Scientific Notation Calculator

Background.

A light-year is 9,460,730,472,580,800 metres. Read that aloud and most people lose count of the digit groups somewhere in the middle, yet the two things worth knowing are simple: it starts 9.46, and it is about ten thousand million million metres. Scientific notation writes exactly those two facts and nothing else. The light-year becomes exactly 9.4607304725808 × 10¹⁵ m, a leading number between 1 and 10 that carries the digits, and a power of ten that carries the scale.

That split is why the notation runs through every science course. Comparing 6.022 × 10²³ with 9.109 × 10⁻³¹ takes a glance at the exponents, where comparing the same numbers in decimal means counting zeros. Multiplying them means multiplying two short numbers and adding two whole numbers. And the digits you write in front show how precisely you know the value, which a trailing string of zeros cannot do.

Enter a number in ordinary or e-notation and you get six results: the mantissa and exponent of scientific notation, the same number in engineering notation, its order of magnitude, and the power of ten of the SI prefix that fits it. Engineering notation is the version electronics uses, with the exponent held to a multiple of three so it lines up with kilo, mega, milli and micro; that is how a resistor gets labelled 4.7 kΩ rather than 4.7 × 10³ Ω.

The results panel shows mantissas to four decimal places, so the light-year's mantissa appears as 9.4607. Nothing is lost by that: a mantissa's digits are the digits you typed, moved past the decimal point, and you can always copy them in full from your own input.

What is scientific notation calculator?

Scientific notation writes a non-zero number as m × 10ⁿ, where n is a whole number and the mantissa m has exactly one non-zero digit before the decimal point, so 1 ≤ |m| < 10. With that rule every non-zero number has one and only one scientific form, which is what makes values easy to sort and compare. The mantissa is also called the significand or the coefficient. Calculators and programming languages print the same thing as e-notation, 6.022E23 meaning 6.022 × 10²³. Engineering notation relaxes the mantissa to 1 ≤ |m| < 1000 in exchange for an exponent that is a multiple of three, matching the SI prefixes. In British schools scientific notation is taught as standard form, while in American schools standard form usually means the ordinary way of writing a number, 47,000,000 rather than 4.7 × 10⁷.

How to use this calculator.

1. 01Type the number in the Value field as you would write it, such as 0.000123 or 299792458. Leave out thousands separators.
2. 02For very large or very small values, e-notation saves typing: 6.022e23 for Avogadro's number or 9.109e-31 for the electron's mass in kilograms.
3. 03Read the mantissa and exponent together as m × 10ⁿ. A negative number keeps its minus sign on the mantissa; a negative exponent means a number smaller than 1.
4. 04For electronics or metric units, read the engineering mantissa and exponent, then name the exponent's prefix: 3 kilo, 6 mega, 9 giga, −3 milli, −6 micro, −9 nano, −12 pico.
5. 05Use the order of magnitude to compare scales: two values whose orders differ by 3 are roughly a thousand times apart. Check the SI prefix exponent last, since it stops at 30 and −30.

The formula.

The exponent is n = ⌊log₁₀|x|⌋, the floor of the common logarithm of the number's size. The floor picks the largest whole number not exceeding the logarithm, which is the same as finding the power of ten at or just below the number: 10ⁿ ≤ |x| < 10ⁿ⁺¹. The mantissa is then m = x ÷ 10ⁿ, which lands between 1 and 10 and keeps the number's sign. The order of magnitude output is the same n under another name.

The floor matters for numbers below 1. For 0.000123 the logarithm is about −3.910, and the floor of that is −4, not −3, because −4 is the next whole number down. So 0.000123 = 1.23 × 10⁻⁴. Truncating toward zero would give −3 and a mantissa of 0.123, which breaks the one-digit rule.

Engineering notation takes the scientific exponent down to a multiple of three: nₑ = ⌊n ÷ 3⌋ × 3, and the engineering mantissa is x ÷ 10ⁿₑ, between 1 and 1000. For 47,000,000, n = 7, so nₑ = 6 and the engineering form is 47 × 10⁶. For 0.000123, n = −4 and ⌊−4 ÷ 3⌋ = ⌊−1.333⌋ = −2, so nₑ = −6 and the number is 123 × 10⁻⁶, or 123 micro.

The SI prefix exponent equals the engineering exponent within the range quecto (10⁻³⁰) to quetta (10³⁰), and is held at −30 or 30 beyond it. That range includes the four prefixes added in 2022, ronna (10²⁷), quetta (10³⁰), ronto (10⁻²⁷) and quecto (10⁻³⁰), so an engineering exponent of −27 reports −27, and one of −33 reports −30. Read a value at either limit as a sign that the number may lie past the quecto–quetta range, not as its prefix.

Going up, the prefixes are kilo (k), mega (M), giga (G), tera (T), peta (P), exa (E), zetta (Z) and yotta (Y); going down, milli (m), micro (µ), nano (n), pico (p), femto (f), atto (a), zepto (z) and yocto (y). Capitals matter: 1 mW is a milliwatt and 1 MW a megawatt, 10⁹ times larger. The 1960 system ran from pico to tera; femto and atto followed in 1964, peta and exa in 1975, and the zetta, yotta, zepto and yocto group in 1991, and ronna, quetta, ronto and quecto in 2022. Hecto, deca, deci and centi are SI prefixes too, but their exponents of 2, 1, −1 and −2 are not multiples of three, so engineering notation never lands on them: 0.05 m comes out as 50 × 10⁻³ m, which is 50 mm rather than 5 cm.

Zero has no logarithm and no scientific form, since no power of ten times a mantissa between 1 and 10 can make it. By convention all six results for zero are 0. Any value from −10³⁰⁰ to 10³⁰⁰ is accepted, which takes in magnitudes as small as 10⁻³⁰⁰, far beyond anything physical.

A worked example.

Frequently asked questions.

References& sources.

1. [1]Thompson, A. and Taylor, B. N. NIST Special Publication 811 — Guide for the Use of the International System of Units (SI), 2008 edition. The US National Institute of Standards and Technology's primary reference for SI usage, including §6 on SI prefixes, §7.10 on scientific notation, and Appendix C on the prefixes and their formal definitions.
2. [2]Bureau International des Poids et Mesures. The International System of Units (SI Brochure), 9th edition, 2019. Table 7 lists the SI prefixes from yocto (10^−24) to yotta (10^24); §3 explains the formation rules and the integer-power-of-ten constraint that underlies engineering notation.
3. [3]International Organization for Standardization. ISO 80000-1:2009 — Quantities and units, Part 1: General. The international standard governing the notation a × 10^n for scientific notation, the rules for significant figures, and the relationship between the SI prefixes and engineering-form exponents.
4. [4]International Union of Pure and Applied Chemistry. Compendium of Chemical Terminology, 2nd edition (the Gold Book), 1997, online updates ongoing. Entries on 'order of magnitude', 'scientific notation', and 'SI prefixes' formalise the chemistry community's usage and harmonise with BIPM.
5. [5]Feynman, R. P., Leighton, R. B., and Sands, M. The Feynman Lectures on Physics, Volume I, Chapter 5 — Time and Distance, §5-1 to §5-4. Feynman's classic exposition of orders of magnitude and the use of scientific notation to compare physical scales from cosmological (10^26 m) to subatomic (10^−15 m).
6. [6]27th General Conference on Weights and Measures (CGPM), Resolution 3 (2022). Ratified four new SI prefixes — ronna (10^27), quetta (10^30), ronto (10^−27), quecto (10^−30) — extending the BIPM prefix table beyond the earlier ±24 range. The Quanta calculator follows this extended range and clamps the SI prefix exponent at ±30.