{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "0404D8TN7Y"
    }
  ],
  "body": "\n## Description\n\nThe golden ratio is the real number\n$$\\varphi = \\frac{1 + \\sqrt{5}}{2} \\approx 1.6180339887.$$\nIt is the positive root of the polynomial $x^2 - x - 1$, so $\\varphi^2 = \\varphi + 1$.\nThe other root is $\\psi = \\frac{1 - \\sqrt{5}}{2} = 1 - \\varphi = -\\frac{1}{\\varphi}$.\n\n## Remarks\n\nTwo positive numbers $a \u003e b$ are in the golden ratio when $\\frac{a + b}{a} = \\frac{a}{b}$. Then $\\frac{a}{b} = \\varphi$.\nThe number $\\varphi$ is irrational, because $\\sqrt{5}$ is irrational.\nThe ratios of consecutive [[Fibonacci numbers|0GA9K57A39]] tend to $\\varphi$. [[Binet's formula|0404D8TN7Y]] uses $\\varphi$ and $\\psi$.\n\n## Explanations\n\n### L1 human\n\nCut a stick into a long part and a short part. Do it so that the whole stick relates to the long part as the long part relates to the short part.\nThe ratio of long to short is then $\\varphi$, about $1.618$.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "kind": "object",
    "msc": [
      "11B39"
    ],
    "name": "Golden ratio",
    "same_as": {
      "wikidata": "Q41690",
      "wikipedia": "Golden ratio"
    },
    "sources": [
      {
        "locator": "Calculation",
        "url": "https://en.wikipedia.org/wiki/Golden_ratio"
      }
    ],
    "tag": "0SC8T7548D"
  },
  "kind": "object",
  "link": "https://math2.tamnd.com/node/0SC8T7548D/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/0SC8T7548D.md",
  "msc": [
    "11B39"
  ],
  "name": "Golden ratio",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0SC/0SC8T7548D.md",
  "status": null,
  "tag": "0SC8T7548D",
  "uri": "m2://0SC8T7548D",
  "url": "https://math2.tamnd.com/wiki/golden-ratio/",
  "writing": {
    "tag": "0SC8T7548D",
    "kind": "object",
    "level": "W1",
    "next": [
      {
        "what": "section Invariants",
        "note": "a table of the main invariants",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge instance_of out",
        "have": 0,
        "want": 1
      }
    ]
  }
}