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  "body": "\n## Summary\n\nA non-Archimedean absolute value on a field is an absolute value that satisfies the strong triangle inequality $\\abs{x + y} \\le \\max(\\abs{x}, \\abs{y})$, see the [[definition|8XDYNK6TV0]].\nIt generalizes the usual [[absolute value|0FFXKTKKQC]] of real numbers in a different direction: small elements stay small under addition.\nThe [[p-adic absolute value|WGN82TCEF2]] is the main example, and the induced metric is an [[ultrametric|A6DPR620B5]].\nThese absolute values are the same as real-valued [[valuations|B85T5VKR3M]], up to a choice of base for the exponent.\n\n## Explanations\n\n### L1 human\n\nThe usual absolute value makes $1 + 1 + \\cdots + 1$ grow without end.\nA non-Archimedean absolute value never lets a sum be larger than its largest term.\nFor the $2$-adic absolute value, a number is small when it is divisible by a high power of $2$: $\\abs{8}_2 = 1/8$ and $\\abs{3}_2 = 1$.\nThen $\\abs{8 + 16}_2 = \\abs{24}_2 = 1/8$, which is not more than the larger of $1/8$ and $1/16$.\nIn such a geometry every triangle is isosceles, and every point of a disc is a centre of it.\n",
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