@AllAnglesMath- I was very happy to see "Road To Reality" in the monoid video. I think the first 380 pages of that book as a higher mathematics primer is great but some worked out examples would be really help. You are effectively bridging that gap for me. Thanks!
Good heavens! You’ve made me giddy in anticipation! This intro is top notch and promises a lot of information I am interested in. I don’t have many subscriptions but I subscribed INSTANTLY. I will eagerly follow your videos. Good luck.
I really love your channel icon. It's really cool that the second symbol representing "Angle" doesn't look like an A, and instead is a triangle with an angle.
@12:20 is EXACTLY whats I've been trying to understand on my own for over 5 years. No one says it as concisely as you did here for some reason, nor do they ever seem to elaborate on WHAT exactly about the maths make them resemble particles, or WHAT about the maths resemble forces (in local symmetries).
I hope I will be able to shed some light on this. It's a long shot, and we have a lot of math to cover first. If you have discovered useful resources during your own 5 years of research, feel free to share them.
@@AllAnglesMath the "representation theory and particle physics" wiki page is very cool but too complicated for me. There are a few full physics lecture series on youtube like Alex Flournoy seems to give extremely cool in depth lectures on particles as irreducible representations of groups, and all the abelian, and non abelian lie groups. They are far too complicated for me. Cohl Furey also has a short series on Octanions and she shows how they apparently admit the charges, spins, and other quantum numbers of fermions. It's very interesting.
@@andrewferris8169 I have watched the Alex Flournoy lectures once. I really enjoyed them, but I understood less than 10% of it. Maybe in the end I will sit down with someone who can explain it to me face to face. We'll see.
@@AllAnglesMath SAME lol. The Octanion one was actually much shorter and she explained it in terms of the ladder creation and anhillation operators and she directly shows how you get the quantum numbers of leptons and the correct variety. I'd give it a watch
havent watched the video yet, but my favorite unexpected connection in 'higher' math is the connection between the group theory Monster and modular forms. its the most unexpected and surprising
@@AllAnglesMath You can use the Monstrous Moonshine as a stepping stone. You can start setting it up by making a series of videos of how to make normal Moonshine, how to consume it correctly etc etc, and while totally drunk slowly teach us more and more advanced mathematical concepts.
@@AllAnglesMath What do you mean "lot of time to set up"? To even start thinking about covering anything related to monster groups or modular forms you would first of all need to know anything about those things, which you clearly don't Worse, you don't even seem to be aware that these things are notoriously difficult. I suppose only a dozen of people have really understood the moonshine, e.g. Borcherds, Conway, Norton It's amusing to see an engineer (?) with a basic understanding of maths light-heartedly contemplating making an educational (!) video on an extremely specialistic pure topic
Hyped for this! I would also like to ask if you would include geometric calculus? TH-cam does not have any good videos on this topic and i believe you would do a great job on it.
Beautiful high level overview of these connections. Simply pointing out the generalisations of the Fundamental theorem of Calculus and what information boundaries of a region have for that region was great. So often, these high level motivations are lost in college classes on these subjects- all how and no why. I honestly don't think I ever heard any lecturer explicitly talk about discreteness/continuity at the heart of maths and what reconciles these. You're left guessing, trying to figure out if e.g. Lie groups and Algebras are the thing that does this. All the while the lecturer is talking about the mechanics of their specific area and not connecting and motivating.
What I have been looking for after years of watching TH-cam science videos. Finally. This would be amazing. I am interested to see all math connections, but when you mentioned foundations and information theory in the same topic, you got me completely. Looking forward to it.
Thank you! Discovered your channel just today! And I find so many areas of mathematics interesting that learning how they are related is an excellent and important subject for me!
At around 8:36, you mentioned that 'the boundry of an area(region) contains information about the inside of an area'. This fact can also be demonstrated by the following simple example. If we have a circle with circumference c, then its area is c^2/(4*pi). In other words, knowing an atribute of the boundry of a region(here the circle's circumference ) will let us infer about some atribute of the region(here circle's area).
