Scientific notation writes any number as a value between 1 and 10 multiplied by a power of ten. It keeps very large and very small quantities readable and makes the precision of a measurement explicit.
The exponent counts how many places the decimal point moved. Moving it right gives a negative exponent, moving it left a positive one. The mantissa always has exactly one non-zero digit before the point.
Engineering notation
A variant that restricts exponents to multiples of three, so the mantissa runs from 1 to 1000 rather than 1 to 10. This exists because it maps directly onto SI prefixes. 4.5678 × 10⁻⁴ becomes 456.78 × 10⁻⁶, which is 456.78 microunits — immediately readable to anyone working with components.
| Exponent | Prefix | Symbol |
|---|---|---|
| 10¹² | tera | T |
| 10⁹ | giga | G |
| 10⁶ | mega | M |
| 10³ | kilo | k |
| 10⁻³ | milli | m |
| 10⁻⁶ | micro | µ |
| 10⁻⁹ | nano | n |
| 10⁻¹² | pico | p |
E-notation
Calculators and programming languages write 4.5678 × 10⁻⁴ as 4.5678e-4, because superscripts are awkward in plain text. It means exactly the same thing. Note that e here is unrelated to Euler’s number.
Significant figures
Scientific notation resolves an ambiguity that plain decimals cannot. Written as 4500, it is unclear whether the trailing zeros are measured or merely placeholders. Written as 4.5 × 10³ it clearly carries two significant figures; as 4.500 × 10³, four.
The rules for counting: all non-zero digits count; zeros between non-zero digits count; leading zeros never count; trailing zeros count only after a decimal point. So 0.00320 has three significant figures — the two leading zeros are placeholders, the trailing one is real.
Arithmetic shortcuts
Multiplying means multiplying mantissas and adding exponents; dividing means dividing mantissas and subtracting exponents. This is often faster mentally than the decimal version, and it makes order-of-magnitude estimation easy — a skill worth more in physics than exact arithmetic, since it tells you within seconds whether an answer is plausible.