Significant Figures

Why significant figures matter

A calculator can display many digits, but a chemistry calculation often begins with measurements that are much less precise. Writing every calculator digit can suggest that the answer is known more precisely than the information allows. Significant figures help you give a useful answer without claiming unjustified precision. If a question states how many significant figures to use, follow that instruction.

Counting significant figures

Start at the first non-zero digit and count from there. Leading zeros only locate the decimal point: $0.003$ has one significant figure, the $3$. Zeros between non-zero digits do count, so $7.04$ has three significant figures. A zero at the end of a decimal can also show precision: $0.0500$ has three significant figures, $5$, $0$ and $0$.

A whole number ending in zeros can be ambiguous. For example, $5800$ might have been rounded to two significant figures or recorded more precisely. If the intended precision matters, it can be made clear by writing $5.8\times10^3$ for two significant figures.

Significant figures are not the same as decimal places. In $0.0346$, there are three significant figures but four digits after the decimal point.

Rounding an answer

Find the last digit you need to keep, then inspect the next digit. If it is $0$–$4$, leave the kept digit unchanged; if it is $5$–$9$, increase it by one. Remove any remaining digits, retaining place-value zeros where needed. For example, $5821$ to two significant figures is $5800$: keep $5$ and $8$, then look at the next digit, $2$. By contrast, $0.03468$ to three significant figures is $0.0347$, because the next digit after $6$ is $8$.

Choosing precision in a chemistry calculation

Suppose, for example, that a product mass is $2.46\,\mathrm{g}$ and its theoretical mass is $3.00\,\mathrm{g}$. The percentage yield is $(2.46\div3.00)\times100=82.0\%$ to three significant figures. The final zero is meaningful: $82.0\%$ has three significant figures, whereas $82\%$ has two.

When no number of significant figures is specified, use a sensible precision for the quantities given rather than copying a long calculator display. In multiplication or division, the least precise measured quantity is a useful guide; in addition or subtraction, consider the least precise decimal place. Exact counts, such as the coefficients in a balanced chemical equation, do not themselves limit the precision of a measured answer. Keep the full calculator value through intermediate steps and round once, at the end.

Count and round

Count from the first non-zero digit. Leading zeros do not count; zeros between significant digits do; final zeros after a decimal point can show precision. To round, check the next digit: $0$–$4$ leave the retained digit unchanged, $5$–$9$ round it up.

In calculations

Follow any stated number of significant figures. Otherwise, match the precision reasonably to the measured inputs. Keep intermediate values unrounded, then round the final answer and include its unit. Significant figures count from the first non-zero digit; decimal places count from the decimal point.