Complying with the electric circuit presented after time 11:45, the followig corrections are essential: Sorry , as the bowser doesn't account for subscripts formatting. On the circuit: Change V = I. R to Vsubscript R = I.R . Change V = - L*dI/dt to Vsubscript L= L*dI/dt . No negative sign. Change I=C dV/dt to I=C*d Vsubscript C/dt. Such changes are needed as every circuit component has its own different voltage which is not the DC source having value V. V is the sum of all such votages since the resistor, inductor, and capacitor are all connected in series as Kirchhoff's Voltage Law (KVL) necessitates. Apart from that: good effort and looking forward for your exposition of higher mathematics.
Very good point. Your approach makes things more precise, and precision is always a good thing. I don't think it's a major issue for this video, but still worth pointing out. Thanks!
To me, this video is old-school. Geometric Algebra and Geometric Calculus enable a larger and more unified overview of these topics and should not be listed last but possibly second after either Group Theory or Complex Numbers(historical introduction).
Oh man I would have greatly benefited from videos like this a few years ago! Is a geometric algebra a kind of exterior algebra? We used the latter a lot in a GR class
Exterior algebra uses a new "wedge product" that combines vectors into areas, volumes, etc. Geometric algebra goes even further than that: it introduces the "geometric product" which combines the advantages of the wedge product and the inner product. Stay tuned ;-)
The exterior algebra is effectively geometric algebra with a degenerate metric. Geometric algebras can be classified by Cl(p, q, r) ("Cl" for "Clifford", the guy who first came up with it. Some people use "G" instead.) Where p, q, and r are the number of basis vectors that have a positive, negative, or zero square respectively. An nD exterior algebra can be modeled as Cl(0,0,n).
In real projective geometry of dimension n, the hyperplane of dimension (n-1) is dual to the 0D point. Eg, in the plane, 2 points define a line and 2 lines intersect at a point. In 3-space, 3 points define the plane and 3 planes intersect at a point. Of course we are talking about general cases and not collinear points or overlapping planes.
Something that would be really useful, is a set of recommendations of relatively friendly textbooks for the self-studier who would like to pursue these areas more. Ones which include lots of intuition and motivation, and are well-written, rather than the terse and austere one's suited to math majors.
This is a great idea. The problem is that such textbooks are extremely rare. I put recommended links under each video, and I will place any good textbooks there if I can find them. People can also share their own favorites in the comments.
@@AllAnglesMath Great, sounds good. I would recommend Blanchard/Devaney for Differential Equations. And Lay for Linear Algebra - it's not the most thrilling, but it is very well organized and 'didactic', especially if you have the Student Guide.
Awesome, can't wait to see more. Keep up the exciting and interesting work. Also, can't help but think of Hegel when you talk about how everything is connected to everything else and how one technique turns out to be a special case of some more general technique. Seems to me Hegel really captured the evolution of logic and the limits of human reasoning to see difference when there is in fact deep similarity waiting to be discovered.
@12:30 - Is That True??? "the fundamental particles and forces of the standard model of physics automatically appear" If that's true. I don't know, but that is what I really want to see explained. please.
We plan to make videos about all the topics in the overview. That's why the current series covers complex numbers and the next one covers group theory. If all goes according to plan, we will get to geometric algebra early next year.
Perhaps a more robust definition of number theory would be less self-referencing and the perceptual illusions of group, category, and set theory wouldn't be such disparate solutions.
@@koenvandamme9409 First, I do applaud this effort. Abstract algebra also does some of this. And that's where there may be 'number" systems where one operator (*) may not be simply be a repeated operation of another operator (+). Yes, with integers under addition and multiplication, repeating one operator to get another it works nicely. Like with 5 and 2. You an have five 2s or two 5s. But in say the real numbers multiplying pi and e, how do you have pi number of e's or the reverse. To work out the answer would take infinite time. There are other number systems like hyper complex numbers or matrices. We might have to invent new number systems and operators to understand the Universe. Addition and multiplication get us close, but might not be the TRUE answer. We need to keep our minds open.
it never bother mathematicians that the square root of negative one has no solution, no value, and is a logical confabulation......so much of math and physics is based on calculations using a number that doesn't exist, or if it exists it exists in the realm of fairy tales and unicorns and gorgons where it has a totem but no result or value or magnitude.....the square root of negative one exists in the same way infinity exists, as a logical absurdity, and both are at the very center of all our mathematical interpretations of 'reality'
I mean, it kind of bothered many mathematicians, and physicists definitely didn't like using imaginary numbers at first, but when they used them, the formulas finally worked and explained the reality. Also, if you really think about it, you could argue, that negative numbers don't exist either, because you can't just show me -5, but it is a useful concept, which helps us with understanding the reality. So despise the unfortunate name complex numbers got, they do exist, the nature uses it, just as much as it uses pi or -⅝ (possibly more than it uses -⅝)
I agree that the "square root of negative one" does not make sense. But I blame that on the instructors. Rather, I view complex numbers as vectors in two dimensions. This is also how they are treated in computers. In this way, everything consists of the usual real numbers.
