Redressing grievances with the treatment of dimensionless quantities in SI ^☆

1. Introduction

A quantity in the International System of Units (SI) can be stated as a mathematical expression–the product of a numerical value and a unit of measurement. The magnitude of a quantity can be expressed in terms of the seven SI base quantities length (m), mass (kg), time (s), electric current (A), thermodynamic temperature (K), amount of substance (mol), and luminous intensity (cd), either individually or in combinations. The base quantities correspond to physical dimensions as used in dimensional analysis.

However, many kinds of quantities have no extent in any of the seven standard dimensions. For example, a counted quantity is a number of some distinguishable kind of thing, such as 23 neutrons. Subdividing a counted quantity consists of dividing a set of things into smaller sets which contain fewer of the kind of thing being counted, rather than less of something that would be measured on a continuous scale. What unit should such a quantity refer to, and what is the dimension corresponding to that unit? These questions are addressed in the International Vocabulary of Metrology (VIM) [1] and the SI brochure [2].

Under the current VIM, all counted quantities and all ratios of two quantities of the same kind are called quantities of dimension one, or alternately dimensionless quantities. The definition is “quantity for which all the exponents of the factors corresponding to the base quantities in its quantity dimension are zero.” The unit of measurement for such quantities is 1, the algebraic result of setting all of the exponents to zero.¹

The VIM does not define a term for the complement of this set of quantities, i.e., those quantities that have some extent in at least one of the dimensions defined by SI. Herein they will be called dimensional quantities.

Quantity calculus as currently accepted protects us from the error of computing nonsensical combinations of quantities that have different dimensions. This is an incomplete but nevertheless valuable consistency check for calculations. On the other hand, there is nothing formal in the system to prevent one from mixing up different kinds of things when combining quantities that are regarded by SI as dimensionless. Aside from the indication that the quantity is dimensionless that the reference to 1 as a unit provides, all of the information that characterizes the kind of the quantity is relegated to informal, informative text.

Not everyone agrees that consistency checking is an important function of quantity calculus, and not everyone is unsatisfied with the current treatment of unit 1. This article will not end that philosophical disagreement. It will, however, provide a mapping from an extended conceptual model back to the standard one, which enables a reduction to accepted SI units.

The proposed resolution is to extend quantity calculus with methods to provide integrity checking for dimensionless quantities that is comparable to what is already provided for dimensional quantities. The remainder of the article proceeds as follows. Section 2 reviews the history of the issues and related work. Section 3 provides the author’s current view of the issues. Section 4 proposes a resolution. Section 5 compares the new proposal with previous proposals. Section 6 describes how the new approach coexists with the current canon. Section 7 explains the limitations of the approach. Finally, Section 8 concludes with future possibilities.

2. History of the issues and related work

Originally, mathematical operations applied only to numbers, and units of measurement were just information that described what the numbers represented. This belief gave way to the practice of including units of measurement within the scope of the mathematical operations, thereby formalizing the method of working with combinations of units [4]. The resulting quantity calculus protects us from the error of computing nonsensical combinations of dimensional quantities that have different dimensions [5], but it does not protect us from equally bad errors that may be committed with quantities of the same dimension.

Dimensional heterogeneity is one indicator that the operation being attempted is confusing different kinds of quantities. Distinguishing different kinds of quantities was important in Maxwell’s discourse, albeit without a formal definition [6,7]. The current VIM defines ‘kind of quantity’ as simply the “aspect common to mutually comparable quantities” [1].² Thus, the definition provides no additional criteria that can be used to verify comparability to a greater degree than simply dimensional homogeneity.

J. de Boer wrote, “All the so-called ‘dimensionless quantities’ belong to one class of quantities of the same kind; they belong to the equivalence class of ‘dimensionless’ quantities…” [5, Section A3.4]. If the word kind in this quote is interpreted using the VIM definition, the results are problematic.

The premise that combinations of units can and should be simplified algebraically has been disputed [8,9]. The consequences for angles and other quantities that have been measured by the ratio of two quantities of the same kind have been particularly troublesome. The International Committee for Weights and Measures (CIPM) decided in 1980 that the radian and steradian, previously considered to be special “supplementary” units that were on the fence between base and derived units, are in fact dimensionless derived units that reduce to the number 1 [10]. This treatment of angles leaves us with a definition of torque that reduces to energy. Some rely on notational conventions, such as writing N m or J/rad for one quantity and J for the other, to avoid confusion.³ The trade-offs of different ways of addressing the issue with angles have been discussed [11].

