Btukfyl

I was more or less day dreaming about whole numbers, and I had an idea which seemed novel to me, but which may in fact be previously known and/or studied. I'm not even sure what to call it, or how to look for previous reports of it. My first question will be: Have I merely rediscovered a way of looking at things that is already known, and if so, where can I find some discussion of it?

I imagine an infinite dimension space, with each orthogonal axis corresponding to a prime number. By the fundamental theorem of arithmetic, natural numbers each have a unique representation of the form $prod p_i^{alpha_i}$. Numbers of this form can be located in $n$-dimensional subspaces as follows: For each prime $p_j$ which has a corresponding exponent $>0$, move $alpha_1$ units from the origin along the $p_1$ axis, then $alpha_2$ units parallel to the $p_2$ axis, then $alpha_3$ units parallel to the $p_3$ axis, etc., for as many primes as are present as factors of that number. The origin would correspond to the number $1$, where all axes intersect and $alpha_i=0$ for all $i$.

Each axis could be extended through the origin, and fractions could be represented as negative integral distances along the axes so extended, allowing the representation of rational numbers. Interestingly, moving "infinitely" out along these negative axes, one approaches $0$.

Next, by moving non-integral distances along axes, non-integral exponents could be represented and algebraic rational numbers could be represented and located in appropriate subspaces. I'm not certain that transcendental numbers can be represented, but neither are they representable in the form $prod p_i^{alpha_i}$. One concern I have is that if one is allowed to move irrational distances along axes, it might conceivably be possible to represent a particular number in two different ways, and locate it in two different subspaces, i.e. if exponents can be identified such that $p_1^{alpha_1}=p_2^{alpha_2}$.

I do not think arithmetic addition/subtraction can be readily performed with numbers so represented, but multiplication/division can be. In this light, this kind of representation bears some kinship to logarithms, but with a unique (prime) base (rather than $e$ or $10$) as the metric along each axis. However, I can think of no particular utility to representing numbers in this way. So my second question is: Can anyone see any merit or utility to thinking of numbers in this way?

For the natural numbers, this is the same as representing each number by its prime factorization. A number $2^a3^b5^cldots$ is represented by the tuple $(a,b,c,ldots)$ This is often a useful way to represent numbers in number theory, but I don't think the geometric picture adds anything.

Once you allow fractional exponents it is less useful. If you require the exponents be rational you still have a unique representation. For example $sqrt 2=2^{1/2}$ can only be expressed that way. The set of numbers you can represent would be a field, but not one I am familiar with. It is larger than the rationals and countable, but smaller than the algebraic numbers.

If you allow real exponents you no longer have a unique point for each number. You could represent $sqrt 5$ as the point $frac 12$ on the $5$ axis, but you could also represent it as the point $frac 12 log_2(5)$ on the $2$ axis or many other ways.