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The Peculiar Math That Could Underlie the Laws of Nature

21 points by galaxyLogic 6y ago

galaxyLogic 6y ago
I found the PDF in this presentation on reals, complex numbers, quaternions and octonions enlightening. What I learned in school was that we have complex numbers and vectors. Vectors have their own peculiar two types of products, dot-product and cross-product. Complex numbers you just "multiply". I always had this question in the back of my brain, aren't complex numbers and vectors really the same. Why do we need both? SEE ALSO: https://nautil.us/blog/the-strange-numbers-that-birthed-modern-algebra https://nautil.us/blog/the-strange-numbers-that-birthed-mode... Turns out Hamilton invented Quaternions but then "Hostilities broke out when a Yale professor named Josiah Gibbs defined the modern vector." So vectors are like handicapped quaternions
- poormystic 6y ago
  All vectors are not complex because complex numbers involve the square root of negative 1. Complex numbers can be treated as vectors, but vectors are not generally complex.
  - deleted 6y ago
    [deleted]
  - galaxyLogic 6y ago
    Thanks for the tip, I'm trying to understand this thing more. If I understood the articles correctly Quaternions subsume complex numbers, but at the same time they are an alternative for the vector representation. I would like to get rid of vectors if all I need is quaternions. It leaves me a bit puzzled then what kind of an entity is a vector of complex numbers? Could that also be replaced by Quaternions?
    - gizmo686 6y ago
      "Vector" is a fairly abstract algebraic term, that is used for a variety of different purposes. For instance, one example of vectors is the functions who's output is a real number. The point here is that "vector" is far more general than quaternions, so quaternions will not fully replace the need for vectors. Also, quaternions are an example of vectors, so it doesn't really make sense to call them an alternative) Getting back to your original question, it might make things easier to look at a simpler example. Imagine all we had was the rational numbers, called Q. Then Pythagoras comes along and proves that sqrt(2) is not rational. To fix this, we introduce the new number √2; and define a new number system where all number take the form a+b√2. To formalize this system, we say that what these numbers are actually 2 dimensional vectors (a,b) with addition and multiplication defined as follows: (a,b) + (x,y) = (a+x, b+y) (a,b) * (x,y) = (ax+2by, bx+ay) We can call this new number system Q(√2), to reflect the fact that we got it by adding √2 to Q. Note that this is almost the exact vector definition for complex numbers, except that we have a "2" instead of "-1" when we define multiplication. Now, having done that, suppose Pythagoras comes back and points out that we are still missing the square root of 3. We can use the same trick to introduce √3 (a,b) + (x,y) = (a+x, b+y) (a,b) * (x,y) = (ax+3by, bx+ay) Giving us Q(√2)(√3) In this case, a,b,x,y are not rational numbers, but elements of Q(√2). In this was, elements of Q(√2)(√3) can themselves be seen as 2 dimensional vectors of elements of Q(√2). On the other hand, if we wanted to, we could also define Q(√2)(√3) directly as a 4 dimensional vector over Q, which we can express as Q(√2,√3) to reflect the fact that we are constructing it in a single step: (a,b,c,d) + (w,x,y,z) = (a+w, b+x, c+y, d+z) (a,b,c,d) * (w,x,y,z) = (aw+2bx+6c7+3dz, ax+bw+3cz+3dy, ay+bz+cw+dx, az+2by+2cx+dw) Assuming I didn't make a mistake typing out the above, you can verify that it does infact represent the number system where all numbers take the form: a+b√2 + c√2√3 + d√3 And addition and mulitplication behave as you would expect. The thing to notice here is that, while it is straightforward to work out what the multiplication rule for Q(√2,√3) is, it is far from a simple rule. On the other hand, by instead thinking of it as Q(√2)(√3), we can break it down into 2 relativly simple multiplication rules. Simmilarly, if we were to start with Q(√2,√3), we could then view Q(√2) not as a 2 dimensional vector over Q, but instead as a portion of Q(√2,√3) where all the elements take the form (a,b,0,0). As a practical matter, the rule for multiplying in Q(√2,√3) is more complicated then the rule for multiplying in Q(√2), so we tend to prefer to stick with the simpler system