Exact Numbers and Significant Figures
Exact numbers are counted or defined values. They do not limit significant figures, but measured, rounded, and approximate values can. The hard part is deciding which type of number you are using before you round.
The direct answer
Exact numbers are treated as having unlimited significant figures. They stay in the calculation, but they are ignored when choosing how many significant figures the final answer should have.
Counted items like samples, atoms in a formula, and trials do not limit the answer.
Unit definitions such as 100 cm = 1 m are exact and do not reduce precision.
Instrument readings, densities, averages, and approximations can limit sig figs.
Exact or measured? Classification table
| Number | Type | Sig fig effect | Why |
|---|---|---|---|
| 12 students | Exact count | Does not limit sig figs | The students are counted, not measured with an instrument. |
| 2 H atoms in H2O | Formula count | Does not limit sig figs | The subscript describes the exact composition of the formula unit. |
| 1 m = 100 cm | Defined metric conversion | Does not limit sig figs | Metric prefix conversions are definitions, so the 100 is exact. |
| 1 in = 2.54 cm | Defined conversion | Does not limit sig figs | The inch-to-centimeter relationship is defined exactly. |
| 6.02214076 x 10^23 entities/mol | SI defining constant | Does not limit sig figs | The mole is defined using this fixed Avogadro constant value. |
| 0.997 g/mL | Measured or tabulated value | Can limit sig figs | Density values come from measurement and should be used with their stated precision. |
| 1 lb approx. 454 g | Rounded conversion | Can limit sig figs | The word approximate and the rounded digits tell you not to treat it as exact. |
| 250.0 mL | Measured volume | Can limit sig figs | The trailing decimal zero reports measurement precision, so it matters. |
Decision checklist before rounding
Was the number counted?
Treat it as exact. Counts such as 4 trials, 12 beakers, or 24 tiles do not set the significant figure limit.
Is it a definition or unit prefix conversion?
Treat it as exact. Metric prefix conversions like 1000 mL = 1 L do not reduce precision.
Is it a coefficient in a formula or balanced equation?
Treat the coefficient as exact, then round based on the measured masses, volumes, concentrations, or moles.
Was it measured, averaged, rounded, or pulled from a table?
Treat it as limited. These values can control the final significant figures.
Worked examples
| Problem | Exact part | Raw value | Report as | Reason |
|---|---|---|---|---|
| 12.5 cm x 2 | 2 is an exact multiplier | 25.0 cm | 25.0 cm | 12.5 cm has 3 sig figs, so the exact 2 does not reduce the answer to 1 sig fig. |
| 3.20 g / 4 samples | 4 samples is an exact count | 0.800 g/sample | 0.800 g/sample | 3.20 g has 3 sig figs, and the exact count does not limit the quotient. |
| 250.0 mL x 1 L / 1000 mL | 1000 mL = 1 L is exact | 0.2500 L | 0.2500 L | The measured 250.0 mL has 4 sig figs, so the converted value keeps 4 sig figs. |
| 2.50 g / 1.2 mL | Neither value is exact | 2.0833... g/mL | 2.1 g/mL | Both numbers are measurements, and 1.2 mL has only 2 sig figs. |
| 14.626 cm / 2.54 cm per inch | 2.54 cm = 1 in is exact | 5.7582677... in | 5.7583 in | The conversion factor is exact, so the 5-sig-fig measurement controls the answer. |
Exact numbers in chemistry and conversions
Chemistry questions often hide exact values inside unit conversions and stoichiometry. Label exact values first, then let only measured values control the final answer.
| Scenario | Exact value | How to use it |
|---|---|---|
| Metric prefix conversion | 1000 mL = 1 L | Convert units without changing the sig fig limit. |
| Counting particles or items | 24 molecules, 6 trials, 3 samples | Use the count as exact unless the problem says it is an estimate. |
| Balanced equation coefficient | 2 H2 + O2 -> 2 H2O | Use stoichiometric coefficients as exact ratios. |
| SI mole definition | 6.02214076 x 10^23 entities per mole | Treat the defining constant as exact; measured sample data still controls rounding. |
| Density or molar mass from a table | 0.997 g/mL or 18.015 g/mol | Use the precision shown unless your class gives a different convention. |
Common mistakes
Letting the 1000 in a metric conversion reduce a 4-sig-fig measurement to 1 sig fig.
Treating a counted value, such as 4 samples, as if it were a measured value with 1 sig fig.
Assuming every conversion factor is exact. Approximate or measured conversion factors can limit the answer.
Dropping trailing zeros after exact-number calculations, such as changing 25.0 cm to 25 cm.
Rounding intermediate values before deciding which measured number actually controls the final answer.
Frequently asked questions
Do exact numbers have infinite significant figures?
Exact numbers are treated as having unlimited significant figures. They do not limit the number of significant figures in a calculated result.
Do counted numbers affect significant figures?
No. Counted values such as 4 trials, 12 students, or 24 tiles are exact counts, so they do not limit the final answer.
Are conversion factors exact for significant figures?
Some conversion factors are exact and some are not. Defined conversions such as 100 cm = 1 m are exact. Measured, rounded, or approximate conversion factors can limit significant figures.
Is 1 m = 100 cm exact?
Yes. It is a metric prefix conversion, so 100 cm in 1 m is exact and does not reduce the precision of a calculation.
Does a balanced equation coefficient limit sig figs?
No. Coefficients in a balanced chemical equation are exact ratios. Round the final answer based on measured quantities such as mass, volume, concentration, or moles.
Can an exact number have a decimal?
Yes. A defined conversion can include decimals, such as 1 inch = 2.54 centimeters. The decimal digits are part of the definition and do not limit the result.