But if it is such a non-existent absurdity, how come physical theories which are extremely empirically successful, make heavy use of it? You might say the same about zero. In fact, you might say the same about a Real number - the ancient Greeks took the same view that you are stating, about irrational numbers, after all. I think you either have to let your standards of what is absurd be set by "what works"; or alternatively, come up with a totally new set of structures for representing mathematical ideas in which things like the "complex numbers" are reinterpreted as something more in line with common sense.
no logical absurdity here. only a very simple construction to characterize roots of polynomials that turns out to underpin pretty much all of algebra. if they seem so out of touch for you, you have not done any relevant math
Sounds like this series will lead up to the generalized Stokes Theorem, which explains why, in any and every case, the boundary of a region contains information about the interior of the region. Looking forward to that!
Notice the failure to mention Geometric Calculus?
@@BlueGiant69202 I don't know enough about geometric calculus yet at this moment. For now, I just lumped it in with differential geometry, but I will probably regret that at some point ;-) If you have any good (= understandable) references, do let me know.
@@BlueGiant69202so.. where are the sources? or did you just want to dunk on the creator for internet points
I wish I had these videos when I was studying this stuff in college...
I thought having even a glimmer of understanding what Lie Group are and do was years away. I feel much better prepared to study them now.
The 3rd video in the complex numbers series will give a (simple!) example of a Lie group. Nothing too advanced yet of course, but it should provide you with a strong first intuition.
@@AllAnglesMath- I was very happy to see "Road To Reality" in the monoid video. I think the first 380 pages of that book as a higher mathematics primer is great but some worked out examples would be really help.
You are effectively bridging that gap for me. Thanks!
@@edvogel56 Eigenchris is currently running a series on Spinors that I can highly recommend.
@@AllAnglesMath Thanks!
Good heavens! You’ve made me giddy in anticipation! This intro is top notch and promises a lot of information I am interested in. I don’t have many subscriptions but I subscribed INSTANTLY. I will eagerly follow your videos. Good luck.
I really love your channel icon. It's really cool that the second symbol representing "Angle" doesn't look like an A, and instead is a triangle with an angle.
Yeah, we spent some time looking for a logo that isn't fully symmetric. And you're right that the angle serves double duty as a letter A.
I can't wait for everything that's to come!
This looks (and sounds) promising.
Also, I think your graphics style is superb; pleasing in terms of both esthetics and didactics.
Thank you!
I'm excited to see how this series plays out!
@12:20 is EXACTLY whats I've been trying to understand on my own for over 5 years. No one says it as concisely as you did here for some reason, nor do they ever seem to elaborate on WHAT exactly about the maths make them resemble particles, or WHAT about the maths resemble forces (in local symmetries).
I hope I will be able to shed some light on this. It's a long shot, and we have a lot of math to cover first. If you have discovered useful resources during your own 5 years of research, feel free to share them.
@@AllAnglesMath the "representation theory and particle physics" wiki page is very cool but too complicated for me. There are a few full physics lecture series on youtube like Alex Flournoy seems to give extremely cool in depth lectures on particles as irreducible representations of groups, and all the abelian, and non abelian lie groups. They are far too complicated for me. Cohl Furey also has a short series on Octanions and she shows how they apparently admit the charges, spins, and other quantum numbers of fermions. It's very interesting.
@@andrewferris8169 I have watched the Alex Flournoy lectures once. I really enjoyed them, but I understood less than 10% of it. Maybe in the end I will sit down with someone who can explain it to me face to face. We'll see.