until we have a need to go to the more complicated system. In the same way, you can define quaternions directly as a 4 dimensional extension of the real numbers. However, it generally makes life easier to view them as a 2 dimensional extension of the complex numbers. As an aside, the complex numbers are often defined not as a simple ordered pair of real numbers, but rather a 2x2 matrix of real numbers that take the form: a b -b a This works out to give us multiplication "for free" as matrix multiplication; where our original definition required us to invent a new defintion of multiplication. The quaternions are similarly defined as complex values matricies of the form: a+bi c+di -c+di a-bi Or, if you are familiar with the complex conjugate a b -b‾ a‾ Again, you get multiplication "for free" from matrix multiplication. In contrast, if you wanted to define the quaternions directly from the real numbers, they would take the form: a b c d -b a -d c -c d a -b d -c b a Which, again is more complicated then what you get by going through the complex numbers.
- adrian_b 6y ago
  No, actually 2-dimensional vectors and 2-dimensional complex numbers are completely distinct entities. The same is true about 3-dimensional vectors and quaternions and the same is true in higher dimensions. The short explanation is that vectors describe additions (i.e. translations), while complex numbers (and their higher dimensional analogs) describe multiplications (i.e. scaling and rotations). The nature of vectors, pseudovectors, complex numbers, quaternions etc. can be understood only in the context of the theory of geometric algebras. For example, an element of a 2-dimensional geometric algebra has 4 components (2^2). 2 components form a vector and the other 2 form a complex number. Similarly, the 8 components of an element of a 3-dimensional geometric algebra are partitioned into a quaternion, a 3-dimensional vector and the last element, which is a pseudoscalar. The confusion is increased by the fact that the word "vector" is used with 2 different meanings. The original, physical and etymological, meaning is that a vector is a translation of an affine space. This is almost always the meaning of "vector" in physics. Many mathematicians also use the word "vector" for the elements of linear spaces a.k.a. vector spaces. The "vectors" of the second meaning have only a subset of the properties of the "vectors" of the first meaning, i.e. only the properties that can be derived from the axioms of a vector space. The complex numbers and the quaternions, octonions etc. are "vectors" in the second meaning, but they are not "vectors" in the first meaning, i.e. in the original physical meaning.
avmich 6y ago
Why stop at octonions? Sedenions and other steps of Cayley-Dickson construction?..
- surfsvammel 6y ago
  If I read the article correctly, it would seem its not needed.
- zamalek 6y ago
  Due to Hurwitz's theorem, as-per the article.
  - avmich 6y ago
    From https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_constru... - "The Hurwitz's theorem (composition algebras) states that the reals, complex numbers, quaternions, and octonions are the only (normed) division algebras (over the real numbers)." So what? From https://en.wikipedia.org/wiki/Sedenion https://en.wikipedia.org/wiki/Sedenion - "Every sedenion is a linear combination of the unit sedenions e_0...e_15 , which form a basis of the vector space of sedenions." - clearly we still have operations to work with. Stopping at octonions looks somewhat arbitrary.
thunderrabbit 6y ago
> it’s very difficult to think of a nonassociative situation.” If, instead of putting on socks then shoes, you first put your socks into your shoes, technically you should still then be able to put your feet into both and get the same result. “The parentheses feel artificial.” I suggest the parenthesis are not artificial: they specify an order of operations. i.e. they specify the timing of operations. Choosing a, b, c as things that change with time makes it easy to think of nonassociative examples: (Crack open an egg) then (cook it) = (scrambled egg) (Cook an egg) then (peel it) = (boiled egg)
- gizmo686 6y ago
  You changed the order, so your example is not commutative. The associative example would be: (Crack an egg) then (cook it then eat it) vs (Crack an egg then cook it) then (eat it)
  - thunderrabbit 6y ago
    Ohhh shoot. I guess that's egg on my face!

