Significant Figures
Significant figures are the digits in a measurement that carry real information, set by the precision of the instrument used.
Definition
Significant figures are the digits in a measurement that you actually know: a ruler marked in millimeters gives a length to about three digits, so you should not write six. They are the digits known with certainty plus one estimated digit; nonzero digits are always significant, zeros between them are significant, leading zeros are not, and trailing zeros after a decimal point are. When multiplying or dividing, the result keeps as many significant figures as the least precise measurement, and when adding or subtracting it keeps the fewest decimal places. The rules are a simplified form of uncertainty propagation, exact numbers such as counts and defined conversion factors never limit a result, and in careful work an explicit uncertainty like $2.70 \pm 0.05$ is preferred.
Example
A bean plant measured as $8.5\ \text{cm}$ has two significant figures, and writing $8.500\ \text{cm}$ would claim a precision the ruler cannot deliver. A rock of mass $13.5\ \text{g}$ (three sig figs) displacing $5.0\ \text{mL}$ (two) has density $13.5 / 5.0 = 2.7\ \text{g/cm}^3$, reported with two, not $2.70$. A measured $1500\ \text{m}$ is ambiguous, while $1.50 \times 10^3\ \text{m}$ states three; converting $2.40\ \text{m/s}$ to $8.64\ \text{km/h}$ keeps three because the factors $1000$ and $3600$ are exact; and a mean of five speeds each read to $0.01\ \text{m/s}$ is reported as $2.40$, not $2.400$, even if the calculator shows $2.4000$.
Key Insight
A calculator gives ten digits, but your ruler only gave you two or three; write down what you measured, not what the calculator says. Significant figures are a form of honesty, because reporting $2.7$ instead of $2.70$ tells the reader exactly how much to trust the number. Rounding at every intermediate step compounds error, so carry extra digits through the calculation and round only the final reported value.