{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "0VB8BFRABH"
    }
  ],
  "body": "\n## Statement\n\nLet $x$ be a real number with $x \\ge 0$.\nThen there is exactly one real number $y$ with $y \\ge 0$ and $y^2 = x$.\n\n## Remarks\n\nExistence uses the [[completeness axiom|0AXWET80YW]]: take $y = \\sup \\{t \\ge 0 : t^2 \\le x\\}$ and show that $y^2 \u003c x$ and $y^2 \u003e x$ are both impossible.\nUniqueness: if $y, z \\ge 0$ and $y^2 = z^2$, then $(y - z)(y + z) = 0$, thus $y = z$.\nIn $\\mathbb{Q}$ the existence fails: $2$ has no rational square root, by the [[irrationality of the square root of 2|0DZGSDHD92]].\n\n## Explanations\n\n### L1 human\n\nEvery nonnegative real number is the area of exactly one square, and the side of that square is the square root.\nThis needs the real numbers: among the fractions, no square has area exactly $2$.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "kind": "statement",
    "msc": [
      "26A09"
    ],
    "name": "Existence of square roots of nonnegative reals",
    "role": "theorem",
    "sources": [
      {
        "locator": "Properties and uses",
        "url": "https://en.wikipedia.org/wiki/Square_root"
      }
    ],
    "tag": "068BZBXE0C",
    "uses": [
      "0AXWET80YW"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/068BZBXE0C/",
  "links": [
    {
      "field": "uses",
      "target": "0AXWET80YW"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/068BZBXE0C.md",
  "msc": [
    "26A09"
  ],
  "name": "Existence of square roots of nonnegative reals",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/068/068BZBXE0C.md",
  "status": null,
  "tag": "068BZBXE0C",
  "uri": "m2://068BZBXE0C",
  "url": "https://math2.tamnd.com/wiki/existence-of-square-roots-of-nonnegative-reals/",
  "writing": {
    "tag": "068BZBXE0C",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}