Adding integer vectors is the same as multiplying positive rational numbers.
By “integer vector” I mean an infinite-dimensional vector of integers which has only a finite number of nonzero components. To add integer vectors, add their components.
Every positive rational number can be expressed uniquely as the ratio between two natural numbers - its numerator and denominator - which share no divisor. By the fundamental theorem of number theory, the numerator and denominator can be expressed uniquely as a product of natural powers of primes. Since no prime can appear both in the numerator and denominator - lest they share a divisor - every rational number is expressible uniquely as a product of primes to integer powers.
The correspondence between positive rational numbers and integer vectors is this: the n’th component of a vector corresponds to the power of the n’th prime in a rational number. Multiplying rational numbers involves adding the powers of their corresponding primes. Take note that multiplying rationals is harder than adding integer vectors; it involves factoring, as does translating a rational into a vector.