Physics · Article

SI units and modern metrology

Connect the SI's exact defining constants with practical unit realization, calibrated results, documented traceability and measurement uncertainty.

Previous article: Variational calculus, tensors, groups, and differential geometry

Back to Physics

After reading, you can

Specify the measurand and relevant measurement conditions before interpreting a number.
Recognize the seven exact SI defining constants and distinguish their values from practical realizations.
Explain why an exact defining constant does not make a derived measurement result exact.
Apply a stipulated calibration correction while retaining its uncertainty and scope.
Check a simple linear uncertainty budget under an explicit absence of correlation.
Distinguish metrological traceability, a stated uncertainty measure and fitness for a particular purpose.

State the measurand

Metrology is the science of measurement and its application. The measurand is the quantity intended to be measured. A unit identifies a reference for expressing its value; it does not identify the object, endpoints, state or procedure. Modern metrology connects that specified quantity with a result whose basis can be examined.

For our invented length record, the measurand is the separation between two designated faces of a stationary spacer, at a specified temperature and under specified contact conditions. A different temperature or contact procedure may require a different model or correction. Specifying the measurand matters before assigning a central value or uncertainty.

A measurement result needs an estimated value and relevant uncertainty information, expressed in compatible units. The earlier lesson introduced quantity values and unit conversion. Here the question is how a definition becomes an available reference and how a practical result remains connected to it. Our numerical records are stipulated teaching cases, not observations or certificates issued by a laboratory.

Read the seven defining constants

The International System of Units, SI, fixes the numerical values of seven defining constants in specified units. These assigned values have no uncertainty. The table follows the ninth-edition SI Brochure, version 4.01, June 2026. Exactness belongs to this definition; it does not transfer automatically to an apparatus or a measurement of another quantity.

The symbols in the table label different constants. Here e means elementary charge and k means the Boltzmann constant. Hz, J, C, lm and W denote hertz, joule, coulomb, lumen and watt. Their SI relationships include Hz = s⁻¹, J = kg·m²·s⁻², C = A·s, lm = cd·sr and W = J/s; sr is the steradian, the unit of solid angle. They are units attached to the constants, not seven additional independently defined base units.

The definitions work together. The second uses the caesium transition frequency; the metre uses c with that second; the kilogram uses h with the metre and second. Defining constant is a role within the SI, rather than a statement that every physical constant is assigned an exact numerical value. The selected set also includes the application-specific luminous efficacy K_cd for the stated radiation frequency.

The seven SI defining constants and their exact assigned values; the conditions in the names are part of the definition
Defining constant	Symbol	Exact value in the stated unit
Unperturbed ground-state hyperfine transition frequency of the caesium-133 atom	Δν_Cs	9 192 631 770 Hz
Speed of light in vacuum	c	299 792 458 m/s
Planck constant	h	6.626 070 15 × 10⁻³⁴ J·s
Elementary charge	e	1.602 176 634 × 10⁻¹⁹ C
Boltzmann constant	k	1.380 649 × 10⁻²³ J/K
Avogadro constant	N_A	6.022 140 76 × 10²³ mol⁻¹
Luminous efficacy of monochromatic radiation of frequency 540 × 10¹² Hz	K_cd	683 lm/W

Realize a definition

To realize a unit is to establish a quantity value and associated uncertainty consistently with the unit's definition. A realization uses a physical method. Fixing a defining constant leaves room for improved methods without changing the definition itself. Guidance on practical realizations is published in the SI Brochure's online Appendix 2.

The current metre guidance uses the vacuum relation L = cΔt: L is the length of the light path and Δt is the time taken along it. Our next case uses a straight one-way path between stationary endpoints in a single fixed inertial frame. It is a deliberately simplified model, with timing as its only uncertain input. It is not a recipe for realizing the metre with any available stopwatch.

Actual length realizations require an appropriate apparatus, procedure and uncertainty evaluation. Light propagation in air requires additional information; the vacuum c must not simply be used as an unqualified speed in a material. A measurement standard used as a reference also has an assigned value and uncertainty. An exact definition and an uncertain reference realization therefore have different roles.

Keep uncertainty when c is exact

A standard uncertainty u is uncertainty expressed as a standard deviation, a measure of dispersion rather than a guaranteed maximum error. For the stipulated vacuum model, supply a time estimate Δt = 10.000 ns with u(Δt) = 0.020 ns. Nano means 10⁻⁹, so both numbers must be converted to seconds before multiplying by c. These assigned inputs make the example definite; their digits are not evidence of an actual timing instrument's performance.

