Units, standards, and physical constants

Keep the quantity, number and unit together

A length is a physical quantity. A metre is an agreed unit for comparing lengths. In the supplied statement L = 2.40 m, L names the particular length, 2.40 is its numerical value in metres, and m is the unit symbol. A unit supplies a reference for quantities of the same kind.

Changing the unit changes the numerical value, not the length being described. The same supplied length can be written as 2.40 m or 2400 mm. A bare number such as 2.40 leaves that reference unstated. The unit also does not tell us which object, endpoints or conditions belong to the measurement; those still need to be specified.

Read base units and combinations of units

The International System of Units, abbreviated SI, provides a shared system. Its seven base units are listed below. Other units can be built from them: area uses m², and speed uses m/s. The slash means division; m² means m × m.

For the cart cases here, average speed means total distance divided by elapsed time. Dividing a distance expressed in metres by a time expressed in seconds gives a result in metres per second. This is a relationship between different quantities; it is not a conversion of metres into seconds.

Use prefixes without changing the quantity

A prefix changes a unit's size by a stated factor. Kilo, symbol k, means 1000; centi, symbol c, means one hundredth; milli, symbol m, means one thousandth. Thus 1 km = 1000 m, 1 cm = 0.01 m, and 1 mm = 0.001 m. In mm, the first m is the prefix and the second m is the metre symbol. These are exact unit relationships.

To express metres as millimetres, multiply the numerical value by 1000 because each metre contains 1000 millimetres. To express millimetres as metres, divide by 1000. You can also multiply the whole quantity by a ratio equal to one: 2.40 m × (1000 mm / 1 m) = 2400 mm. The metre units cancel while the length is preserved.

For area, both length factors change. From 1 m = 100 cm, derive 1 m² = (100 cm) × (100 cm) = 10 000 cm². Multiplying an area number by only 100 would miss one factor. An exact conversion changes how a result and its uncertainty are expressed; it does not remove measurement uncertainty.

Distinguish a defining value from a measured value

The SI is defined by fixing the numerical values of seven selected constants in specified units. For example, the speed of light in vacuum, c, is exactly 299 792 458 m/s in the SI. This fixes the metre in relation to the second. The exact value belongs to the definition; it is not a claim that every experiment measures all those digits without uncertainty.

Other physical constants are not assigned exact SI defining values. The Newtonian constant of gravitation, G, has a recommended value with a stated standard uncertainty in the CODATA table. A physical constant can therefore have an experimentally determined value. The word constant alone does not mean its numerical value is exact by definition.

A supplied value in a teaching case has a third role. We can stipulate a cart's distance as exactly 2.40 m within an invented calculation. That makes the arithmetic definite; it does not turn 2.40 m into a natural constant or establish the distance of a real cart journey. Always identify why a number is being treated as exact.

Put a definition into practice

To realize a unit is to implement its definition physically, using a suitable method and apparatus. Different practical realizations can use the same definition. An exact defining value does not make a laboratory setup free of uncertainty.

A measurement standard is a reference realization with an assigned quantity value and associated uncertainty. It may be a measuring system, a material measure or a reference material. A reference mass used to calibrate a balance is an example. Here, standard means a measurement reference, distinct from a written technical specification.

Connect readings with references

Calibration relates a standard's values and uncertainties to an instrument's indications under specified conditions, then provides a way to obtain measurement results from indications. It can supply a correction or a table of corrections. Calibration and adjustment have different roles: adjusting the instrument changes it, while a calibration establishes the relationship used to interpret its readings.

Metrological traceability is a property of a measurement result. A documented, unbroken chain of calibrations connects that result with a stated reference, and each calibration contributes uncertainty. Traceability does not guarantee that the uncertainty is small enough for your task or that no mistake occurred.

AIST's account from Japan gives a practical mass example: users' weights and scales are connected through calibrations to national standards maintained by NMIJ, with uncertainty evaluated at each calibration. This illustrates the use of shared references; it does not imply that every instrument gives identical readings.

Report what the number means

Write a space between a numerical value and its unit symbol, as in 2.40 m. Unit symbols retain their form in the plural: write 3 kg, not 3 kgs.

For a real measurement, identify the quantity and conditions, the unit, the value and relevant uncertainty information. If a correction is used, retain the conditions and range for which it was established. A conversion, a calibration label and extra displayed digits answer different questions; none by itself supplies the missing information.

For a teaching calculation, label the supplied inputs and the idealized model. Keep that arithmetic separate from a unit's exact definition and from evidence about a measuring system. This habit makes a numerical result easier to interpret and check.

Worked examples

Try it yourself

1. A supplied length is written as 0.004 m and as 4 mm. Are these different lengths? Explain the conversion.

   Show the answer and reasoning

   No. They express the same length in different units.

   One metre contains 1000 millimetres, so 0.004 m × (1000 mm / 1 m) = 4 mm. The larger numerical value accompanies the smaller unit; it does not indicate a longer object.

2. A supplied area is 6 cm². Express it in m². Why is multiplying by 0.01 insufficient?

   Show the answer and reasoning

   6 cm² = 0.0006 m². The conversion factor must apply to both length factors.

   Since 1 cm = 0.01 m, 1 cm² = (0.01 m) × (0.01 m) = 0.0001 m². Multiplying by 6 gives 0.0006 m². A length factor of 0.01 applied only once would not convert a squared unit correctly.

3. Classify these three uses of numbers: c = 299 792 458 m/s exactly in SI; a recommended value of G with standard uncertainty; and a lesson that stipulates an exact 20 m track. Does exactness have the same basis in all three?

   Show the answer and reasoning

   SI defining value; experimentally determined physical constant; supplied teaching input. Their bases differ.

   The SI fixes c's numerical value in m/s as part of its definition. The recommended G value carries measurement uncertainty. The exact 20 m track is an assumption of that invented case, not a unit definition or a measured finding about a real track.

4. A task requires a reported uncertainty no greater than 0.002 m. A traceable result reports 0.020 m uncertainty using the same reporting convention. Does traceability alone make this result suitable for the task?

   Show the answer and reasoning

   No. Its reported uncertainty exceeds the stated requirement by a factor of ten.

   Compare the supplied values in the same unit: 0.020 m / 0.002 m = 10. Traceability describes the documented connection to a reference, while suitability also depends on the uncertainty requirement. This exercise uses supplied uncertainties; it does not calculate an uncertainty budget.

What to carry forward

• A quantity value needs its reference unit as well as its number.
• Changing units changes the numerical expression, not the quantity described.
• Exact defining values, measured constants and supplied teaching inputs have different roles.
• Practical standards and calibrated instruments retain measurement uncertainty.
• A traceable result still needs conditions and uncertainty suitable for its intended use.