Are p-adic numbers real or not?

Question

Are p-adic numbers real numbers?Why or why not?I came across the idea that it is not a real number from-Is 0.9999... equal to -1? (last comment)

Answer 1

No, the field of $p$-adic numbers, $\Bbb{Q}_p$, is not isomorphic to a subfield of $\Bbb{R}$, and thus I don't see a way of calling $p$-adic numbers "real". A way of seeing this is that, by Hensel lifting, the negative integer $-4p+1$ has a square root that is a $p$-adic integer. It is well known that $-4p+1$ has no square roots in $\Bbb{R}$.

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As others pointed out the $p$-adics really are something more or less independent from the reals. A construction called completion takes us from $\Bbb{Q}$ to $\Bbb{R}$ as well as to $\Bbb{Q}_p$, but the related metrics are all independent (in a sense that can be made precise), and the completions look very different.

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The only reason I used $-4p+1$ above instead of $-p+1$ is to make the argument also work for $p=2$.

Answer 2

There are two possible types of completion of $\mathbb Q$. With one you get $\mathbb R$ and with the other you get the p-adics. I don't have much experience in the area, but I don't think that it makes sense to say $\mathbb Q_p \subset \mathbb R$ or $\mathbb R \subset \mathbb Q_p.$