Short Answer
In the GUM method, uncertainty propagation begins with a measurement model that relates input quantities to an output quantity. Some inputs are measured and carry uncertainty; others are exact. This reference page is part of a precision-and-rounding site, not just a calculator. Use our significant figures calculator to check rounding, but the rules below explain why exact numbers behave differently.
Rule Statement
An exact number is a number whose value is known with no uncertainty in the context of the measurement model. In GUM, exact inputs are assigned u(xi) = 0. The law of propagation of uncertainty (JCGM 100:2008, clause 5.2.2) is:
uc2(y) = Σ (∂f/∂xi)2 u2(xi) + 2 Σ Σ (∂f/∂xi)(∂f/∂xj) u(xi, xj)
For exact xi, u(xi) = 0, so variance and covariance terms involving only exact inputs vanish. However, exact numbers still appear in sensitivity coefficients ∂f/∂xi and can scale the uncertainty of other inputs. For example, if y = ax and a is exact, then u(y) = |a| u(x). If y = x/a, then u(y) = u(x)/|a|.
Exact numbers do not add uncertainty; they transmit and scale uncertainty.
Key points:
- Mathematical constants such as π and e are exact in formula use.
- Defined conversion factors, such as 1 in = 25.4 mm, are exact.
- Counts are exact only when they are complete enumerations, not random samples.
- A rounded exact constant may introduce error; use enough digits or treat the rounding as a correction.
Worked Examples
Example 1: Circle area with exact π
Let A = πr2, with r = 2.00 cm ± 0.01 cm. π is exact. The sensitivity coefficient is ∂A/∂r = 2πr. Thus u(A) = 2π(2.00 cm)(0.01 cm) = 0.12566 cm2. The result is A = 12.566 cm2 ± 0.13 cm2, reported as 12.57 cm2 ± 0.13 cm2 after rounding.
Example 2: Defined unit conversion
Convert L = 3.00 in ± 0.02 in to millimetres. The factor 1 in = 25.4 mm is exact. Then L = 76.2 mm and u(L) = 25.4 × 0.02 mm = 0.508 mm. Report 76.2 mm ± 0.5 mm. No uncertainty term is added for the conversion factor.
Example 3: Exact count as divisor
Twelve items are counted exactly, and their total mass is M = 60.00 g ± 0.05 g. The mean mass is M/12. The count is exact, so u(mean) = 0.05 g / 12 = 0.00417 g. Report 5.000 g ± 0.004 g.
Example 4: Density with exact volume conversion
Let ρ = m/V. Suppose m = 10.00 g ± 0.01 g and V = 5.00 mL ± 0.02 mL. The conversion 1 mL = 1 cm3 is exact and adds no uncertainty. Relative uncertainty is sqrt[(0.01/10.00)2 + (0.02/5.00)2] = 0.004123. With ρ = 2.00 g/mL, u(ρ) = 0.00825 g/mL, reported as 2.00 g/mL ± 0.008 g/mL.
Counter-Examples
These are common but incorrect treatments.
- Assigning uncertainty to π because 3.14 was used. If 3.14 replaces π, the truncation error is about 0.0016 relative and may dominate. Use more digits or include the rounding as a correction.
- Adding uncertainty for 1 in = 25.4 mm. This conversion is exact by international agreement. Do not add a variance term.
- Treating a random count as exact. Counts from radioactive decay, particle detection, or sampling often follow Poisson statistics, where u(N) ≈ √N.
- Rounding exact constants before propagation. Carry full precision through intermediate calculations, then round the final uncertainty and result.
- Double-counting an exact scale factor. If y = ax and a is exact, the relative uncertainty of y is u(x)/|x|, not u(x)/|x| plus u(a)/|a|.
Convention Comparison Table
| Number type | Example | GUM treatment | Uncertainty term | Notes |
|---|---|---|---|---|
| Mathematical constant | π, e | Exact in formula | u = 0 | Use enough digits to avoid truncation error. |
| Defined conversion | 1 in = 25.4 mm | Exact scale factor | u = 0 | No variance contribution. |
| Exact enumeration | 12 items counted | Exact integer | u = 0 | Valid only if the count is complete. |
| Random count | Poisson counts | Measured input | u ≈ √N | Do not treat as exact. |
| Defined physical constant | c = 299 792 458 m/s | Exact in SI | u = 0 | Definition-based exactness. |
| Measured physical constant | G, h before redefinition | Measured input | u from CODATA | Include uncertainty. |
| Rounded exact value | 3.14 for π | Approximation | Correction or uncertainty | Use full precision when possible. |
Standards Citation
- JCGM 100:2008 (GUM), clause 5.1.2: defines the measurement model Y = f(X1, …, XN) and the input quantities.
