My current intuition about the p-adic numbers comes from the following three facts:

  1. You can describe $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ with the $Gal(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$ groups.
  2. Hensel's lemma
  3. We find that the p-adic integers are formal neighborhoods of closed points in $\mathbb{Z}$, so they naturally show up in the deformation theory of arithmetic objects (Schemes over $\mathbb{Z}$)

What are other reasons to consider p-adic integers/numbers?

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    $\begingroup$ Because they're fun. $\endgroup$ – G Tony Jacobs May 19 '17 at 20:36
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    $\begingroup$ @GTonyJacobs You should unpack the entertainment for the audience! $\endgroup$ – 54321user May 19 '17 at 20:38
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    $\begingroup$ Ostrowski's theorem tells us that every nontrivial absolute value on $\mathbb{Q}$ is equivalent either to the usual one or to one of the $p$-adic absolute values. $\endgroup$ – Kaj Hansen May 19 '17 at 20:46
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    $\begingroup$ It is natural to consider the natural injective map $\mathbb{Z} \to \mathbb{Z}_p$ and to say that $\mathbb{Z}_p[p^{-1}]$ is a field. Then it is a matter of understanding this new field. It becomes the prototypal example of a completion of a field by a non-archimedean absolute value. $\endgroup$ – reuns May 19 '17 at 20:48
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    $\begingroup$ @GTonyJacobs To paraphrase Feynman, p-adics make arithmetic deformation theory possible, but that's not why we study them. :) $\endgroup$ – Neal May 19 '17 at 22:12

On one hand, the $p$-adic numbers are extremely natural objects of study: by Ostrowski's theorem every nontrivial absolute value on $\mathbf Q$ is equivalent to either the usual absolute value or the $p$-adic absolute value for some $p$. So the $p$-adic numbers, together with the real numbers, give all the posible completions of $\mathbf Q$.

On the other hand, the $p$-adic numbers are also extremely useful, even if you only care about $\mathbf Q$. A great example of this is the Hasse principle, that says (for example) that a homogeneous quadratic equation has a nontrivial solution over $\mathbf Q$ if and only if it does over $\mathbf R$ and $\mathbf Q_p$ for each $p$, and the latter question turns out to be straightforward to answer.

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  • $\begingroup$ So are p-adic numbers just the first example of where we see the local to global principle? $\endgroup$ – 54321user May 27 '17 at 20:51

About 40 years ago, a fellow who was doing C-star Algebras was teasing me about how none of my Number Theory would ever find applications in "the real world". I told him to just wait and see – some day, there would be crucial applications of $p$-adic numbers in Physics. I was kidding, but apparently there is a serious pursuit of $p$-adic quantum mechanics. See here.

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Quoting Wikipedia, "the Skolem–Mahler–Lech theorem states that if a sequence of numbers is generated by a linear recurrence relation, then with finitely many exceptions the positions at which the sequence is zero form a regularly repeating pattern. More precisely, this set of positions can be decomposed into the union of a finite set and finitely many full arithmetic progressions.... Its proofs use p-adic analysis."

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