The central value is L = (299 792 458 m/s)(10.000 × 10⁻⁹ s) = 2.99792458 m. For multiplication by an exact positive constant, standard uncertainty scales by the same factor: u(L) = c u(Δt) = 0.00599584916 m. This follows because each deviation of the input from its central value is multiplied by c. No uncertainty contribution is assigned to the exact SI value of c.

For a compact display, round this case to L = 2.998 m with standard uncertainty u(L) = 0.006 m. The longer arithmetic values above are calculation intermediates, not a claim of additional measurement precision. This uncertainty describes only the stipulated model; a real experiment must include other significant inputs and effects. The exact definition does not eliminate uncertainty in timing, geometry or corrections.

Use calibration with its scope

Calibration first relates reference values and their uncertainties to an instrument's indications under specified conditions, then uses that relation to obtain results from indications. It may supply a correction with uncertainty. Adjustment changes the measuring system's indications; it is a different operation. Calibration is also distinct from a decision that specified requirements are satisfied.

In a separate invented spacer record, supply an indication r = 12.008 mm and an additive calibration correction δ = −0.006 mm for this reading, stated range and conditions. The chosen model is L = r + δ, so the corrected central value is 12.002 mm. The negative correction reduces the indication; it does not become a negative length.

Also supply standard uncertainties u(r) = 0.004 mm and u(δ) = 0.003 mm. A correction's uncertainty describes incomplete knowledge of the correction; it is not the correction's signed value or the systematic error itself. These two supplied components represent the complete budget only within this invented model. Applying the correction does not make the corrected value exact, and this record does not justify extrapolating it to other ranges or conditions.

Check one linear uncertainty budget

An uncertainty budget identifies contributing inputs, their uncertainty measures and how the model combines them. For L = r + δ, stipulate that the two input uncertainty components are uncorrelated: their covariance, the measure of their paired variation, is zero. Under this condition the variances add, so u_c(L) = √[u(r)² + u(δ)²]. The subscript c here means combined; u_c is the combined standard uncertainty.

In this exact linear model, square the standard uncertainties, add, and take the positive square root: u_c(L) = √[(0.004 mm)² + (0.003 mm)²] = √(0.000025 mm²) = 0.005 mm. Equivalently, the components are 4 µm and 3 µm, giving 5 µm; micro means 10⁻⁶. The squared-unit terms combine before the square root restores a length unit.

The simple root-sum-of-squares rule is conditional. Correlated inputs require their covariance contribution, and other models can require different sensitivity factors or a more complete calculation. A list of calibration links does not justify adding all uncertainty numbers by this rule: the model must identify distinct contributions, dependencies and any omitted effects.

Complete stipulated budget for this additive linear model, with zero covariance between its two inputs
Input or result	Standard uncertainty	Role
Indication r	0.004 mm = 4 µm	Supplied input component
Correction δ	0.003 mm = 3 µm	Supplied input component
Corrected L = 12.002 mm	u_c(L) = 0.005 mm = 5 µm	Combined result under the stated model and covariance assumption

Document metrological traceability

Metrological traceability belongs to a measurement result. It relates that result to a stated reference through documented, unbroken calibrations, each contributing uncertainty. A practical SI realization can provide the reference at one end. For a model with several inputs, the supporting calibrations can form branches rather than a single simple ladder.

A useful record identifies the references, calibration results, relevant dates, conditions, uncertainties and how those inputs are used in the final model. Writing SI units or the name of a national laboratory on a result does not supply missing calibration evidence. NIST places responsibility for supporting a traceability claim on the provider of the result, and responsibility for assessing that claim on its user.

Shared definitions are supported by international comparison and national provision of standards. NMIJ's institutional account describes developing SI-based national standards, comparing them internationally, and supplying references through calibration services and reference materials in Japan. This supports comparability within evaluated measurement capabilities; it does not promise identical instrument readings. Traceability alone also does not guarantee freedom from mistakes or uncertainty small enough for a task.

Report uncertainty and check the purpose

A report must say what its uncertainty number means. An expanded uncertainty is U = k_cov u_c, where the positive dimensionless coverage factor k_cov multiplies the combined standard uncertainty. We use k_cov to distinguish it from the Boltzmann constant k. The associated coverage probability needs a justified basis; k_cov = 2 does not give exactly 95% coverage for every distribution or every evaluated result. The SI Brochure calls for both coverage factor and coverage probability when expanded uncertainty is reported.