- JCGM 100:2008, clause 5.2.2: gives the law of propagation of uncertainty. Exact inputs have u(xi) = 0.
- JCGM 200:2012 (VIM): defines quantity value and exact quantity value; exact values are not assigned measurement uncertainty.
- NIST TN 1297, §7: provides guidance on evaluating and expressing uncertainty, including treatment of exact constants.
- ISO 80000-1:2009, clause 6.4: covers rounding of numerical values and significant digits.
- ASTM E29-22, clause 6: standard practice for using significant digits in test data and conformance.
Always verify the edition and clause in your controlled copy.
Common Mistakes
- Adding an uncertainty term for a defined conversion factor.
- Using a low-precision value of π and ignoring the resulting error.
- Treating a sampled count as an exact count.
- Rounding exact numbers before the propagation step.
- Forgetting that exact numbers still affect sensitivity coefficients.
- Confusing exact by definition with exact by measurement.
- Including exact numbers in relative uncertainty sums when they should not appear.
Practice Problems
- F = ma, with m = 2.00 kg ± 0.01 kg and a = 9.81 m/s2 ± 0.02 m/s2. Compute F and u(F).
- C = 2πr, with r = 10.0 cm ± 0.1 cm. Compute C and u(C).
- c = n/V, with n = 0.0500 mol ± 0.0001 mol and V = 0.2500 L ± 0.0002 L. Compute c and u(c).
- Convert V = 0.2500 L ± 0.0002 L to m3, using 1 L = 0.001 m3 exactly. Report the result and uncertainty.
Selected answers: (1) F = 19.62 N, u(F) = 0.05 N. (2) C = 62.8 cm, u(C) = 0.6 cm. (3) c = 0.2000 M, u(c) = 0.00043 M. (4) V = 0.0002500 m3 ± 0.0000002 m3.
Quick Reference Table
| Situation | Action | Example |
|---|---|---|
| Mathematical constant in formula | Use full precision; u = 0 | π in A = πr2 |
| Defined conversion | Multiply or divide; u = 0 | 1 in = 25.4 mm |
| Exact count | Use as integer; u = 0 | 12 items |
| Measured constant | Include u | G, measured g |
| Rounded exact constant | Use enough digits or add correction | 3.14 for π |
| Nonlinear function | Sensitivity coefficient still applies | ∂f/∂xi |
Sources & Further Reading
- JCGM 100:2008. Evaluation of measurement data — Guide to the expression of uncertainty in measurement (GUM). BIPM. Clauses 5.1.2 and 5.2.2.
- JCGM 200:2012. International vocabulary of metrology — Basic and general concepts and associated terms (VIM), 3rd ed.
- NIST Technical Note 1297. Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results. NIST, 1994.
- ISO 80000-1:2009. Quantities and units — Part 1: General.
- ASTM E29-22. Standard Practice for Using Significant Digits in Test Data to Determine Conformance with Specifications. ASTM International.
For related rules, see Exact Numbers, Error Propagation, GUM Guide, and Rounding Rules.
FAQ
Do exact numbers have zero uncertainty in GUM?
Yes. If an input quantity is truly exact in the measurement model, its standard uncertainty is u = 0, and it contributes no variance term.
How do exact numbers affect uncertainty?
They affect uncertainty through sensitivity coefficients. An exact scale factor can multiply or divide the uncertainty of a measured input, even though it adds no uncertainty of its own.
Are all counts exact?
No. A complete enumeration is exact. A random or sampled count, such as detected particles or radioactive decays, is a measured quantity and often has Poisson uncertainty.
Should I include uncertainty for 1 in = 25.4 mm?
No. That conversion is exact by definition. Do not add an uncertainty term for it.
How many digits of π should I use?
Use enough digits that the truncation error is negligible compared with the other uncertainty components. Six to eight digits are usually sufficient for typical laboratory work.
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