@@AllAnglesMath SAME lol. The Octanion one was actually much shorter and she explained it in terms of the ladder creation and anhillation operators and she directly shows how you get the quantum numbers of leptons and the correct variety. I'd give it a watch
Just 1 minute of the video make me click the subscribe button
Bravo ! Marvellous
IF only I had seen heard this in 1970 age 15 !!! , or even better aged 12 (1967)
keep these videos coming please
Thanks
So excited for the series! I hope that more people can see how cool and beautiful math is.
excited to follow you on your journey
A very in depth explanation, thanks. Keep on uploading, cause finding math and physics channels are so rare.
this is a very ambitious video. I subscribe right away so I don't miss any. 👍👍👍
10:53 can't wait until that little d 😍
havent watched the video yet, but my favorite unexpected connection in 'higher' math is the connection between the group theory Monster and modular forms. its the most unexpected and surprising
That is definitely an "out of nowhere" connection. It would take a lot of time to set up, so I don't know if I will ever get to it.
@@AllAnglesMath You can use the Monstrous Moonshine as a stepping stone. You can start setting it up by making a series of videos of how to make normal Moonshine, how to consume it correctly etc etc, and while totally drunk slowly teach us more and more advanced mathematical concepts.
@@AllAnglesMath What do you mean "lot of time to set up"? To even start thinking about covering anything related to monster groups or modular forms you would first of all need to know anything about those things, which you clearly don't
Worse, you don't even seem to be aware that these things are notoriously difficult. I suppose only a dozen of people have really understood the moonshine, e.g. Borcherds, Conway, Norton
It's amusing to see an engineer (?) with a basic understanding of maths light-heartedly contemplating making an educational (!) video on an extremely specialistic pure topic
@@adamzajkowski9669 this comment is unnessesarily hostile out of nowhere?
shhh let him have his rant
Looking forward to future videos!
Hyped for this!
I would also like to ask if you would include geometric calculus? TH-cam does not have any good videos on this topic and i believe you would do a great job on it.
I can't make any promises.
Alan Macdonald would be surprised to see that. He has good videos on both Geometric Algebra and Geometric Calculus.
Looking forward to this series!
Thoughtful and beautifully executed. I'm excited to see more.
Beautiful high level overview of these connections. Simply pointing out the generalisations of the Fundamental theorem of Calculus and what information boundaries of a region have for that region was great. So often, these high level motivations are lost in college classes on these subjects- all how and no why. I honestly don't think I ever heard any lecturer explicitly talk about discreteness/continuity at the heart of maths and what reconciles these. You're left guessing, trying to figure out if e.g. Lie groups and Algebras are the thing that does this. All the while the lecturer is talking about the mechanics of their specific area and not connecting and motivating.
Thanks! Making these kinds of connections very explicit is one of the main goals of the channel.
What I have been looking for after years of watching TH-cam science videos. Finally. This would be amazing. I am interested to see all math connections, but when you mentioned foundations and information theory in the same topic, you got me completely. Looking forward to it.
This is what I am looking for ❤️💯
Wonderful intro n mission statement
Interesting premise. Subscribed. I will be waiting for the series to unfold
Thank you! Discovered your channel just today! And I find so many areas of mathematics interesting that learning how they are related is an excellent and important subject for me!
I think I'm hooked and you haven't started yet!
The animations and explanations were great! I will certainly be along for the ride!
At around 8:36, you mentioned that 'the boundry of an area(region) contains information about the inside of an area'. This fact can also be demonstrated by the following simple example.
If we have a circle with circumference c, then its area is c^2/(4*pi). In other words, knowing an atribute of the boundry of a region(here the circle's circumference ) will let us infer about some atribute of the region(here circle's area).
You might also want to check out the integral definition of the derivative in higher dimensional Geometric Calculus.
Complying with the electric circuit presented after time 11:45, the followig corrections are essential:
Sorry , as the bowser doesn't account for subscripts formatting.
On the circuit:
Change V = I. R to Vsubscript R = I.R .
Change V = - L*dI/dt to Vsubscript L= L*dI/dt . No negative sign.
Change I=C dV/dt to I=C*d Vsubscript C/dt.
Such changes are needed as every circuit component has its own different voltage which is not the DC source having value V.
V is the sum of all such votages since the resistor, inductor, and capacitor are all connected in series as Kirchhoff's Voltage Law (KVL) necessitates.
Apart from that: good effort and looking forward for your exposition of higher mathematics.