For another fictional length record, supply y = 7.500 mm, u_c(y) = 0.012 mm and k_cov = 2. Arithmetic gives U = 0.024 mm and endpoints y − U = 7.476 mm, y + U = 7.524 mm. No probability model or coverage probability is supplied, so these calculations alone do not complete an expanded-uncertainty report or establish a guaranteed interval. Report the supplied standard uncertainty and retain the missing coverage information as missing.

Suppose that fictional result has a valid documented traceability chain, and the stated task requires u_c ≤ 0.010 mm. Since 0.012 mm exceeds 0.010 mm, it fails this uncertainty requirement despite traceability. This is only the supplied task's uncertainty check, not a full conformity decision about the spacer. Traceability, uncertainty size and suitability answer different questions. An interpretable result keeps the measurand, model, conditions, reference evidence and stated uncertainty together.

From an exact definition to a usable result

1. Definition
The SI fixes seven defining numerical values. The vacuum c is exactly 299 792 458 m/s; this defines a unit relationship rather than an uncertainty-free timing apparatus.
2. Realization and reference
A practical method establishes a value consistently with the definition, with associated uncertainty. A measurement reference carries that value and uncertainty into calibrations.
3. Calibrated model
For the supplied local model L = r + δ, 12.008 mm plus −0.006 mm gives 12.002 mm. The correction's standard uncertainty remains part of the result's budget.
4. Uncertainty and traceability
The stipulated uncorrelated 4 µm and 3 µm components give a combined standard uncertainty of 5 µm. Traceability also needs documented calibrations to the stated reference; units and arithmetic alone do not establish it.
5. Purpose
The separate fictional record has u_c = 0.012 mm, exceeding its task limit of 0.010 mm. Its supplied k_cov = 2 permits calculating U = 0.024 mm, while a justified coverage probability is still missing.

The steps have different roles. An exact SI defining value does not remove practical uncertainty; documented traceability does not by itself satisfy a task's uncertainty requirement. The numerical cases are stipulated teaching records.

Worked examples

Use exact c with uncertain timing

In the stipulated straight one-way vacuum path model L = cΔt with stationary endpoints in a fixed inertial frame, supply Δt = 10.000 ns and u(Δt) = 0.020 ns. Treat timing as the only uncertain input. Find the length estimate and its standard uncertainty, then round them as in the lesson.

Convert nano to 10⁻⁹: Δt = 10.000 × 10⁻⁹ s and u(Δt) = 0.020 × 10⁻⁹ s. This is a one-way travel time; there is no round-trip factor of one half.
Multiply the time estimate by the exact SI c: L = (299 792 458 m/s)(10.000 × 10⁻⁹ s) = 2.99792458 m.
The exact positive scale factor multiplies the standard uncertainty too: u(L) = (299 792 458 m/s)(0.020 × 10⁻⁹ s) = 0.00599584916 m.
Round the uncertainty here to 0.006 m and the central value to the same decimal place, 2.998 m. This display expresses the stipulated calculation; it is not evidence of actual instrument precision.

L = 2.998 m with standard uncertainty u(L) = 0.006 m after the specified rounding. The exact value of c does not make the uncertain travel-time input exact. Other uncertainty contributions were excluded only by the teaching model.

Correct a reading and retain uncertainty

For the stipulated spacer model L = r + δ, use r = 12.008 mm, δ = −0.006 mm, u(r) = 0.004 mm and u(δ) = 0.003 mm. The two components are uncorrelated and are the entire supplied budget. Find the corrected central value and combined standard uncertainty.

Add the signed correction: L = 12.008 mm − 0.006 mm = 12.002 mm, at the supplied conditions and range.
For this additive model with zero covariance, use u_c(L)² = u(r)² + u(δ)².
The squared sum is (0.000016 + 0.000009) mm² = 0.000025 mm². Its positive square root is 0.005 mm.
Equivalently, the input components are 4 µm and 3 µm, and √(4² + 3²) µm = 5 µm. The correction and its standard uncertainty have different signs and roles: δ is negative, while u(δ) is nonnegative.

The corrected length estimate is 12.002 mm with combined standard uncertainty 0.005 mm = 5 µm. This is the complete result of the stated linear teaching budget, not a zero-uncertainty result or a calibration valid at arbitrary conditions.

