Scientific notation explained, with e-notation and engineering notation

Scientific notation writes a number as a value between 1 and 10 multiplied by a power of ten: 299,792,458 becomes 2.99792458 × 10⁸, and 0.000047 becomes 4.7 × 10⁻⁵. It makes very large and very small numbers short to write, easy to compare and clear about their precision. The scientific notation converter switches between forms instantly, and this guide explains how to do it yourself.

What scientific notation is

A number in scientific notation has two parts: a coefficient (or mantissa) with exactly one non-zero digit before the decimal point, and an exponent telling you how many places to move that point. In 6.02 × 10²³, the coefficient is 6.02 and the exponent is 23. A positive exponent means a large number, a negative exponent a small one, and 10⁰ = 1 means no shift at all. Compare sizes by the exponent first: 3 × 10⁸ is a thousand times bigger than 3 × 10⁵, whatever the coefficients look like.

Converting in both directions

To write a number in scientific notation, move the decimal point until one non-zero digit is left in front of it, and count the moves:

• 299,792,458 m/s, the speed of light: move the point 8 places left to get 2.99792458 × 10⁸.
• 0.000047: move the point 5 places right to get 4.7 × 10⁻⁵.
• 5.29 × 10⁻¹¹ m, the approximate Bohr radius of a hydrogen atom, back to decimal: move the point 11 places left to get 0.0000000000529.

Moving left makes the exponent go up; moving right makes it go down. A quick sanity check: numbers of 10 or more get positive exponents, and numbers less than 1 get negative ones.

E-notation on calculators

Calculators, spreadsheets and programming languages write the same thing as e-notation: 6.02214076e23 means 6.02214076 × 10²³, which is the Avogadro constant, fixed exactly at that value since 2019. A display reading 1.5E-4 means 0.00015. The e has nothing to do with Euler's number e ≈ 2.718. JavaScript switches to e-notation by itself for numbers of 10²¹ and above, and below 10⁻⁶, which is why a very large result can suddenly appear as 1e+21. The converter accepts any of these styles, including 6.02 × 10^23 and numbers with commas, and returns scientific, e-notation, engineering and plain decimal forms; you can click any row to copy it.

Engineering notation and prefixes

Engineering notation is a variant that only uses exponents that are multiples of three, so the coefficient ranges from 1 to just under 1000. That lines the numbers up with SI prefixes:

• 10³ kilo, 10⁶ mega, 10⁹ giga, 10¹² tera;
• 10⁻³ milli, 10⁻⁶ micro, 10⁻⁹ nano, 10⁻¹² pico.

A capacitor of 4.7 × 10⁻⁵ farads in scientific notation is 47 × 10⁻⁶ F in engineering notation, which reads directly as 47 µF. In 2022 the prefix list was extended to ronna (10²⁷) and quetta (10³⁰), and ronto (10⁻²⁷) and quecto (10⁻³⁰). The unit converter handles the prefixes when you change units, and Metric vs imperial: how we got two systems explains the wider system.

Calculating with it

The rules follow from the laws of exponents:

• Multiply: multiply the coefficients and add the exponents. (3 × 10⁴) × (5 × 10⁶) = 15 × 10¹⁰, then tidy to 1.5 × 10¹¹.
• Divide: divide the coefficients and subtract the exponents. (6 × 10⁸) ÷ (2 × 10⁻³) = 3 × 10¹¹.
• Add or subtract: first rewrite one number so the exponents match. 3.2 × 10⁵ + 4.1 × 10⁴ = 3.2 × 10⁵ + 0.41 × 10⁵ = 3.61 × 10⁵.

On the scientific calculator you can type these directly using the exponent key, which saves counting zeros.

Showing precision

Scientific notation also shows how precise a number is, which ordinary notation can hide. Written as 4560, it is unclear whether the final zero was measured or is just a placeholder. Written as 4.56 × 10³, the value has three significant figures; written as 4.560 × 10³, it has four. That is why scientists report measurements this way. The rules for counting digits are in Significant figures: the rules, with examples, and the significant figures calculator counts and rounds them for you. The converter rounds the coefficient to six decimal places, so trim it to the precision your data supports.