Why are the p-adic Integers $\mathbb{Z}_p$ called integers? I looked online but I can't find any reasoning as to why the p-adic integers are called integers. Are there certain properties that $\mathbb{Z}_p$ shares with $\mathbb{Z}$, and is there sufficeint overlap to really associate the name integers with $\mathbb{Z}_p$? Or is this just a name that was used and stuck?
Understanding the similarity between (or lack thereof) will help with some of the intuition, or at least I hope it will. Right now I am finding things a bit confusing because $\mathbb{Z}_p$ emphatically contains things that are not integers.

$\mathbb{Q}$ is the quotient field of the ring of (rational) integers $\mathbb{Z}$
And $\mathbb{Q}_p$ is the quotient field of the ring of p-adic integers $\mathbb{Z}_p$
