{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "4J9JJ3DY5R"
    },
    {
      "field": "uses",
      "target": "942BKZYGQS"
    },
    {
      "field": "uses",
      "target": "FKF9YGBKHF"
    },
    {
      "field": "uses",
      "target": "GBN9F8FAQ3"
    },
    {
      "field": "uses",
      "target": "GCQ23XT6VD"
    },
    {
      "field": "uses",
      "target": "NYJFYZ0B4M"
    },
    {
      "field": "uses",
      "target": "PHT3TJCS90"
    },
    {
      "field": "uses",
      "target": "R8HCWA981S"
    },
    {
      "field": "uses",
      "target": "RDV3C0PXTH"
    },
    {
      "field": "uses",
      "target": "SP8H3S2XWX"
    },
    {
      "field": "defines",
      "target": "CHT5JYM6QW"
    },
    {
      "field": "about",
      "target": "PHT3TJCS90"
    }
  ],
  "body": "\n## Summary\n\nFor a [[prime number|01826KMG2H]] $p$, the $p$-adic numbers form the field $\\Q_p$, the completion of $\\Q$ for the [[p-adic absolute value|WGN82TCEF2]].\nIn $\\Q_p$, a number is small when it is divisible by a high power of $p$.\nThe field $\\Q_p$ is a [[local field|J85FZEQG38]] of characteristic $0$.\nQuestions about congruences modulo all powers of $p$ become questions about equations in $\\Q_p$.\n\n## Explanations\n\n### L1 human\n\nA real number can be written with infinitely many digits after the decimal point.\nA $p$-adic number is written in base $p$ with infinitely many digits to the left instead, and only finitely many to the right.\nTwo numbers are close when their difference is divisible by a large power of $p$.\nFor example, in $\\Q_5$ the powers $5, 25, 125, \\ldots$ get smaller and smaller and tend to $0$.\nThe sum $4 + 4 \\cdot 5 + 4 \\cdot 5^2 + \\cdots$ converges in $\\Q_5$, and its value is $-1$.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "12J25",
      "11S99"
    ],
    "name": "p-adic number",
    "same_as": {
      "wikidata": "Q311627",
      "wikipedia": "P-adic number"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/P-adic_number"
      }
    ],
    "tag": "BT6T702C28"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/BT6T702C28/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/BT6T702C28.md",
  "msc": [
    "12J25",
    "11S99"
  ],
  "name": "p-adic number",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/BT6/BT6T702C28.md",
  "status": null,
  "tag": "BT6T702C28",
  "uri": "m2://BT6T702C28",
  "url": "https://math2.tamnd.com/wiki/p-adic-number/",
  "writing": {
    "tag": "BT6T702C28",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}