Unique Factorisation in the Natural Numbers

Abstract

We are so used to working with the natural numbers from infancy onwards that we take it for granted that natural numbers may be factorised uniquely into prime numbers. For example, $360=2^{3}3^{2}5$ is the prime factorisation of 360. However, we should notice that there are already senses in which this factorisation is not really unique; we can write $360=2\times 3\times 5\times 2\times 3\times 2$, or even $360=(-2)\times 5\times 3\times (-3)\times 2\times 2$. Nevertheless, we can see that all these factorisations are “essentially the same”, in a way which we could make precise, and we will do so later.