{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "00HPT88ZX0"
    },
    {
      "field": "uses",
      "target": "0GTEHJQRFK"
    },
    {
      "field": "uses",
      "target": "0SPYEFH7H4"
    }
  ],
  "body": "\n## Definition\n\nA Gaussian integer is a complex number $a + bi$ such that $a$ and $b$ are integers. Here $i$ is a complex number with $i^2 = -1$.\nThe set of Gaussian integers is written $\\mathbb{Z}[i] = \\{ a + bi : a, b \\in \\mathbb{Z} \\}$.\n\n## Remarks\n\nThe set $\\mathbb{Z}[i]$ is closed under addition, subtraction and multiplication, so it is a subring of $\\mathbb{C}$. It is an integral domain.\nThe units of $\\mathbb{Z}[i]$ are $1$, $-1$, $i$ and $-i$.\nIn the complex plane, the Gaussian integers are the points of the square lattice.\nThe [[norm|00HPT88ZX0]] $N(a + bi) = a^2 + b^2$ links them to [[sums of two squares|04NPWJMA6H]].\n\n## Explanations\n\n### L1 human\n\nGaussian integers are the complex numbers whose real part and imaginary part are both integers, such as $3 - 2i$.\nThey behave much like the ordinary integers: you can add them, multiply them and factor them into primes.\n",
  "commit": "545acf83a9b75c8ddc375ef8b23d2ae101a4d7e4",
  "front": {
    "defines": "074P7C85XH",
    "kind": "definition",
    "msc": [
      "11R11"
    ],
    "name": "Gaussian integer",
    "sources": [
      {
        "locator": "Basic definitions",
        "url": "https://en.wikipedia.org/wiki/Gaussian_integer"
      }
    ],
    "tag": "0B268GVX95"
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/0B268GVX95/",
  "links": [
    {
      "field": "defines",
      "target": "074P7C85XH"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0B268GVX95.md",
  "msc": [
    "11R11"
  ],
  "name": "Gaussian integer",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0B2/0B268GVX95.md",
  "status": null,
  "tag": "0B268GVX95",
  "uri": "m2://0B268GVX95",
  "url": "https://math2.tamnd.com/wiki/gaussian-integer-definition/",
  "writing": {
    "tag": "0B268GVX95",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}