The prime factor module

Every integer can be written as a product of its prime factors, for example: 20 = 2 ^2 5^1.

This product of prime powers is a unique representation of that number, up to the order of the factors. This is known as the fundamental theorem of arithmetic.

Note that this also holds for rational numbers, for example: \frac{100}{15} = \frac{2^2 5^2}{3^1 5^1} = 2^2  3^{-1}  5^1.

The interesting thing is that you can consider the prime factorization of a rational number as a vector, where the elements of the vector represent how many times a particular prime number appears. For example, 20 = (2, 0, 1) and \frac{100}{15} = (2, -1, 1). These

It turns out that these prime factor vectors, with vector addition and scalar multiplication defined in the usual way, form a module over the integers. I’m not the first to have spotted this idea either.

I had a bit of fun formalising these ideas in this document.

I’m curious to know whether thinking of numbers in this way can give us some more insights than the usual number line. What do quantities like an angle or a volume in this module represent?

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