How to Treat Exact Numbers in Uncertainty Propagation (GUM): Zero Uncertainty and Combined Uncertainty

In metrology, an exact number is a quantity known with no uncertainty. Under the Guide to the Expression of Uncertainty in Measurement (GUM), exact inputs are assigned zero standard uncertainty. They therefore contribute nothing to the combined standard uncertainty of a result, no matter how large their sensitivity coefficients are. This article is part of our precision-and-rounding reference library, not merely a calculator page. It explains the rule, cites the standards, works through examples, and flags the mistakes that most often corrupt uncertainty budgets.

Rule Statement

Let a measurement model be y = f(x₁, x₂, …, x_N). Each input estimate x_i has a standard uncertainty u(x_i). The GUM combined standard uncertainty is:

GUM 5.1.2: u_c²(y) = Σ (∂f/∂x_i)² u²(x_i).

If x_i is exact, then u(x_i) = 0. Its term vanishes. Exact numbers include defined conversion factors (for example, 1 in = 25.4 mm exactly), defined SI constants (for example, c = 299 792 458 m/s), mathematical constants used symbolically (π, e), and counted integers when the count itself is the quantity of interest. Exact numbers have infinite significant figures; they do not limit the precision of a result. Only inexact inputs limit uncertainty.

Standards Citation

• JCGM 100:2008 (GUM), 2.3.1: defines standard uncertainty as a parameter characterizing the dispersion of values.
• JCGM 100:2008 (GUM), 5.1.2: gives the combined standard uncertainty formula used above.
• JCGM 100:2008 (GUM), 5.2.2: extends the formula to correlated inputs; exact inputs still have zero variance and zero covariance.
• NIST Technical Note 1297, Section 2.1: states that exact quantities have zero uncertainty.
• ISO 80000-1:2009, clause 7: distinguishes exact numbers from measured values and discusses significant figures and rounding.

Worked Examples

Example 1: Exact conversion factor

A length is measured as L = 12.00 in with u(L) = 0.05 in. Convert to millimetres using the exact factor 1 in = 25.4 mm. The model is L_mm = 25.4 L_in. The sensitivity coefficient is c = 25.4 mm/in. Thus u(L_mm) = |25.4| × 0.05 = 1.27 mm. The conversion factor contributes zero uncertainty. If the length were exact at 12 in, then u(L_mm) = 0.

Example 2: Counting number in concentration

Suppose 145 colonies are counted on a plate, and the volume is V = 1.00 mL with u(V) = 0.02 mL. The concentration is C = N/V. The sensitivity for N is 1/V = 1.00 mL^-1, but u(N) = 0 because the count is exact as a count. The sensitivity for V is −N/V² = −145 mL^-2. Therefore u(C) = |145| × 0.02 = 2.9 mL^-1. The exact count does not add measurement uncertainty. In biology, however, sampling variability may be modelled separately as a Poisson component; that is not the same as measurement uncertainty of the count.

Example 3: Mathematical constant π

For a circle, A = πr². The constant π is exact in the model, so u(π) = 0. If r = 2.00 cm with u(r) = 0.01 cm, then u(A) = |2πr| u(r) = 2 × 3.14159 × 2.00 × 0.01 = 0.126 cm². Using 3.14 instead of π creates a computational rounding error of about 0.0064 cm² in the area, but that is not a measurement uncertainty and should not be combined with u(A) unless the approximation is a deliberate part of the model.

Counter-Examples

• Rounding because the exact number has few digits. The exact value 12 has infinite significant figures. It does not force a result to two significant figures.
• Assigning uncertainty to 25.4 mm/in. The inch-to-millimetre factor is exact by international agreement. It has zero uncertainty.
• Treating 3.14 as π and propagating its rounding error as Type B uncertainty. Better practice is to use sufficient digits or carry π symbolically.
• Confusing exactness with trailing zeros. A measured value reported as 12.0 mm is inexact and has an uncertainty; the trailing zero indicates precision, not exactness.
• Double-counting an exact factor. Do not include a conversion factor as both an exact multiplier and an uncertain input.

Convention Comparison Table

Common Mistakes

1. Forgetting the sensitivity coefficient. If u(x_i) = 0, the contribution is zero regardless of ∂f/∂x_i.
2. Using exact numbers to justify excessive digits. Exact inputs do not reduce the uncertainty of measured inputs.
3. Mixing significant-figure rules with GUM propagation. GUM uses uncertainties, not sig-fig counting, for the uncertainty budget.
4. Treating a rounded exact constant as exact. The symbol π is exact; the decimal 3.14 is an approximation.
5. Adding floating-point noise to u_c. Binary representation errors are computational, not measurement uncertainties.

Quick Reference Table

Software Behavior Note

Most software (Python, R, MATLAB, Excel, Google Sheets) stores numbers as binary floating point. Exact decimal constants such as 0.1 are not exactly representable, so tiny floating-point errors can appear. These are computational artifacts, not measurement uncertainties. Do not add them to u_c. Use decimal libraries or rational arithmetic when exactness matters. See our related articles on JavaScript floating-point and Python for more detail. In uncertainty calculators, exact numbers should be entered with u = 0; the calculator should not force significant-figure truncation on them.

Discipline Note

In physics and engineering, defined constants such as c, h, and e are exact in the SI. In chemistry, standard atomic weights are not exact; they carry uncertainties. In biology, counts may be exact but sampling variability is often modelled separately, for example with a Poisson distribution. In astronomy, some astronomical constants are exact by definition while others are measured. Always check the discipline-specific convention. GUM provides the general framework, but field guides such as ISO/IEC 17025 and NIST TN 1297 may add requirements.