Is there ever a case where it’s useful to view the natural numbers as points in an infinite dimensional vector space, the bases of which are the prime numbers and where each unit increment along any basis corresponds to that prime raised to the next higher power.
To illustrate, in this space, the number 90 (which equals 2[sup]1[/sup] X 3[sup]2[/sup] X 5[sup]1[/sup]) would be represented by (1, 2, 1, 0, 0, . . . ).
Likewise, 231 (which equals 2[sup]0[/sup]X 3[sup]1[/sup] X 5[sup]0[/sup] X 7[sup]1[/sup] X 11[sup]1[/sup]) would be represented by (0, 1, 0, 1, 1, 0, 0, . . . )
It seems like a pretty obvious thing to do, but I am too stupid to figure out whether there’s any utility in doing so.
That’s interesting. I’d never thought of something like that before and it is intriguing.
I looked at them a little bit, and one thing that quickly became apparent, although I’m not sure it’s particularly relevant to whether or not it’s usefulness, is that there are multiple ways to represent the same number.
For example:
3 = (0,1,0…) but also (1.58496250072, 0, 0…).
This problem can be avoided if instead of using them to generate a real vector space you use them to generate an abelian group (or at least a commutative monoid). Just allow only integers (or just natural numbers) as exponents. In this guise they come up all the time.
Euler’s expansion for the Riemann zeta function as a product essentially uses this version, and this is instrumental in showing that zeta is the partition function for the “free primon gas”, which is a toy quantum field theory built out of “primons” in pretty much the way the OP suggests.
Another place they come up is in Gödel-numbering. Assign each symbol of your formal system an index number, and then write down any formula as a sequence of those numbers. Read that list (filled out with 0s) like in the OP and you have a number for each formula in your formal system, which then you can use your formal system to talk about.
Well, for a moment I thought that you could use this representation of the integers to prove that the integers are uncountable (which would be a tremendous contradiction): you could, in principle, map every integer to a string of integers, and map that to the decimal expansion of a real number. Then use some variant on Cantor’s diagonal trick to prove that the set of such numbers is uncountable. The problem with this argument, of course, is that every integer is mapped to a terminating string of integers (those which are zero after a given point), and Cantor’s argument relies on the existence of non-terminating strings. So mathematics is saved.
I should know better than to contemplate set theory after half a pitcher of Three Floyds Gumballhead Ale.
True enough, but my tacit assumption (in this case, at least) was that the components of each vector were strictly integral. Phrased differently, I was trying to “exploit” the fundamental theorem of arithmetic (i.e. every natural number > 1 has a unique factorization in primes).
Interesting, I thought about this representation for natural numbers once or twice too. Like you said it seems pretty obvious when you pay attention to the fundamental theorem of arithmetics. I played with it a bit then put it aside figuring someone in the eighteenth century figured all of this stuff. Mainly I just tried to use some Elementary Linear Algebra ideas to study the “space of integers” and hopefully prove some useful theorems in number theory or translate number theory problems into linear algebra problems. But beyond my limitations I don’t know enough advanced algebra to dwell further than that.
I’m fairly sure that wouldn’t be a vector space because integers don’t make for a very good field. How are you defining your addition and scalar product operations anyway?
Also, we can still get a full vector space by letting components range over the entire scalar field of rationals; the vectors, in this case, will no longer correspond to merely the positive integers, but, rather, will correspond 1-1 to those numbers which are nth roots of positive rationals. (The subset of vectors with only natural number components will, of course, still correspond to the positive integers)
Yeah, I think the best way to get a genuine vector space is by taking the scalar field of rationals, as above. Presumably, addition of vectors would be component-wise addition (and thus correspond to multiplication of the represented numbers), with scalar multiplication being component-wise as well (corresponding to exponentiation of the represented number).
Of course, purely as a vector space, this is no different from any other space of countably infinite dimension over the scalar field of rationals. If there is any use to this idea, it would be in the simultaneous thinking of the structure as both a vector space and the multiplicative group of nth roots of rationals.
I missed the edit window, but I wanted to change the end of my last post slightly:
Of course, purely as a vector space, this is no different from any other space of countably infinite dimension over the scalar field of rationals. If there is any use to this idea, it would have to be found in the simultaneous viewing of the structure as both a vector space and as the multiplicative group of nth roots of positive rationals equipped with a rational exponentiation operator. Then concepts which are more natural to think of under the one interpretation can be explored under the other (e.g., the vector space concepts of linear independence and spans can be explored in the multiplicative context).
There is a free (has a basis) commutative group consisting of the eventually 0 sequences of integers, added term-wise, that can be thought of representing the positive rational numbers by letting a sequence n_1,n_2,…,n_k,0,0,… correspond to the rational number 2^n_13^n_2…*p_k^n_k. p_k is the k-th prime. The sequences of all non-negative integers correspond to the positive integers. The fact that only eventually-zero sequences are used saves the set from being uncountable. Incidentally, the fact that the primes are a basis is equivalent to unique prime factorization.