Very good point. Your approach makes things more precise, and precision is always a good thing. I don't think it's a major issue for this video, but still worth pointing out. Thanks!
Nice overview !
Thanks for what looks like a great series showing how different areas of math are related. Great work thus far!
Very much looking forward to this series. Thank you : )
I'm excited for this! one tip I suggest is making the videos have a black background (much easier on the eyes)
Thanks for the tip, I will consider it.
Looking forward to watch how your exposition plan develops, congrats on a very promising start! 🙂
love the animations! what software do you use?
I developed my own Python library over time. It uses OpenCV for rendering.
I see a great promise in this channel. I have already subscribed. I am waiting for your content!
I love these animations Keep it up :)
looking forward to seeing the evolution of this series
The explanations along with the animations were top notch. Great job! Will surely be following this channel and I hope it grows!!
Very excited for the content and especially group theory. Hope you introduced in a way that it inspires to learn more about them.
After complex numbers, group theory is next. We're already working on those videos.
Instant subscriptio. Quality work.
Here before this channel blows up.
Note to future self: I was the 56th subscriber.
Yeah let’s hope! 👏
Let's do it
Excellent video. Just subscribed to join for the ride.
I needed this thank you!
Look forward to some very interesting content!
Subscribing.
great intro to your content not sure if that was intentional but i am subbed to you now and pairing this with quantum sense
I agree that Quantum Sense has some brilliant videos about quantum physics.
Thank you for the video.
Russel's paradox is a result of applying set theory to math, especially sets of sets.
Looking forward to seeing your videos
So hyped rn
Looking forward to watching your videos. So far this video seemed promising.
Cant wait for the playlist!
Thank you
So glad to see this type of content. You make really quality content.
To me, this video is old-school. Geometric Algebra and Geometric Calculus enable a larger and more unified overview of these topics and should not be listed last but possibly second after either Group Theory or Complex Numbers(historical introduction).
3blue1brown has a competition now :D
Oh man I would have greatly benefited from videos like this a few years ago! Is a geometric algebra a kind of exterior algebra? We used the latter a lot in a GR class
Exterior algebra uses a new "wedge product" that combines vectors into areas, volumes, etc. Geometric algebra goes even further than that: it introduces the "geometric product" which combines the advantages of the wedge product and the inner product. Stay tuned ;-)
The exterior algebra is effectively geometric algebra with a degenerate metric. Geometric algebras can be classified by Cl(p, q, r) ("Cl" for "Clifford", the guy who first came up with it. Some people use "G" instead.) Where p, q, and r are the number of basis vectors that have a positive, negative, or zero square respectively. An nD exterior algebra can be modeled as Cl(0,0,n).
I can't wait to see more!
good stuff
Looking forward to your videos!
In my life I was capable to pursue higher math study. Now my life is quite happy and I am glad i didnt choose that path.
Full support brother
I think you shouldve speficied that linear algebra studies finite dimensional vector spaces
Wonderful
Looks awesome, Im waiting for more
In real projective geometry of dimension n, the hyperplane of dimension (n-1) is dual to the 0D point.
Eg, in the plane, 2 points define a line and 2 lines intersect at a point.
In 3-space, 3 points define the plane and 3 planes intersect at a point.
Of course we are talking about general cases and not collinear points or overlapping planes.
Something that would be really useful, is a set of recommendations of relatively friendly textbooks for the self-studier who would like to pursue these areas more. Ones which include lots of intuition and motivation, and are well-written, rather than the terse and austere one's suited to math majors.
This is a great idea. The problem is that such textbooks are extremely rare. I put recommended links under each video, and I will place any good textbooks there if I can find them. People can also share their own favorites in the comments.
@@AllAnglesMath Great, sounds good. I would recommend Blanchard/Devaney for Differential Equations. And Lay for Linear Algebra - it's not the most thrilling, but it is very well organized and 'didactic', especially if you have the Student Guide.
Awesome, can't wait to see more. Keep up the exciting and interesting work. Also, can't help but think of Hegel when you talk about how everything is connected to everything else and how one technique turns out to be a special case of some more general technique. Seems to me Hegel really captured the evolution of logic and the limits of human reasoning to see difference when there is in fact deep similarity waiting to be discovered.
This is amazing!
