Definition · Plain-language
Scientific notation
Scientific notation is a compact way of writing very large or very small numbers as a coefficient between 1 and 10 multiplied by a power of ten.
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A coefficient times a power of ten
Scientific notation expresses any number as a coefficient multiplied by ten raised to a power. The coefficient is written with one non-zero digit before the decimal point — a value of at least 1 and less than 10 — and the power of ten records the scale. So 4,500 becomes 4.5 × 10³ and 0.0072 becomes 7.2 × 10⁻³. In Britain this form is often called standard form. Its purpose is economy: a number that would otherwise need a long string of zeros is captured in a few symbols, with the exponent carrying the magnitude.
Reading the exponent
The exponent tells you how far, and in which direction, the decimal point has moved. A positive exponent means a large number: 10⁶ is a million, so 2 × 10⁶ is 2,000,000. A negative exponent means a small number: 10⁻⁶ is a millionth, so 3 × 10⁻⁶ is 0.000003. To convert from scientific notation back to ordinary form, move the decimal point to the right for a positive exponent and to the left for a negative one, by the number of places the exponent gives. This direct link to powers of ten is why scientific notation pairs so neatly with metric prefixes.
Why scientists use it
Scientific notation earns its place for three reasons. It makes extreme quantities manageable — the number of atoms in a sample, or the size of a virus, would be unwieldy written out in full. It makes magnitudes easy to compare at a glance, since you simply read off the exponents. And it removes ambiguity about significant figures: writing 1,500 leaves the precision unclear, but 1.50 × 10³ states unambiguously that there are three significant figures. For all these reasons it is the default way of writing numbers across the physical sciences.
Key facts
At a glance
- Definition: a number as a coefficient (1–10) times a power of ten
- British name: also called standard form
- Example large: 3 × 10⁸ = 300,000,000
- Example small: 5 × 10⁻⁷ = 0.0000005
- Exponent: counts the places the decimal point moves
- Bonus: removes ambiguity about significant figures
Common misconceptions
What people often get wrong
Often heard: The coefficient in scientific notation can be any number.
Actually: By convention it is at least 1 and less than 10 — a single non-zero digit before the decimal point. Writing 45 × 10² is not proper scientific notation; 4.5 × 10³ is.
Often heard: A negative exponent means the number itself is negative.
Actually: It does not. A negative exponent signals a number smaller than one, such as 5 × 10⁻³ = 0.005. The sign of the exponent is about size, not whether the number is positive or negative.
Often heard: Scientific notation is only useful for very large numbers.
Actually: It is equally suited to very small ones, where negative exponents replace long strings of leading zeros, as in 1.6 × 10⁻¹⁹ for the charge on an electron.
