{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Proof\n\n1. Let $z = a + b i$ be a [[complex number|09FTAET5Z4]], with $a$ and $b$ real.\n2. Its [[conjugate|0BJV24C4SY]] is $\\bar{z} = a + (-b) i$.\n3. By the rule for multiplying complex numbers,\n$$z \\bar{z} = \\bigl(a \\cdot a - b \\cdot (-b)\\bigr) + \\bigl(a \\cdot (-b) + b \\cdot a\\bigr) i = (a^2 + b^2) + 0 i.$$\n4. By the definition of the [[modulus|0KW2X0TFBX]], $|z| = \\sqrt{a^2 + b^2}$.\nHere $\\sqrt{a^2 + b^2}$ is the nonnegative real number whose square is $a^2 + b^2$, by [[existence of square roots|068BZBXE0C]].\n5. So $|z|^2 = a^2 + b^2$. With step 3 and [[equality of complex numbers|05V7J4WD1C]], this gives $z \\bar{z} = |z|^2$.\n6. Since $a^2 \\ge 0$ and $b^2 \\ge 0$, the number $z \\bar{z} = a^2 + b^2$ is a nonnegative real number.\n\n## Idea\n\nThe product $(a + b i)(a - b i)$ is a difference of squares, $a^2 - (b i)^2 = a^2 + b^2$.\nThe imaginary parts cancel, and what remains is the square of the distance from $z$ to $0$.\n",
  "commit": "68a8c3dc73fd55004b2796431c607c30960d1a85",
  "front": {
    "kind": "proof",
    "msc": [
      "30A99"
    ],
    "name": "Proof that a complex number times its conjugate is the square of its modulus",
    "proves": "0ZB7J071WQ",
    "sources": [
      {
        "locator": "Properties",
        "url": "https://en.wikipedia.org/wiki/Complex_conjugate"
      }
    ],
    "tag": "67KN8TZJ5C",
    "uses": [
      "09FTAET5Z4",
      "0BJV24C4SY",
      "0KW2X0TFBX",
      "068BZBXE0C",
      "05V7J4WD1C"
    ]
  },
  "kind": "proof",
  "link": "https://math2.tamnd.com/node/67KN8TZJ5C/",
  "links": [
    {
      "field": "proves",
      "target": "0ZB7J071WQ"
    },
    {
      "field": "uses",
      "target": "09FTAET5Z4"
    },
    {
      "field": "uses",
      "target": "0BJV24C4SY"
    },
    {
      "field": "uses",
      "target": "0KW2X0TFBX"
    },
    {
      "field": "uses",
      "target": "068BZBXE0C"
    },
    {
      "field": "uses",
      "target": "05V7J4WD1C"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/67KN8TZJ5C.md",
  "msc": [
    "30A99"
  ],
  "name": "Proof that a complex number times its conjugate is the square of its modulus",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/67K/67KN8TZJ5C.md",
  "status": null,
  "tag": "67KN8TZJ5C",
  "uri": "m2://67KN8TZJ5C",
  "url": "https://math2.tamnd.com/wiki/proof-that-a-complex-number-times-its-conjugate-is-the-square-of-its-modulus/",
  "writing": {
    "tag": "67KN8TZJ5C",
    "kind": "proof",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}