{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "031BG625Y3"
    },
    {
      "field": "uses",
      "target": "05V7J4WD1C"
    },
    {
      "field": "uses",
      "target": "0BJV24C4SY"
    },
    {
      "field": "uses",
      "target": "0KW2X0TFBX"
    },
    {
      "field": "uses",
      "target": "0T71SSYZ39"
    }
  ],
  "body": "\n## Definition\n\nLet $z = a + b i$ be a [[complex number|09FTAET5Z4]], with $a$ and $b$ real.\nThe real part of $z$ is $\\operatorname{Re} z = a$.\nThe imaginary part of $z$ is $\\operatorname{Im} z = b$.\n\n## Remarks\n\nThe imaginary part is a real number: $\\operatorname{Im}(2 + 3 i) = 3$, not $3 i$.\nThe maps $\\operatorname{Re}$ and $\\operatorname{Im}$ are additive and $\\mathbb{R}$-linear, but they do not respect products.\nWith the [[conjugate|0BJV24C4SY]], $\\operatorname{Re} z = (z + \\bar{z})/2$ and $\\operatorname{Im} z = (z - \\bar{z})/(2 i)$.\nA complex number $z$ is real when $\\operatorname{Im} z = 0$, and purely imaginary when $\\operatorname{Re} z = 0$.\n\n## Explanations\n\n### L1 human\n\nEach complex number has two real coordinates.\nFor $z = 4 - 7 i$, the real part is $4$ and the imaginary part is $-7$.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "defines": "06J8TEPJPY",
    "kind": "definition",
    "msc": [
      "30A99"
    ],
    "name": "Real and imaginary parts",
    "sources": [
      {
        "locator": "Definition and basic operations",
        "url": "https://en.wikipedia.org/wiki/Complex_number"
      }
    ],
    "tag": "02ZE6JHJ7G",
    "uses": [
      "09FTAET5Z4"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/02ZE6JHJ7G/",
  "links": [
    {
      "field": "defines",
      "target": "06J8TEPJPY"
    },
    {
      "field": "uses",
      "target": "09FTAET5Z4"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/02ZE6JHJ7G.md",
  "msc": [
    "30A99"
  ],
  "name": "Real and imaginary parts",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/02Z/02ZE6JHJ7G.md",
  "status": null,
  "tag": "02ZE6JHJ7G",
  "uri": "m2://02ZE6JHJ7G",
  "url": "https://math2.tamnd.com/wiki/real-and-imaginary-parts-definition/",
  "writing": {
    "tag": "02ZE6JHJ7G",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}