21 points by galaxyLogic 6y ago

I found the PDF in this presentation on reals, complex numbers, quaternions and octonions enlightening. What I learned in school was that we have complex numbers and vectors. Vectors have their own peculiar two types of products, dot-product and cross-product. Complex numbers you just "multiply". I always had this question in the back of my brain, aren't complex numbers and vectors really the same. Why do we need both? SEE ALSO: https://nautil.us/blog/the-strange-numbers-that-birthed-modern-algebra https://nautil.us/blog/the-strange-numbers-that-birthed-mode... Turns out Hamilton invented Quaternions but then "Hostilities broke out when a Yale professor named Josiah Gibbs defined the modern vector." So vectors are like handicapped quaternions

poormystic 6y ago
All vectors are not complex because complex numbers involve the square root of negative 1. Complex numbers can be treated as vectors, but vectors are not generally complex.
- deleted 6y ago
  [deleted]
- galaxyLogic 6y ago
  Thanks for the tip, I'm trying to understand this thing more. If I understood the articles correctly Quaternions subsume complex numbers, but at the same time they are an alternative for the vector representation. I would like to get rid of vectors if all I need is quaternions. It leaves me a bit puzzled then what kind of an entity is a vector of complex numbers? Could that also be replaced by Quaternions?
  - gizmo686 6y ago
    "Vector" is a fairly abstract algebraic term, that is used for a variety of different purposes. For instance, one example of vectors is the functions who's output is a real number. The point here is that "vector" is far more general than quaternions, so quaternions will not fully replace the need for vectors. Also, quaternions are an example of vectors, so it doesn't really make sense to call them an alternative) Getting back to your original question, it might make things easier to look at a simpler example. Imagine all we had was the rational numbers, called Q. Then Pythagoras comes along and proves that sqrt(2) is not rational. To fix this, we introduce the new number √2; and define a new number system where all number take the form a+b√2. To formalize this system, we say that what these numbers are actually 2 dimensional vectors (a,b) with addition and multiplication defined as follows: (a,b) + (x,y) = (a+x, b+y) (a,b) * (x,y) = (ax+2by, bx+ay) We can call this new number system Q(√2), to reflect the fact that we got it by adding √2 to Q. Note that this is almost the exact vector definition for complex numbers, except that we have a "2" instead of "-1" when we define multiplication. Now, having done that, suppose Pythagoras comes back and points out that we are still missing the square root of 3. We can use the same trick to introduce √3 (a,b) + (x,y) = (a+x, b+y) (a,b) * (x,y) = (ax+3by, bx+ay) Giving us Q(√2)(√3) In this case, a,b,x,y are not rational numbers, but elements of Q(√2). In this was, elements of Q(√2)(√3) can themselves be seen as 2 dimensional vectors of elements of Q(√2). On the other hand, if we wanted to, we could also define Q(√2)(√3) directly as a 4 dimensional vector over Q, which we can express as Q(√2,√3) to reflect the fact that we are constructing it in a single step: (a,b,c,d) + (w,x,y,z) = (a+w, b+x, c+y, d+z) (a,b,c,d) * (w,x,y,z) = (aw+2bx+6c7+3dz, ax+bw+3cz+3dy, ay+bz+cw+dx, az+2by+2cx+dw) Assuming I didn't make a mistake typing out the above, you can verify that it does infact represent the number system where all numbers take the form: a+b√2 + c√2√3 + d√3 And addition and mulitplication behave as you would expect. The thing to notice here is that, while it is straightforward to work out what the multiplication rule for Q(√2,√3) is, it is far from a simple rule. On the other hand, by instead thinking of it as Q(√2)(√3), we can break it down into 2 relativly simple multiplication rules. Simmilarly, if we were to start with Q(√2,√3), we could then view Q(√2) not as a 2 dimensional vector over Q, but instead as a portion of Q(√2,√3) where all the elements take the form (a,b,0,0). As a practical matter, the rule for multiplying in Q(√2,√3) is more complicated then the rule for multiplying in Q(√2), so we tend to prefer to stick with the simpler system until we have a need to go to the more complicated system. In the same way, you can define quaternions directly as a 4 dimensional extension of the real numbers. However, it generally makes life easier to view them as a 2 dimensional extension of the complex numbers. As an aside, the complex numbers are often defined not as a simple ordered pair of real numbers, but rather a 2x2 matrix of real numbers that take the form: a b -b a This works out to give us multiplication "for free" as matrix multiplication; where our original definition required us to invent a new defintion of multiplication. The quaternions are similarly defined as complex values matricies of the form: a+bi c+di -c+di a-bi Or, if you are familiar with the complex conjugate a b -b‾ a‾ Again, you get multiplication "for free" from matrix multiplication. In contrast, if you wanted to define the quaternions directly from the real numbers, they would take the form: a b c d -b a -d c -c d a -b d -c b a Which, again is more complicated then what you get by going through the complex numbers.
adrian_b 6y ago
No, actually 2-dimensional vectors and 2-dimensional complex numbers are completely distinct entities. The same is true about 3-dimensional vectors and quaternions and the same is true in higher dimensions. The short explanation is that vectors describe additions (i.e. translations), while complex numbers (and their higher dimensional analogs) describe multiplications (i.e. scaling and rotations). The nature of vectors, pseudovectors, complex numbers, quaternions etc. can be understood only in the context of the theory of geometric algebras. For example, an element of a 2-dimensional geometric algebra has 4 components (2^2). 2 components form a vector and the other 2 form a complex number. Similarly, the 8 components of an element of a 3-dimensional geometric algebra are partitioned into a quaternion, a 3-dimensional vector and the last element, which is a pseudoscalar. The confusion is increased by the fact that the word "vector" is used with 2 different meanings. The original, physical and etymological, meaning is that a vector is a translation of an affine space. This is almost always the meaning of "vector" in physics. Many mathematicians also use the word "vector" for the elements of linear spaces a.k.a. vector spaces. The "vectors" of the second meaning have only a subset of the properties of the "vectors" of the first meaning, i.e. only the properties that can be derived from the axioms of a vector space. The complex numbers and the quaternions, octonions etc. are "vectors" in the second meaning, but they are not "vectors" in the first meaning, i.e. in the original physical meaning.