Show the beauty of mathematics 💪
Will you also cover the curvature of a circle?
great job
The big plex appears at 12:10 if All Angles allows this shortcut for watchers who want to spot this once more.
Well presented.
awesome video! thx 1000x
already subbed
Sick as hell
@12:30 - Is That True???
"the fundamental particles and forces of the standard model of physics automatically appear"
If that's true. I don't know, but that is what I really want to see explained. please.
The only thing I can say right now is that it will take some time until we get there ;-)
Hyped as fuck
math itself is a language, logic provides the rule for the language (contradictions are forbidden).
Im very excited lol
Fundamental group of SO3 = Zmod2. My mind was blown 🤯 Haven’t watched the vid yet.
verry cool
Hype! ✨🙌✨
How did I make it through advanced calculus. I feeling like I did when I was sitting there in class.
What do you use to render these animations? Manim?
I use my own python library with OpenCV for the rendering.
Nice
Topology has left the chat.
12:06 this is not necessarily true. You can also have linearly polarized waves
Can you do a video on geometric algebra?
We plan to make videos about all the topics in the overview. That's why the current series covers complex numbers and the next one covers group theory. If all goes according to plan, we will get to geometric algebra early next year.
@@AllAnglesMath keep it up
looks promising....
Perhaps a more robust definition of number theory would be less self-referencing and the perceptual illusions of group, category, and set theory wouldn't be such disparate solutions.
At the same time, I didn't find this surprising. Even though it was. Math is weird.
I think it's a mistake to think of multiplication as simply repeated addition. Same with multiplication and exponentiation.
What would be your alternative way of defining it?
@@koenvandamme9409 First, I do applaud this effort. Abstract algebra also does some of this. And that's where there may be 'number" systems where one operator (*) may not be simply be a repeated operation of another operator (+).
Yes, with integers under addition and multiplication, repeating one operator to get another it works nicely. Like with 5 and 2. You an have five 2s or two 5s. But in say the real numbers multiplying pi and e, how do you have pi number of e's or the reverse. To work out the answer would take infinite time. There are other number systems like hyper complex numbers or matrices. We might have to invent new number systems and operators to understand the Universe. Addition and multiplication get us close, but might not be the TRUE answer. We need to keep our minds open.
do people really learn Calc in high school ?
to me it was in college that we got pre-calc and calc 1 & 2
I didn't get the "surprising" part.
Please stop saying I should have known something since high school, we are not all the same.
it never bother mathematicians that the square root of negative one has no solution, no value, and is a logical confabulation......so much of math and physics is based on calculations using a number that doesn't exist, or if it exists it exists in the realm of fairy tales and unicorns and gorgons where it has a totem but no result or value or magnitude.....the square root of negative one exists in the same way infinity exists, as a logical absurdity, and both are at the very center of all our mathematical interpretations of 'reality'
I mean, it kind of bothered many mathematicians, and physicists definitely didn't like using imaginary numbers at first, but when they used them, the formulas finally worked and explained the reality. Also, if you really think about it, you could argue, that negative numbers don't exist either, because you can't just show me -5, but it is a useful concept, which helps us with understanding the reality. So despise the unfortunate name complex numbers got, they do exist, the nature uses it, just as much as it uses pi or -⅝ (possibly more than it uses -⅝)
I agree that the "square root of negative one" does not make sense. But I blame that on the instructors.
Rather, I view complex numbers as vectors in two dimensions. This is also how they are treated in computers.
In this way, everything consists of the usual real numbers.
But if it is such a non-existent absurdity, how come physical theories which are extremely empirically successful, make heavy use of it? You might say the same about zero. In fact, you might say the same about a Real number - the ancient Greeks took the same view that you are stating, about irrational numbers, after all. I think you either have to let your standards of what is absurd be set by "what works"; or alternatively, come up with a totally new set of structures for representing mathematical ideas in which things like the "complex numbers" are reinterpreted as something more in line with common sense.
@@MichaelFJ1969 yeah dude. completely natural to multiply vectors as (a,b)×(c,d)=(ac-bd,ad+bc)
no logical absurdity here. only a very simple construction to characterize roots of polynomials that turns out to underpin pretty much all of algebra. if they seem so out of touch for you, you have not done any relevant math
traitor
Engineering is based on science according to crappy education system. Engineering is the mother of everything in real life.