What's is more interesting is that all operations involving multiplication with imaginaries, exist because we assumed long ago that operations have to be symmetric (commutative), but group theory down the line shows us that is not necessarily the case. That's what makes operation on 2D with complexes and in 3D with quaternions be not only isomorphic with Linear Algebra, but they're actually fathers of it. As uncovered more explicitly by Geometric Algebra.
Really incredible insight on a lot of higher concepts. Losing structure as you go up in dimensions for number systems in particular was something I never considered
While the Cayley-Dickson Construction, which generates the quaternions from the ℂomplex numbers, and octonions from the quaternions, loses structure as you repeat the process, Clifford Algebras, which for the algebraic basis for Geometric Algebras, lose commutativity and inverses for every element, but _never_ lose the other desirable properties like associativity. This is very important, especially for geometry, since associativity is the ability to compose two actions, represented as discrete elements of the algebra, and get a new element that preforms the previous two actions in sequence. Rotations are easy to confirm geometrically or physically that they are commutative in 2D, but not in 3D or higher, but any geometric transformation is associative almost by-definition. Clifford Algebra can generate a 2^N dimensional associative algebra no matter how large N is. You never need to worry about losing that structure.
Sedenions (16D) lose alternativity and they have zero divisors. Trigintaduonions (32D) lose distributivity as well. Power associativity is presumably lost at the sexagintaquatronions (64D).
Question: in the section @10:30, you mentioned that complex numbers don't have ordering, which suggests that if there are numbers with a defined inequality, due to the 'rotation' property, the inequality doesn't hold between the two numbers after rotation. But can't we use the norm to define distance? this could apply to all numbers in the complex plane right? Like all the numbers on a circle of unit length 1 will be less any other bigger circle, isn't that an order? (although all the number on the circle would be equal, but I am not sure if that breaks ordering)
Hi, thanks for your excellent question. The short answer is: it's totally possible to define an ordering for the complex numbers, but you can't make it cooperate with the arithmetic operations. The long answer: you could define an ordering based on the norm like you propose. For example, a complex number with norm 1 is less than a complex number with norm 2. So far so good. This definition isn't complete yet, because you also have to say what happens when two numbers have the same norm. OK, we could use polar coordinates and say that a complex number with a smaller angle is less than a number with a larger angle. You then combine the angle and the norm, and you get a perfectly reasonable ordering. There is no problem at all. But then you want to multiply your complex numbers. Say you have two numbers with the same norm, but one of them has a smaller angle: c < d. You then multiply both of them with a unit complex number that rotates them both around the origin. Depending on the exact angle of rotation, you may now get c < d or d < c. The angles may or may not wrap around back to zero, depending on how much you rotate. So you have no way to predict the behavior of the ordering in advance. So it's not the ordering in itself that causes problems, but its interaction with arithmetic operations.
This is exactly what math is made of: Asking creative questions, and then working out all the results. Can you turn your ideas into a number system? Which properties does it have? Which ones does it miss?
Nice video! I want to become a Patron, but currently it seems I'm not able to because your lowest tier is $5 and I'm not in a position to give that much at this time. (Not sure if this is a fault of the Patreon site design and there's some no-tier workaround.)
Thank you for your interest in the channel. I don't think there is currently a tier available below $5. In the meantime, you can already support us by liking and sharing the videos.
What's is more interesting is that all operations involving multiplication with imaginaries, exist because we assumed long ago that operations have to be symmetric (commutative), but group theory down the line shows us that is not necessarily the case. That's what makes operation on 2D with complexes and in 3D with quaternions be not only isomorphic with Linear Algebra, but they're actually fathers of it. As uncovered more explicitly by Geometric Algebra.
Enlighten and delightful are the words. Excellent video. Keep going.
Really incredible insight on a lot of higher concepts. Losing structure as you go up in dimensions for number systems in particular was something I never considered
While the Cayley-Dickson Construction, which generates the quaternions from the ℂomplex numbers, and octonions from the quaternions, loses structure as you repeat the process, Clifford Algebras, which for the algebraic basis for Geometric Algebras, lose commutativity and inverses for every element, but _never_ lose the other desirable properties like associativity. This is very important, especially for geometry, since associativity is the ability to compose two actions, represented as discrete elements of the algebra, and get a new element that preforms the previous two actions in sequence. Rotations are easy to confirm geometrically or physically that they are commutative in 2D, but not in 3D or higher, but any geometric transformation is associative almost by-definition.
Clifford Algebra can generate a 2^N dimensional associative algebra no matter how large N is. You never need to worry about losing that structure.
Excellent video. It gives a great overview of higher mathematical concepts.
Excellent presentation. The graphics are great and the pace is pleasant. And mist if all the subject matter is fascinating. Thank you
Thanks! Glad to hear that you enjoyed the video.
We gain algebraic completeness when going from real to complex. Is there anything we gain when going to quaternions and octonions?
Great question! Unfortunately, I don't know enough about quaternions or octonions to have the answer.
Sedenions (16D) lose alternativity and they have zero divisors.
Trigintaduonions (32D) lose distributivity as well.
Power associativity is presumably lost at the sexagintaquatronions (64D).
Question: in the section @10:30, you mentioned that complex numbers don't have ordering, which suggests that if there are numbers with a defined inequality, due to the 'rotation' property, the inequality doesn't hold between the two numbers after rotation.
But can't we use the norm to define distance?
this could apply to all numbers in the complex plane right? Like all the numbers on a circle of unit length 1 will be less any other bigger circle, isn't that an order? (although all the number on the circle would be equal, but I am not sure if that breaks ordering)
Hi, thanks for your excellent question.
The short answer is: it's totally possible to define an ordering for the complex numbers, but you can't make it cooperate with the arithmetic operations.
The long answer: you could define an ordering based on the norm like you propose. For example, a complex number with norm 1 is less than a complex number with norm 2. So far so good. This definition isn't complete yet, because you also have to say what happens when two numbers have the same norm. OK, we could use polar coordinates and say that a complex number with a smaller angle is less than a number with a larger angle. You then combine the angle and the norm, and you get a perfectly reasonable ordering. There is no problem at all.
But then you want to multiply your complex numbers. Say you have two numbers with the same norm, but one of them has a smaller angle: c < d. You then multiply both of them with a unit complex number that rotates them both around the origin. Depending on the exact angle of rotation, you may now get c < d or d < c. The angles may or may not wrap around back to zero, depending on how much you rotate. So you have no way to predict the behavior of the ordering in advance.
So it's not the ordering in itself that causes problems, but its interaction with arithmetic operations.
Nice
Excellent video!
Hmmm if that so what about a number m : M^2=M and M≠1 M≠0 ?
And we have: (a+bM)(c+dM)=ac+(ad+bc+cd)M
Or let M^2=-M
To get
(a+bM)(c+dM)=ac+(ad+bc-cd)M
This is exactly what math is made of: Asking creative questions, and then working out all the results. Can you turn your ideas into a number system? Which properties does it have? Which ones does it miss?
Nice video. Just a typo on one of your slides where you have 4^2 + 1 = 5.
Oops! You're absolutely right. I will put a remark in the description. Thanks for your vigilance.
Nice video! I want to become a Patron, but currently it seems I'm not able to because your lowest tier is $5 and I'm not in a position to give that much at this time. (Not sure if this is a fault of the Patreon site design and there's some no-tier workaround.)
Thank you for your interest in the channel. I don't think there is currently a tier available below $5. In the meantime, you can already support us by liking and sharing the videos.