Short Answer
On May 20, 2019, the International System of Units (SI) underwent its most significant revision since the metre was redefined in 1983. The 2019 SI redefinition fixed the numerical values of four additional defining constants: the Planck constant h, the elementary charge e, the Boltzmann constant k, and the Avogadro constant NA. These constants are now exact by definition, with zero uncertainty. For anyone who works with significant figures, rounding, and measurement uncertainty, this change alters how constants enter calculations and how precision is reported.
Rule Statement
The core rule is simple: exact numbers have unlimited significant figures and contribute no uncertainty. Since 2019, the seven SI defining constants are exact. They do not limit the precision of a calculation. The final result is limited by the least precise measured input, not by the exact constants.
- Exact constants: Use them at full precision. Do not round them to match the significant figures of measured data.
- Measured quantities: Still carry uncertainty and significant figures. They determine the final rounding.
- Unit conversions: Exact conversions, such as 1 in = 2.54 cm exactly or 1 L = 10−3 m3 exactly, also have unlimited significant figures.
- Distinguish exact from measured: The gravitational constant G, the fine-structure constant α, and the electron mass me are measured, not exact. They have uncertainties and finite significant figures.
Before 2019, h, e, k, and NA were measured constants with uncertainties. After 2019, their numerical values are fixed by the SI Brochure and are exact. Older textbook values are approximations, not definitions.
Worked Examples
Example 1: Photon Energy with Exact h and c
Calculate the energy of a photon with wavelength λ = 532 nm, given to three significant figures. Use E = hc/λ. Both h and c are exact.
- Convert wavelength: λ = 532 nm = 5.32 × 10−7 m (three significant figures).
- Use exact constants: h = 6.62607015 × 10−34 J s; c = 299792458 m/s.
- Compute: E = (6.62607015 × 10−34 × 299792458) / (5.32 × 10−7) = 3.734… × 10−19 J.
- Round the final answer to three significant figures because λ has three: 3.73 × 10−19 J.
Do not round h or c to three significant figures. Their exactness means they do not reduce the precision of the result.
Example 2: Avogadro Constant and Mole Calculations
How many atoms are in 0.500 mol of a pure substance? NA is exact: 6.02214076 × 1023 mol−1. The amount 0.500 mol has three significant figures.
- Compute: N = nNA = 0.500 × 6.02214076 × 1023 = 3.01107038 × 1023.
- Round to three significant figures: 3.01 × 1023 atoms.
The exact constant supplies as many digits as needed; the measured amount limits the result.
Example 3: Boltzmann Constant and Thermal Energy
Find kT for T = 300.0 K (four significant figures). k = 1.380649 × 10−23 J/K exactly.
- Compute: kT = 1.380649 × 10−23 × 300.0 = 4.141947 × 10−21 J.
- Round to four significant figures: 4.142 × 10−21 J.
Counter-Examples
The following mistakes are common when exact constants are misunderstood.
- Rounding exact constants to match measured data: Using h = 6.63 × 10−34 J s because λ has three significant figures is wrong. It injects avoidable rounding error. Use the exact value or a calculator that carries full precision.
- Adding uncertainty to exact constants: Exact constants have zero uncertainty. Do not propagate uncertainty for h, e, k, or NA when they are used as SI defining constants.
- Assuming all constants are exact: G, α, and particle masses are measured. They still have uncertainties and finite significant figures.
- Using pre-2019 CODATA values as exact: Values such as h = 6.62606957 × 10−34 J s were measured approximations. They are not the current SI definition.
- Double rounding: Rounding an exact constant in an intermediate step and then rounding again at the end can shift the final digit. Carry exact constants at full precision until the final rounding.
