Scientific notation

The oldest surviving attempt to write genuinely large numbers is Archimedes' The Sand Reckoner, composed around 250 BCE and addressed to King Gelon of Syracuse. Greek numerals effectively stopped at the myriad, 10,000, so Archimedes built his own scaffolding: numbers up to a myriad myriads (10⁸) formed his first order, products of those the second order, and so on upward. With it he answered a deliberately absurd question — how many grains of sand would fill the universe — arriving at a bound on the order of 10⁶³ grains, using Aristarchus' heliocentric estimates to size the sphere he was filling. He also stated a rule for multiplying his orders together, which is the law of exponents in disguise. It is an overstatement to say he invented exponential notation, though: he had no algebraic symbolism for it, and the claim that he "anticipated logarithms" is a modern gloss rather than something in the text.

The symbols came much later. Descartes used raised numerals for powers in La Géométrie (1637), and Newton, in his two 1676 letters to Henry Oldenburg at the Royal Society, extended them to negative and fractional exponents. Writing a measurement as a coefficient times a power of ten became ordinary practice in nineteenth-century physics and astronomy, where quantities ranged over dozens of orders of magnitude and the alternative was counting zeros by eye.

Two twentieth-century conventions produced the forms on this page. FORTRAN's E format descriptor, from the 1957 compiler, wrote 6.022E+23 because punched cards and line printers had no superscripts; the notation outlived both and is still what spreadsheets and CSV exports emit. And the metric prefixes — kilo, hecto, deca, deci, centi and milli date from the French law of 1795 — were folded into the International System of Units at the 11th General Conference on Weights and Measures in 1960, with peta and exa added in 1975, zetta and yotta in 1991, and ronna, quetta, ronto and quecto in November 2022. Engineering notation exists to line up with that table: keep the exponent a multiple of three and every value has a prefix waiting for it.

Key properties

• Normalized scientific notation writes a value as a × 10ᵇ with 1 ≤ |a| < 10 and b a whole number; zero has no normalized form.
• The number of digits in the coefficient is exactly the number of significant figures being claimed.
• Engineering notation restricts b to multiples of three, which keeps the coefficient in [1, 1000) and maps the exponent straight onto an SI prefix.
• E notation (6.022E+23) means precisely the same thing as 6.022 × 10²³ — the E is a separator inherited from early programming languages, not a variable or a base.
• Leading zeros are never significant and trailing zeros after a decimal point always are; trailing zeros in a whole number like 1200 are ambiguous, which is one reason measurements are reported in scientific notation.
• Multiplying two values multiplies the coefficients and adds the exponents; dividing subtracts them.
• The SI prefix set spans 10⁻³⁰ (quecto) to 10³⁰ (quetta); hecto, deca, deci and centi exist but are not powers of 1000, so engineering notation skips them.
• A double-precision float carries only about 15–17 significant decimal digits, so this page rounds digit strings directly instead of converting through a floating-point number.

Where they turn up

• The speed of light in vacuum is exactly 299,792,458 m/s — 2.99792458 × 10⁸ — because the metre has been defined from it since 1983.
• Since the 2019 revision of the SI, the Avogadro constant is exactly 6.02214076 × 10²³ mol⁻¹ and the Planck constant exactly 6.62607015 × 10⁻³⁴ J·s; both are now definitions rather than measurements.
• Spreadsheets switch long numbers to E notation automatically, which is how identifiers and accession codes get silently mangled into values like 1.23457E+14 — a routine data-cleaning hazard rather than a rounding error you can recover from.
• Programming languages accept the same notation as a literal: 1e-9 in JavaScript, Python or C is a nanosecond expressed as a fraction of a second.
• Storage marketing uses SI prefixes (a terabyte is 10¹² bytes) while operating systems have historically counted in powers of two, which is why the IEC introduced the separate kibi-, mebi- and gibi- prefixes in 1998.