{
  "id": "scientific-notation",
  "version": "2fb3ad50b4be",
  "status": "published",
  "name": "Scientific Notation Calculator",
  "question": "What is this in scientific notation?",
  "summary": "Converts a number to scientific notation (a × 10ⁿ with 1 ≤ |a| < 10), E notation, engineering notation, and plain decimals, with its significant figures and optional rounding.",
  "category": "math",
  "subcategory": "arithmetic",
  "url": "https://www.acalculator.org/math/scientific-notation-calculator",
  "markdown": "https://www.acalculator.org/math/scientific-notation-calculator.md",
  "kind": "function",
  "method": "Write the number as a × 10ⁿ with 1 ≤ |a| < 10: n is the number of places the decimal point moves (left is positive), and a keeps the significant digits.",
  "assumptions": [
    "Leading zeros are never significant. Trailing zeros are significant after a decimal point (0.0010 has 2), and not in a whole number without one (1200 has 2; type 1200. to keep 4).",
    "Rounding to significant figures drops digits: more than half rounds up, less rounds down, and exactly half rounds to the even digit (2.45 → 2.4, 2.35 → 2.4).",
    "Asking for more significant figures than the number has adds zeros (1.2 to 4 figures is 1.200).",
    "Digits are exact as typed: nothing is rounded through binary floating point.",
    "Zero is written as 0 in every form. An empty box counts as 0."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "n": {
        "title": "Number",
        "description": "The number: a decimal such as 0.00045 or 123,400, E notation such as 4.5e-4, or 4.5 × 10^-4.",
        "type": "string",
        "maxLength": 100
      },
      "sf": {
        "title": "Round to significant figures",
        "description": "Optional: how many significant figures to keep, from 1 to 30. Leave empty to keep them all.",
        "type": "integer",
        "minimum": 1,
        "maximum": 30
      }
    }
  },
  "outputs": {
    "scientific": {
      "label": "Scientific notation",
      "description": "The number as a × 10ⁿ, with one non-zero digit before the point of a.",
      "format": "text"
    },
    "e": {
      "label": "E notation",
      "description": "The same number as calculators and spreadsheets write it: a e n means a × 10ⁿ.",
      "format": "text"
    },
    "engineering": {
      "label": "Engineering notation",
      "description": "The number with a power of ten that is a multiple of 3, matching the SI prefixes.",
      "format": "text"
    },
    "decimal": {
      "label": "Decimal notation",
      "description": "The number written out in full, for powers from 10⁻³⁰ to 10³⁰.",
      "format": "text"
    },
    "coefficient": {
      "label": "Coefficient a",
      "description": "The number before × 10ⁿ.",
      "format": "text"
    },
    "exponent": {
      "label": "Exponent n",
      "description": "The power of ten.",
      "format": "integer"
    },
    "figures": {
      "label": "Significant figures",
      "description": "How many significant figures the result has.",
      "format": "integer"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "n": "123,400"
    },
    "outputs": {
      "scientific": "1.234 × 10⁵",
      "e": "1.234e5",
      "engineering": "123.4 × 10³",
      "decimal": "123,400",
      "coefficient": "1.234",
      "exponent": 5,
      "figures": 4
    },
    "text": "123,400 in scientific notation is 1.234 × 10⁵."
  },
  "examples": [
    {
      "given": {
        "n": "123,400"
      },
      "expect": {
        "scientific": "1.234 × 10⁵",
        "e": "1.234e5",
        "engineering": "123.4 × 10³",
        "decimal": "123,400",
        "exponent": 5,
        "sf": 4
      },
      "source": "hand calculation in content.mdx: move the point 5 places left; trailing zeros of a whole number do not count"
    },
    {
      "given": {
        "n": "37,000"
      },
      "expect": {
        "scientific": "3.7 × 10⁴"
      },
      "source": "OpenStax Prealgebra 2e, section 10.5: 37,000 = 3.7 × 10⁴"
    },
    {
      "given": {
        "n": "0.00045"
      },
      "expect": {
        "scientific": "4.5 × 10⁻⁴",
        "e": "4.5e-4",
        "engineering": "450 × 10⁻⁶",
        "decimal": "0.00045",
        "sf": 2
      },
      "source": "hand calculation in content.mdx: move the point 4 places right; leading zeros do not count"
    },
    {
      "given": {
        "n": "299792458",
        "sf": 3
      },
      "expect": {
        "scientific": "3.00 × 10⁸",
        "e": "3.00e8",
        "sf": 3
      },
      "source": "NIST CODATA: the speed of light is exactly 299 792 458 m/s; hand calculation in content.mdx: 2.99|79… rounds up to 3.00"
    },
    {
      "given": {
        "n": "6.02214076e23"
      },
      "expect": {
        "scientific": "6.02214076 × 10²³",
        "engineering": "602.214076 × 10²¹",
        "decimal": "602,214,076,000,000,000,000,000",
        "sf": 9
      },
      "source": "NIST CODATA 2022: the Avogadro constant is exactly 6.022 140 76 × 10²³ per mole"
    },
    {
      "given": {
        "n": "9.1093837139 × 10^-31",
        "sf": 4
      },
      "expect": {
        "scientific": "9.109 × 10⁻³¹",
        "e": "9.109e-31",
        "exponent": -31
      },
      "source": "NIST CODATA 2022: the electron mass is 9.109 383 7139 × 10⁻³¹ kg; rounded to 4 figures by hand"
    },
    {
      "given": {
        "n": "2.45",
        "sf": 2
      },
      "expect": {
        "scientific": "2.4 × 10⁰"
      },
      "source": "OpenStax Chemistry 2e, section 1.5: a dropped digit of exactly 5 rounds to the even digit, so 2.45 → 2.4"
    },
    {
      "given": {
        "n": "-0.0010"
      },
      "expect": {
        "scientific": "−1.0 × 10⁻³",
        "e": "-1.0e-3",
        "decimal": "−0.0010",
        "sf": 2
      },
      "source": "OpenStax Chemistry 2e, section 1.5: a trailing zero after the decimal point is significant"
    }
  ],
  "sources": [
    "OpenStax, Prealgebra 2e, §10.5 Integer Exponents and Scientific Notation (a × 10ⁿ with 1 ≤ a < 10 and n an integer; 37,000 = 3.7 × 10⁴). https://openstax.org/books/prealgebra-2e/pages/10-5-integer-exponents-and-scientific-notation",
    "OpenStax, Chemistry 2e, §1.5 Measurement Uncertainty, Accuracy, and Precision (significant figures; rounding a dropped 5 to the even digit). https://openstax.org/books/chemistry-2e/pages/1-5-measurement-uncertainty-accuracy-and-precision",
    "NIST, CODATA 2022 values of the fundamental physical constants: speed of light 299 792 458 m/s (exact), Avogadro constant 6.022 140 76 × 10²³ mol⁻¹ (exact), electron mass 9.109 383 7139 × 10⁻³¹ kg. https://physics.nist.gov/cuu/Constants/",
    "NIST, Metric (SI) Prefixes (kilo 10³, mega 10⁶, milli 10⁻³, micro 10⁻⁶). https://www.nist.gov/pml/owm/metric-si-prefixes"
  ],
  "related": [
    "rounding",
    "exponent",
    "remainder"
  ],
  "changelog": []
}