"Gaussian integer" meaning in All languages combined

{
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      "form": "Gaussian integers",
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      "expansion": "Gaussian integer (plural Gaussian integers)",
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  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
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      "categories": [
        {
          "kind": "other",
          "name": "English entries with incorrect language header",
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      "examples": [
        {
          "ref": "1997, Bernard L. Johnston, Fred Richman, Numbers and Symmetry: An Introduction to Algebra, CRC Press, page 44:",
          "text": "We could say that a Gaussian integer is larger than another if its norm is larger, that is, if its distance from the origin is larger. That's all right, although it has the peculiarity that 5 is larger than -7 as an integer, but smaller than -7 as a Gaussian integer! In a way, this notion of larger makes more sense: shouldn't -7 be considered larger than 5?",
          "type": "quote"
        },
        {
          "text": "2000, André Weilert, Asymptotically fast GCD Computation in ℤ[i], Wieb Bosma (editor), Algorithmic Number Theory: 4th International Symposium, ANTS-IV, Proceedings, Springer, LNCS 1838, page 595,\nWe present an asymptotically fast algorithm for the computation of the greatest common divisor (GCD) of two Gaussian integers."
        },
        {
          "text": "2008, Timothy Gowers, June Barrow-Green, Imre Leader (editors), The Princeton Companion to Mathematics, Princeton University Press, page 319,\nFor example, in the ring of Gaussian integers, R_-1, we have the factorizations\n2=(1+i)(1-i),\n5=(1+2i)(1-2i),\n13=(2+3i)(2-3i),\n17=(1+4i)(1-4i),\n29=(2+5i)(2-5i),\n⋮\nwhere all the Gaussian integer factors in brackets above are irreducible elements of the ring of Gaussian integers."
        }
      ],
      "glosses": [
        "Any complex number of the form a + bi, where a and b are integers."
      ],
      "hypernyms": [
        {
          "word": "complex number"
        },
        {
          "word": "Gaussian rational number"
        },
        {
          "word": "Gaussian rational"
        },
        {
          "sense": "quadratic integer",
          "word": "algebraic integer"
        }
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      "hyponyms": [
        {
          "word": "Gaussian prime"
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          "word": "integer"
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      "raw_glosses": [
        "(algebra) Any complex number of the form a + bi, where a and b are integers."
      ],
      "related": [
        {
          "word": "Eisenstein integer"
        },
        {
          "word": "Blum integer"
        }
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    },
    {
      "sense": "quadratic integer",
      "word": "algebraic integer"
    },
    {
      "word": "Gaussian rational number"
    },
    {
      "word": "Gaussian rational"
    }
  ],
  "hyponyms": [
    {
      "word": "Gaussian prime"
    },
    {
      "word": "integer"
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  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "related": [
    {
      "word": "Eisenstein integer"
    },
    {
      "word": "Blum integer"
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  ],
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        "en:Numbers"
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        {
          "ref": "1997, Bernard L. Johnston, Fred Richman, Numbers and Symmetry: An Introduction to Algebra, CRC Press, page 44:",
          "text": "We could say that a Gaussian integer is larger than another if its norm is larger, that is, if its distance from the origin is larger. That's all right, although it has the peculiarity that 5 is larger than -7 as an integer, but smaller than -7 as a Gaussian integer! In a way, this notion of larger makes more sense: shouldn't -7 be considered larger than 5?",
          "type": "quote"
        },
        {
          "text": "2000, André Weilert, Asymptotically fast GCD Computation in ℤ[i], Wieb Bosma (editor), Algorithmic Number Theory: 4th International Symposium, ANTS-IV, Proceedings, Springer, LNCS 1838, page 595,\nWe present an asymptotically fast algorithm for the computation of the greatest common divisor (GCD) of two Gaussian integers."
        },
        {
          "text": "2008, Timothy Gowers, June Barrow-Green, Imre Leader (editors), The Princeton Companion to Mathematics, Princeton University Press, page 319,\nFor example, in the ring of Gaussian integers, R_-1, we have the factorizations\n2=(1+i)(1-i),\n5=(1+2i)(1-2i),\n13=(2+3i)(2-3i),\n17=(1+4i)(1-4i),\n29=(2+5i)(2-5i),\n⋮\nwhere all the Gaussian integer factors in brackets above are irreducible elements of the ring of Gaussian integers."
        }
      ],
      "glosses": [
        "Any complex number of the form a + bi, where a and b are integers."
      ],
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        "(algebra) Any complex number of the form a + bi, where a and b are integers."
      ],
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