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  "body": "\n## Summary\n\nThe completion of a field with an absolute value is a larger such field, see the [[definition|KFYW0WN9AD]].\nIn it every Cauchy sequence converges, and the original field is dense.\nIt is the [[completion of the metric space|06RW4RKWH7]] of the field, with the field operations extended by continuity.\nThe real numbers are the completion of $\\Q$ for the usual absolute value, and the [[p-adic numbers|BT6T702C28]] are its completion for the [[p-adic absolute value|WGN82TCEF2]].\nMany problems over a valued field are solved first in its completion, where limits exist and [[Hensel's lemma|010QVB71T2]] applies.\n\n## Explanations\n\n### L1 human\n\nA field such as $\\Q$ has holes: the decimal approximations $1, 1.4, 1.41, 1.414, \\ldots$ of $\\sqrt 2$ get closer and closer to each other but have no limit in $\\Q$.\nCompleting means adding a limit for every such sequence, and doing arithmetic with these limits.\nWhich sequences count as \"getting closer\" depends on the absolute value.\nFor the $5$-adic absolute value, the integers $1, 1 + 5, 1 + 5 + 25, \\ldots$ get closer, and their limit in the completion is $-1/4$, because $1 + 5 + 25 + \\cdots = 1/(1 - 5)$ in this sense.\n",
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