Convention Comparison Table
| Constant | Symbol | Exact SI Value | Status Before 2019 | Unit Defined or Affected |
|---|---|---|---|---|
| Hyperfine transition frequency of Cs-133 | ΔνCs | 9192631770 Hz | Exact since 1967 | second |
| Speed of light in vacuum | c | 299792458 m/s | Exact since 1983 | metre |
| Planck constant | h | 6.62607015 × 10−34 J s | Measured | kilogram |
| Elementary charge | e | 1.602176634 × 10−19 C | Measured | ampere |
| Boltzmann constant | k | 1.380649 × 10−23 J/K | Measured | kelvin |
| Avogadro constant | NA | 6.02214076 × 1023 mol−1 | Measured | mole |
| Luminous efficacy of monochromatic 540 THz radiation | Kcd | 683 lm/W | Exact since 1979 | candela |
Standards Citation
Authoritative sources for exact constants and significant-figure practice include:
- NIST SP 330 (2019), The International System of Units (SI), 9th ed. See §2.3.1 and Table 1 for the seven defining constants and their exact values.
- ISO 80000-1:2022, Quantities and units — Part 1: General. See §6.4 for rounding and significant-figure conventions.
- ASTM E29-13a, Standard Practice for Using Significant Digits in Test Data to Determine Conformance with Specifications. See §§6–8 for rounding and conformance rules.
- JCGM 100:2008 (GUM), Evaluation of measurement data — Guide to the expression of uncertainty in measurement. See §2.3 for exact numbers and §4 for uncertainty propagation.
- BIPM SI Brochure, 9th edition (2019). The definitive international statement of the redefinition.
Common Mistakes
Even experienced practitioners sometimes mishandle exact constants. Watch for these pitfalls:
- Confusing exactness with infinite decimal digits: An exact constant is exact by definition, but its decimal representation may be finite or repeating. The key is zero uncertainty, not endless digits.
- Letting exact constants inflate final precision: A result does not gain significant figures just because h or NA is exact. The measured input still limits the answer.
- Forgetting that derived units inherit exactness: The kilogram, ampere, kelvin, and mole are now defined through exact constants, but measurements made with them still have uncertainty.
- Mixing old and new constants: Using a pre-2019 h value with a post-2019 e value can create inconsistency. Use the current SI values.
- Over-rounding intermediate results: Keep extra digits for exact constants and measured values until the final step. Then round once according to the least precise input.
Quick Reference Table
| Question | Answer |
|---|---|
| Are h, e, k, and NA exact? | Yes, since May 20, 2019, by SI definition. |
| How many significant figures do exact constants have? | Unlimited; they contribute zero uncertainty. |
| Do exact constants change the basic sig fig rules? | No. The rules are unchanged; the set of exact constants expanded. |
| What limits the final precision? | The least precise measured input, not the exact constants. |
| Should I round exact constants in calculations? | No. Carry them at full precision until final rounding. |
| Are all physical constants exact? | No. Only the seven SI defining constants are exact by definition. |
Related Rules
- Exact Numbers and Significant Figures
- Rounding Rules
- Significant Figures in Scientific Notation
- Error Propagation
- Why Sig Figs Are an Approximation
Sources & Further Reading
- BIPM, The International System of Units (SI), 9th ed., 2019.
- NIST Special Publication 330, The International System of Units (SI), 2019.
- ISO 80000-1:2022, Quantities and units — Part 1: General.
- ASTM E29-13a, Standard Practice for Using Significant Digits in Test Data to Determine Conformance with Specifications.
- JCGM 100:2008, Evaluation of measurement data — Guide to the expression of uncertainty in measurement (GUM).
FAQ
Did the 2019 SI redefinition change significant-figure rules?
No. The fundamental rules for significant figures and rounding are unchanged. What changed is that four additional constants—h, e, k, and NA—became exact and now behave like exact numbers in calculations.
Are exact constants the same as exact numbers like 2 or π?
They are similar in that they have no uncertainty and unlimited significant figures. However, π is a mathematical constant, while SI defining constants are fixed by international agreement. Both are exact for significant-figure purposes.
How should I report a result that uses exact constants?
Round the final result to the number of significant figures justified by the measured inputs. Do not add digits because an exact constant was used. For uncertainty budgets, assign zero uncertainty to exact constants.
What about Avogadro’s number—is 6.022 × 1023 still acceptable?
As an approximation, yes, if the context allows. But the exact SI value is 6.02214076 × 1023 mol−1. When exactness matters, use the full defined value.
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