{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "047KEVEZ9Q"
    },
    {
      "field": "uses",
      "target": "065S5B9HF1"
    }
  ],
  "body": "\n## Definition\n\nLet $x$ be a real number with $x \\ge 0$.\nA decimal expansion of $x$ is a natural number $a_0$ and a sequence $(d_k)_{k \\ge 1}$ of digits $d_k \\in \\{0, 1, \\dots, 9\\}$ with\n$$x = a_0 + \\sum_{k=1}^{\\infty} \\frac{d_k}{10^k}.$$\nThe notation is $x = a_0.d_1 d_2 d_3 \\dots$, with $a_0$ in base $10$.\n\n## Remarks\n\nThe series always converges, because its partial sums are increasing and bounded above by $a_0 + 1$.\nA number can have two expansions. This happens exactly when one expansion ends in infinitely many $0$ and the other in infinitely many $9$, as for [[0.999... and 1|0NN5JEMTVE]].\nFor a negative number $x$, one writes $-$ followed by an expansion of $-x$.\n\n## Explanations\n\n### L1 human\n\nThe digits after the point say how many tenths, hundredths, thousandths, and so on, the number has.\nFor example, $3.14$ means $3 + 1/10 + 4/100$.\nFor a number such as $\\pi$, the digits go on forever.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "defines": "0VM8B2SB4P",
    "kind": "definition",
    "msc": [
      "11A63"
    ],
    "name": "Decimal expansion",
    "sources": [
      {
        "locator": "lead section",
        "url": "https://en.wikipedia.org/wiki/Decimal_representation"
      },
      {
        "locator": "Integer and fractional parts",
        "url": "https://en.wikipedia.org/wiki/Decimal_representation"
      }
    ],
    "tag": "06XY4X9430",
    "uses": [
      "08J2M1JYAG",
      "0AXWET80YW"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/06XY4X9430/",
  "links": [
    {
      "field": "defines",
      "target": "0VM8B2SB4P"
    },
    {
      "field": "uses",
      "target": "08J2M1JYAG"
    },
    {
      "field": "uses",
      "target": "0AXWET80YW"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/06XY4X9430.md",
  "msc": [
    "11A63"
  ],
  "name": "Decimal expansion",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/06X/06XY4X9430.md",
  "status": null,
  "tag": "06XY4X9430",
  "uri": "m2://06XY4X9430",
  "url": "https://math2.tamnd.com/wiki/decimal-expansion-definition/",
  "writing": {
    "tag": "06XY4X9430",
    "kind": "definition",
    "level": "W1",
    "next": [
      {
        "what": "edge not_instance_of in",
        "note": "one non-example that fails one clause",
        "have": 0,
        "want": 1
      }
    ]
  }
}