Check reporting and fitness separately

A fictional traceable length result has y = 7.500 mm and u_c(y) = 0.012 mm. Supply k_cov = 2 but no probability model or coverage probability. Calculate U and y ± U. Does this complete an expanded-uncertainty report? Does the result meet a task requiring u_c ≤ 0.010 mm?

Calculate U = k_cov u_c = 2(0.012 mm) = 0.024 mm. The coverage factor is dimensionless.
The arithmetic endpoints are 7.500 mm − 0.024 mm = 7.476 mm and 7.500 mm + 0.024 mm = 7.524 mm.
No justified coverage probability is provided. Retain y and its supplied standard uncertainty; do not label these endpoints a guaranteed interval or invent 95% coverage. The expanded-uncertainty reporting information remains incomplete.
Compare the requested measure with its limit: 0.012 mm > 0.010 mm. The uncertainty requirement is not met even under the stipulated valid traceability claim.

U = 0.024 mm and the arithmetic endpoints are 7.476 mm and 7.524 mm. Coverage information is missing, and the supplied standard uncertainty exceeds the task's limit. A traceable result can still be unsuitable for this stated measurement need.

Try it yourself

The SI assigns h its exact numerical value in J·s. Does that make a laboratory's practical realization of a mass value free of uncertainty?
Show the answer and reasoning
No. Exactness of the defining value does not make the practical realization exact.
The definition and its realization have different roles. A physical method requires measured inputs, an apparatus, conditions and an uncertainty evaluation. The kilogram definition uses h together with the metre and second.
In the same stipulated one-way vacuum model, keep Δt = 10.000 ns but halve its standard uncertainty to 0.010 ns. Before display rounding, what happens to L and u(L)?
Show the answer and reasoning
L remains 2.99792458 m; u(L) halves to 0.00299792458 m.
The central time estimate is unchanged, while multiplication by the same exact positive c scales the standard uncertainty. This conclusion uses the timing-only model; it does not establish a real method for halving all experimental uncertainty.
For r = 12.008 mm and δ = −0.006 mm, find r + δ. Can u(δ) = 0.003 mm be replaced by −0.006 mm in the uncertainty budget?
Show the answer and reasoning
r + δ = 12.002 mm. No: the signed correction is not its standard uncertainty.
The correction changes the central value. Its nonnegative standard uncertainty quantifies dispersion associated with incomplete knowledge of that correction. Squaring the correction itself would use the wrong quantity in the budget.
For L = r + δ, the supplied standard uncertainties are 4 µm and 3 µm. If the inputs may be correlated and their covariance is unknown, is 5 µm justified as the combined standard uncertainty?
Show the answer and reasoning
No. The zero-covariance condition needed for that calculation is missing.
√(4² + 3²) = 5 is correct arithmetic for the uncorrelated additive model. Correlation adds a covariance contribution; information about that dependence is needed before asserting the result.
For u_c = 0.012 mm and k_cov = 2, find U. Does k_cov alone prove exactly 95% coverage or a guaranteed maximum error?
Show the answer and reasoning
U = 0.024 mm. No: neither claim follows from k_cov alone.
An expanded uncertainty multiplies the standard uncertainty by a stated factor. Its coverage interpretation depends on the distribution and evaluation. The lesson supplies no basis for an exact coverage percentage or a hard bound.
Assume the fictional y record has valid documented SI traceability and u_c = 0.012 mm. A task requires u_c ≤ 0.010 mm. Does traceability satisfy this requirement, and would a unit label alone establish the traceability claim?
Show the answer and reasoning
No to both. The uncertainty exceeds the task limit, and a unit label does not document a calibration chain.
The numerical comparison is 0.012 > 0.010 in the same unit. Traceability concerns the documented connection of a result to a reference, with uncertainty at each calibration. Suitability also requires the uncertainty measure to meet the particular stated need.

What to carry forward

Specify the measurand, conditions and model before interpreting a measurement result.
The seven SI defining values are exact; practical realizations and calibrated references retain uncertainty.
A calibration correction changes an estimate and carries its own uncertainty and applicable scope.
The simple root-sum-of-squares calculation here requires a linear additive model and zero covariance.
Metrological traceability connects a result to a stated reference through documented calibrations that contribute uncertainty.
State the uncertainty measure and its basis; traceability alone does not guarantee a result meets a task's uncertainty requirement.

References

Sources for the concepts and teaching approaches in this article.