{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "04J6G4H78J"
    }
  ],
  "body": "\n## Statement\n\nThe [[square root|0VB8BFRABH]] $\\sqrt{2}$ of $2$ is an [[irrational number|080T2TXVQV]].\nThat is, there are no integers $a$ and $b$ with $b \\neq 0$ and $(a / b)^2 = 2$.\n\n## Remarks\n\nA classical proof: suppose that $a / b = \\sqrt{2}$ in [[lowest terms|0R64A2CVN4]]. Then $a^2 = 2 b^2$, thus $a$ is even. Then $b$ is also even, which is a contradiction.\nThe theorem is a special case of the [[irrationality of the square root of a prime|0YHFMXJJAZ]].\n\n## History\n\nThe ancient Greeks knew this result. Tradition attributes its discovery to the Pythagoreans.\n\n## Explanations\n\n### L1 human\n\nNo fraction squared gives exactly $2$.\nFractions such as $7/5$ and $141/100$ come close, but there is always an error.\nThis means that the diagonal of a square with side $1$ has a length that is not a ratio of whole numbers.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "kind": "statement",
    "msc": [
      "11J72"
    ],
    "name": "Irrationality of the square root of 2",
    "role": "theorem",
    "sources": [
      {
        "locator": "Proofs of irrationality",
        "url": "https://en.wikipedia.org/wiki/Square_root_of_2"
      },
      {
        "locator": "History",
        "url": "https://en.wikipedia.org/wiki/Square_root_of_2"
      }
    ],
    "tag": "0DZGSDHD92",
    "uses": [
      "080T2TXVQV",
      "0VB8BFRABH"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0DZGSDHD92/",
  "links": [
    {
      "field": "uses",
      "target": "080T2TXVQV"
    },
    {
      "field": "uses",
      "target": "0VB8BFRABH"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0DZGSDHD92.md",
  "msc": [
    "11J72"
  ],
  "name": "Irrationality of the square root of 2",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0DZ/0DZGSDHD92.md",
  "status": null,
  "tag": "0DZGSDHD92",
  "uri": "m2://0DZGSDHD92",
  "url": "https://math2.tamnd.com/wiki/irrationality-of-the-square-root-of-2/",
  "writing": {
    "tag": "0DZGSDHD92",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}