I’ve recently been learning about p-adic numbers, and I’ve run across a question I can’t seem to find the answer to online...
We talk about some rational number n having a p-adic expansion, and often times I hear people talk as if the p-adic expansion of n IS that rational number. But at the same time, p-adic numbers have different properties than the rationals, so it seems as if the p-adic expansion of n is a new number, and not just a different representation of n.
Further, I’ve heard people say that much like the irrationals, the p-adics “fill in the gaps” between the rationals. Does this mean that there are p-adic numbers which do not correspond to rational numbers? And if so, how do we define “correspond”?
Over all, I suppose I do not understand what we mean by saying there is a p-adic expansion of a rational number, and any guidance on this topic would be much appreciated.