Writing Extreme Numbers
Scientific notation writes any number as a coefficient between 1 and 10 multiplied by a power of ten. The mass of an electron is 9.109 × 10−31 kg rather than a decimal point followed by thirty zeros. The convention removes both the tedium and the counting errors.
Converting
Move the decimal point until exactly one non-zero digit sits in front of it, and count the moves. Moving left gives a positive exponent, moving right a negative one:
| Decimal | Scientific notation | Moves |
|---|---|---|
| 452,000 | 4.52 × 10⁵ | 5 left |
| 0.000000452 | 4.52 × 10−7 | 7 right |
| 7.3 | 7.3 × 10⁰ | none |
Arithmetic Rules
- Multiplication — multiply the coefficients, add the exponents. (3 × 10⁴)(2 × 10³) = 6 × 10⁷.
- Division — divide the coefficients, subtract the exponents.
- Addition and subtraction — the exponents must match first. 3 × 10⁴ + 2 × 10³ becomes 3 × 10⁴ + 0.2 × 10⁴ = 3.2 × 10⁴.
After any operation, renormalise if the coefficient falls outside 1 to 10.
Engineering Prefix Chart
Engineering notation restricts exponents to multiples of three so they align with SI prefixes:
| Prefix | Symbol | Power | Prefix | Symbol | Power |
|---|---|---|---|---|---|
| tera | T | 1012 | milli | m | 10−3 |
| giga | G | 10⁹ | micro | µ | 10−6 |
| mega | M | 10⁶ | nano | n | 10−9 |
| kilo | k | 10³ | pico | p | 10−12 |
This is why a resistor is labelled 4.7 kΩ rather than 4.7 × 10³ Ω, and why file sizes are quoted in megabytes rather than powers of ten.
E Notation
Calculators and programming languages write 4.52e-7 for 4.52 × 10−7. The e stands for exponent and has nothing to do with Euler's number — a genuine source of confusion, since ln and e appear in the same contexts.
Significant Figures Travel With It
Scientific notation resolves an ambiguity decimal cannot. Is 1200 precise to two significant figures or four? Written as 1.2 × 10³ it is unambiguously two; 1.200 × 10³ is four. This is a large part of why scientific writing uses it as standard.
Frequently Asked Questions
What is the difference between scientific and engineering notation?
Scientific notation keeps the coefficient between 1 and 10. Engineering notation allows 1 to 1000 so that the exponent is always a multiple of three, matching SI prefixes.
How do I enter scientific notation on a calculator?
Use the EE or EXP key rather than typing ×10^ — on many calculators the latter is parsed differently and produces a wrong result.
Can the coefficient be negative?
Yes. −4.52 × 10³ is −4,520. The sign belongs to the coefficient; the exponent controls magnitude only.