Information and expertise like this is why hanging out at Less Wrong is worth the time. I estimate that I value the information in your comment at about $35, meaning my present self would advise my former self to pay up to $35 to read it.
So, I get it. My brain is more wired for analysis than algebra; so this isn’t the first time that linear algebra has been a useful bridge for me. I see that we could have a ‘vector space’ of infinite-dimensional vectors where each vector (a1, a2, …, an, …) represents a number N where N = (P1^a1)(P2^a2)...(Pn^an)… and Pi are the ordered primes. Clearly 1 is the zero element and would never be a basis element.
I should admit here that my background in algebra is weak and I have no idea how you would need to modify the notion of ‘vector space’ to make certain things line up. But I can already speculate on how the choice of the “scalar field” for specifying the a_i would have interesting consequences:
non-negative integer ‘scalar field’ --> the positive integers,
all integers ‘scalar field’ --> positive rational numbers,
complex integers --> finally include the negative rationals.
I’d like to read more. What sub-field of mathematics is this?
I see that we could have a ‘vector space’ of infinite-dimensional vectors where each vector (a1, a2, …, an, …) represents a number N where N = (P1^a1)(P2^a2)...(Pn^an)… and Pi are the ordered primes.
Information and expertise like this is why hanging out at Less Wrong is worth the time. I estimate that I value the information in your comment at about $35, meaning my present self would advise my former self to pay up to $35 to read it.
So, I get it. My brain is more wired for analysis than algebra; so this isn’t the first time that linear algebra has been a useful bridge for me. I see that we could have a ‘vector space’ of infinite-dimensional vectors where each vector (a1, a2, …, an, …) represents a number N where N = (P1^a1)(P2^a2)...(Pn^an)… and Pi are the ordered primes. Clearly 1 is the zero element and would never be a basis element.
I should admit here that my background in algebra is weak and I have no idea how you would need to modify the notion of ‘vector space’ to make certain things line up. But I can already speculate on how the choice of the “scalar field” for specifying the a_i would have interesting consequences:
non-negative integer ‘scalar field’ --> the positive integers,
all integers ‘scalar field’ --> positive rational numbers,
complex integers --> finally include the negative rationals.
I’d like to read more. What sub-field of mathematics is this?
Oh! And orthogonal vectors are relatively prime!
I’m not sure that the idea of orthogonality is defined for modules, is it? Is there a standard definition of an inner product for a Z-module?
Yes; the same definition works. See here.
Yay! I actually got something right!
Number theory, ne? Or is that too general?
It looks like it’s more abstract algebra (possibly applied to number theory) that byrnema is interested in. Check out Wikipedia on module.)
Precisely! Thanks also.