{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "0SPYEFH7H4"
    }
  ],
  "body": "\n## Definition\n\nLet $\\alpha = a + bi$ be a [[Gaussian integer|0B268GVX95]], with integers $a$ and $b$.\nThe norm of $\\alpha$ is the integer $N(\\alpha) = a^2 + b^2 = \\alpha \\bar{\\alpha}$, where $\\bar{\\alpha} = a - bi$ is the complex conjugate of $\\alpha$.\n\n## Remarks\n\nThe norm is multiplicative: $N(\\alpha \\beta) = N(\\alpha) N(\\beta)$ for all Gaussian integers $\\alpha$ and $\\beta$.\nThis product rule is the [[Brahmagupta–Fibonacci identity|0K2DMACEZ4]].\nThe norm is a nonnegative integer, and $N(\\alpha) = 0$ only for $\\alpha = 0$. The units are the elements of norm $1$.\nFor example, $N(2 + i) = 5$ and $N(3 - 2i) = 13$.\n\n## Explanations\n\n### L1 human\n\nThe norm of $a + bi$ is the square of its distance from $0$ in the complex plane.\nIt turns a question about Gaussian integers into a question about ordinary integers.\n",
  "commit": "545acf83a9b75c8ddc375ef8b23d2ae101a4d7e4",
  "front": {
    "defines": "0HYRJF4WNQ",
    "kind": "definition",
    "msc": [
      "11R11"
    ],
    "name": "Norm of a Gaussian integer",
    "sources": [
      {
        "locator": "Basic definitions",
        "url": "https://en.wikipedia.org/wiki/Gaussian_integer"
      }
    ],
    "tag": "00HPT88ZX0",
    "uses": [
      "0B268GVX95"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/00HPT88ZX0/",
  "links": [
    {
      "field": "defines",
      "target": "0HYRJF4WNQ"
    },
    {
      "field": "uses",
      "target": "0B268GVX95"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/00HPT88ZX0.md",
  "msc": [
    "11R11"
  ],
  "name": "Norm of a Gaussian integer",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/00H/00HPT88ZX0.md",
  "status": null,
  "tag": "00HPT88ZX0",
  "uri": "m2://00HPT88ZX0",
  "url": "https://math2.tamnd.com/wiki/norm-of-a-gaussian-integer-definition/",
  "writing": {
    "tag": "00HPT88ZX0",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}