core27 min studyLesson 102 in Physics

SI units and exact defining constants

Connect units to the constants defining the SI while distinguishing exact definitions from uncertain experimental realizations.

Before you begin

On this page

The question for today

What you’ll understand

Explain why fixed defining constants do not make measurements exact.
Convert numerical values while preserving the physical quantity.
Distinguish a quantity equation from an equation for numerical values.

A common unit system lets laboratories compare quantities without shipping the same ruler between continents. The SI defines its units through fixed numerical values of seven specified constants. This replaces dependence on a single material artifact with definitions that can be realized through suitable experiments. The definitions are exact by convention; every practical realization still has physical imperfections and an uncertainty that must be evaluated.

A quantity is more than its numerical value

Writing a quantity as a number multiplied by a unit separates the physical property from its representation. Change the unit and the number changes reciprocally, while the quantity remains the same. This is why unit conversion is multiplication by a ratio equal to one. Confusing numerical values with quantities produces avoidable mistakes, particularly when a computer silently assumes metres while a data file contains millimetres. Store and document the unit alongside each physically meaningful field.

A quantity equation such as distance equals speed multiplied by duration holds for any consistent choice of units. An equation written only for numerical values may contain conversion factors tied to particular units. If a spreadsheet calculates distance in kilometres from speed in metres per second and time in seconds, its numerical formula needs a factor of one thousand in the denominator. That factor belongs to representation, not to the underlying physical law.

Q=\{Q\}_{u}\,u,\qquad 2.5\,\mathrm m=250\,\mathrm{cm}

The symbol {Q}ᵤ denotes the numerical value of quantity Q when expressed in unit u. Changing the unit changes this numerical value without changing Q.↔ Scroll wide equations horizontally.

Exact constants define a system of units

The defining constants include the caesium-133 hyperfine transition frequency, the speed of light in vacuum, Planck's constant, elementary charge, Boltzmann's constant, Avogadro's constant, and a specified luminous efficacy. Their fixed numerical values jointly define the SI. For example, the second is connected to the specified caesium transition, and the metre follows through the exact speed of light together with the second. These definitions do not assert that every physical constant has an exact numerical value.

A practical realization builds an experiment that connects to a definition. A laser distance measurement must determine time or frequency and account for propagation conditions and instrument geometry. A frequency standard must control perturbations of the reference transition. Fixing a constant removes uncertainty from its defining numerical value, but not from every input in those experiments. Distinguish the definition of the unit, the method used to realize it, and the uncertainty assigned to a particular realization.

c=299\,792\,458\,\mathrm{m\,s^{-1}}\quad\text{exactly},\qquad d=\frac{ct}{2}

The displayed c is the defining speed of light in vacuum. The round-trip distance formula assumes a stationary reflector, a vacuum optical path, and corrected timing; an air path requires a suitable propagation model.↔ Scroll wide equations horizontally.

Coherent units keep the algebra visible

Derived units express combinations of base units. A newton is a kilogram metre per second squared, and a joule is a newton metre. Converting inputs into coherent SI units before calculating often prevents accidental conversion factors. Prefixes are scale factors, and their powers matter: one square centimetre is one ten-thousandth of a square metre, rather than one hundredth. Write the conversion factor before squaring or cubing so that geometry and numerical scaling remain connected.

Exact conversion factors do not add uncertainty, but uncertain input quantities retain theirs. A length of one hundred centimetres has the same relative standard uncertainty when expressed in metres. Likewise, using the exact defining value of a constant does not entitle a result to unlimited significant figures. Keep adequate guard digits during arithmetic and report the final value consistently with its evaluated uncertainty. Unit notation communicates the scale; uncertainty communicates what the experiment supports.

Work it through

A vacuum round-trip distance

A corrected round-trip light travel time is $t=(20.0\pm0.2)\,\mathrm{ns}$ , with the quoted uncertainty a standard uncertainty. Assume a stationary reflector in vacuum and neglect other uncertainty contributions. Calculate the one-way distance and its standard uncertainty.

Convert time to seconds using the exact prefix factor: $t=20.0\times10^{-9}\,\mathrm s$ and $u(t)=0.2\times10^{-9}\,\mathrm s$ . These conversions do not introduce additional uncertainty.
The path is traveled twice, so $d=ct/2=299792458(20.0\times10^{-9})/2=2.99792458\,\mathrm m$ . The defining value of c contributes no uncertainty in this model.
Because the model is linear in time, $u(d)=c\,u(t)/2=0.0299792458\,\mathrm m$ . Round to $0.030\,\mathrm m$ and round the estimated distance to the matching decimal place.

The result

Report $d=(2.998\pm0.030)\,\mathrm m$ with standard uncertainty under the stated assumptions. An actual air-path measurement would require more than substituting the vacuum value of c.

The living notebook

Explore the model

Interactive lab

Interactive lab · vectors

Compare magnitude and representation

This lab changes one dimensionless vector's magnitude and angle and shows its components. Set angle to 30 degrees and magnitude to 0.75. Predict that increasing magnitude to 1.50 doubles both components without changing their ratio, then check. Imagine reporting each component in another unit while preserving the same vector. The lab itself performs no unit conversion and does not realize an SI standard.

Vectorx / y projectionsTrace marker

x component: 1.393
y component: 0.975
Magnitude: 1.7

Angle35°Magnitude1.7

t = 0 s

Playback traces a fixed vector and its projections. Angle and magnitude set the geometry; the moving marker does not change them. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forA unit conversion changes numerical representation; a magnitude change changes the represented quantity.

Pause & reason

Check your understanding

Interactive checks need JavaScript. You can still read the explanation.

Read the checkpoint explanation

The defining value of c is exact, but the experiment still supplies uncertain timing and other input quantities. Unit definitions establish a common reference; they do not remove imperfections from a practical realization.

Your turn

Try it yourself

A detector area is $(4.00\pm0.08)\,\mathrm{cm^2}$ , where the uncertainty is standard. Express the result in square metres and state whether its relative uncertainty changes. Explain a factor-of-one-hundred error that could arise.

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

Convert the length unit and square that entire conversion factor.

Compare with the solution

Since $1\,\mathrm{cm}=10^{-2}\,\mathrm m$ , one square centimetre equals $10^{-4}\,\mathrm{m^2}$ . The result is $(4.00\pm0.08)\times10^{-4}\,\mathrm{m^2}$ . Its relative standard uncertainty remains $0.08/4.00=0.02$ , or two percent, because both value and uncertainty receive the same exact factor. Multiplying area by $10^{-2}$ instead of $10^{-4}$ would treat the prefix as though it occurred only once and produce an area one hundred times too large.

Keep the idea

Take it with you

A quantity remains unchanged when its numerical value and unit are converted together.
The SI fixes selected constant values; practical realizations still require uncertainty evaluation.
Apply prefix conversion factors to the full power carried by the unit.

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