avmich 6y ago

Why stop at octonions? Sedenions and other steps of Cayley-Dickson construction?..

surfsvammel 6y ago
If I read the article correctly, it would seem its not needed.
zamalek 6y ago
Due to Hurwitz's theorem, as-per the article.
- avmich 6y ago
  From https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_constru... - "The Hurwitz's theorem (composition algebras) states that the reals, complex numbers, quaternions, and octonions are the only (normed) division algebras (over the real numbers)." So what? From https://en.wikipedia.org/wiki/Sedenion https://en.wikipedia.org/wiki/Sedenion - "Every sedenion is a linear combination of the unit sedenions e_0...e_15 , which form a basis of the vector space of sedenions." - clearly we still have operations to work with. Stopping at octonions looks somewhat arbitrary.

thunderrabbit 6y ago

> it’s very difficult to think of a nonassociative situation.” If, instead of putting on socks then shoes, you first put your socks into your shoes, technically you should still then be able to put your feet into both and get the same result. “The parentheses feel artificial.” I suggest the parenthesis are not artificial: they specify an order of operations. i.e. they specify the timing of operations. Choosing a, b, c as things that change with time makes it easy to think of nonassociative examples: (Crack open an egg) then (cook it) = (scrambled egg) (Cook an egg) then (peel it) = (boiled egg)

gizmo686 6y ago
You changed the order, so your example is not commutative. The associative example would be: (Crack an egg) then (cook it then eat it) vs (Crack an egg then cook it) then (eat it)
- thunderrabbit 6y ago
  Ohhh shoot. I guess that's egg on my face!