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  "body": "\n## Summary\n\n$p$-adic analysis is the analysis of functions of [[$p$-adic numbers|BT6T702C28]], where nearness is measured by the [[$p$-adic absolute value|WGN82TCEF2]] instead of the usual one.\nThis absolute value is ultrametric, which makes many things simpler than in real analysis: a series converges as soon as its terms tend to $0$.\nOther things behave differently: the $p$-adic integers form a compact, totally disconnected space, and the exponential series converges only on a small disc.\nThe subject is a main tool of modern number theory, for example in $p$-adic $L$-functions and in the study of congruences.\n\n## Explanations\n\n### L1 human\n\nIn the $p$-adic world, a number is small when it is divisible by a high power of the prime $p$.\nSo $p, p^2, p^3, \\ldots$ tend to $0$, and the series $1 + p + p^2 + \\cdots$ converges, to $1/(1 - p)$.\nOne can then do calculus with limits, series, continuity and derivatives, as over the reals.\nMuch of it works the same way, but some facts change: for example, a function with derivative $0$ everywhere on $\\Z_p$ need not be constant.\n",
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