{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "4JAEKN22Y5"
    },
    {
      "field": "uses",
      "target": "8Y4C6N09YQ"
    },
    {
      "field": "uses",
      "target": "955ZNEXEF0"
    },
    {
      "field": "uses",
      "target": "JQBG1GT0XW"
    },
    {
      "field": "uses",
      "target": "NDS0H727BA"
    },
    {
      "field": "uses",
      "target": "PA1B7DJ0C4"
    },
    {
      "field": "uses",
      "target": "PSNW0ZQ0E9"
    },
    {
      "field": "uses",
      "target": "SVSVMWA9NQ"
    },
    {
      "field": "specializes",
      "target": "GNDWPYYFW4"
    },
    {
      "field": "defines",
      "target": "R8S821AD6T"
    }
  ],
  "body": "\n## Summary\n\nFloating-point arithmetic is the way computers represent real numbers and compute with them.\nA floating-point number is a fixed number of significant digits times a power of a base, such as $1.0110 \\times 2^{-3}$.\nThe result of each operation is [[rounded|5NPVX29Y0C]] to a nearby floating-point number, so most computations carry a small [[round-off error|CPN6SD3BTV]].\nIts relative precision is measured by the [[machine epsilon|JA3QXVF2B8]], and the [[IEEE 754 standard|GNDWPYYFW4]] fixes the formats used in practice.\n\n## Explanations\n\n### L1 human\n\nFloating point is scientific notation for a computer.\nThe number $6.02 \\times 10^{23}$ keeps three significant digits and an exponent, so it can describe huge and tiny numbers with the same relative accuracy.\nA computer does the same in base $2$, with a fixed number of binary digits.\nBecause only finitely many numbers fit, most results must be rounded.\nFor example, $0.1$ has no finite binary expansion, so in a computer $0.1 + 0.2$ is not exactly equal to $0.3$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "65G50",
      "65Y04"
    ],
    "name": "Floating-point arithmetic",
    "same_as": {
      "wikidata": "Q117879",
      "wikipedia": "Floating-point arithmetic"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Floating-point_arithmetic"
      }
    ],
    "tag": "RTKRSC3KPH"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/RTKRSC3KPH/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/RTKRSC3KPH.md",
  "msc": [
    "65G50",
    "65Y04"
  ],
  "name": "Floating-point arithmetic",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/RTK/RTKRSC3KPH.md",
  "status": null,
  "tag": "RTKRSC3KPH",
  "uri": "m2://RTKRSC3KPH",
  "url": "https://tamnd.github.io/math2/wiki/floating-point-arithmetic/",
  "writing": {
    "tag": "RTKRSC3KPH",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}