The current SI brochure unambiguously specifies that the coherent derived unit for any quantity that is defined as the ratio of two quantities of the same kind is always the number one, 1. It addresses counted quantities similarly but with some ambiguity, saying first that they are “usually regarded as dimensionless quantities, or quantities of dimension one, with the unit one, 1” (emphasis added) [2, Section 1.3], and later that they “are taken to have the SI unit one, although the unit of counting quantities cannot be described as a derived unit expressed in terms of the base units of the SI. For such quantities, the unit one may instead be regarded as a further base unit” [2, Section 2.2.3]. The extra words are explained by Mills: “Those dimensionless quantities defined as a ratio of two other quantities of the same kind (e.g., angles, mole fractions, etc.) suggest we should think of 1 as a derived unit, whereas counting suggests the concept of 1 as a base unit” [12].

The wiggle room provided by “usually” in the SI brochure is absent from the current edition of National Institute of Standards and Technology (NIST) Special Publication 811, Guide for the use of the International System of Units, where the use of counting units other than 1 is deemed an unacceptable mingling of the formal units with other, less formal information concerning the quantity. Specific examples indicate that it is unacceptable to use molecules, neutrons, or atoms as counting units in expressions such as n/s or atoms/cm³ because they are neither SI units nor units that are accepted for use with the SI by CIPM or NIST policy [13, Section 7.5].

Use of the unit 1 for all kinds of counts and all possible ratios of two quantities of the same kind has led to confusion and inconsistencies. Most prominently, the de facto standard practice of identifying the types of entities counted (e.g., neutrons or bits⁴) as if they were units is inconsistent with official guidance. In addition, Mohr and Phillips noted that the absence of an explicit specification of cycles in the definition of hertz gives rise to errors of a factor of 2π and misuse of the unit hertz for rates of non-periodic events [14–16].

Software libraries and packages that implement “quantities and units” functions need to work with angles measured in radians, amounts of data measured in bits or bytes, and other quantities of SI dimension 1 in some way that is not astonishing to users [17]. As a result, they apply workarounds such as adding an explicit base unit for 1, treating radians as a special case, and allowing users to introduce arbitrary irreducible units. Different software has applied different workarounds, creating subtle problems for transfer of scientific data.

3. Discussion

Scientists are sometimes advised to display units in unsimplified form just to avoid confusion. By convention, different expressions that are algebraically equivalent are interpreted as indicating different kinds of quantities. This is true of the example given in Section 2, where units expressed as N m or J/rad instead of just J are a hint that one is dealing with torque rather than energy. It is also true of mass fractions, when the units are expressed as kg/kg, and many similar quantities.⁵

The author sees these conventions, along with those surrounding hertz and becquerel, as workarounds that avoid addressing quantity kinds formally. The impact is more serious for dimensionless quantities.

Emerson wrote, “Any system of units, including the SI, that fails to recognize countable quantities as base quantities, and their entities as base units, can be considered to be incomplete” [22]. While agreeing that the system is incomplete, one can regard dimensionless units as a distinct complement to dimensional units, a different category of units.

The unit 1 behaves like a singularity of the function that we expect dimensional analysis to perform; it is a point of failure. The singularity is where units vanish and all different kinds of quantities become formally indistinguishable. One arrives at the singularity when all of the exponents in dimensional analysis become zero or when the type of things being counted in a counted quantity is discarded. The result is valid at some level of abstraction, but only informal, descriptive text remains to distinguish different kinds of dimensionless quantities from one another. The onus is on the scientist to do all of the type-checking, rather than relying on a rigorous method to avoid mistakes. As alternatively described in [9], “such abstract entity additions…are of no more practical use than additions of purely mathematical numbers, since on this topmost abstract level no kinds of things are differentiated from other kinds of things.”

While the argument can be made that preventing mistakes involving quantities of dimension 1 is not the purpose of quantity calculus, the same argument can be made regarding units for dimensional quantities. In principle, the scientist could obtain correct results by reasoning with numbers alone. Nevertheless, the utility of specifying and managing units methodically in calculations has been accepted.

In order for quantity calculus to provide the most value in terms of ensuring the integrity of calculations, we must cease to discard relevant information about the kinds of quantities being calculated. This guiding principle applies to ratios in the same way that it applies to counting units. The methods of quantity calculus should be extended to distinguish atoms, molecules, particles, bits, pixels, and other countable things from one another in a rational and rigorous way.

4. Proposed resolution

The resolution proposed here is a formalization, extension, and modification of the approach that was initiated by Mohr and Phillips, and is related to the conceptual analysis of quantities of dimension 1 that was done by Josef Kogan [23, Section 1.3]. Mohr and Phillips’ initial set of counting units is formalized through the application of type theory. The resulting type system is then extended to include additional counting units, ratios of two quantities of the same kind, and unitized scaling factors.

Within several scientific disciplines, formal definitions of type theory have been axiomatized and extensively developed. This is a body of work that can be relied upon when first principles are required. However, for this discussion, it suffices to have a simple, intuitive understanding of the application.

The idea behind this model is to extend quantity calculus to support subtyping of the special unit 1. This enables traceability to SI to occur not only through direct reference to SI units (the identity relationship), but also through subtyping (the generalization/specialization relationship). For example, the radian becomes a specialization of the SI unit 1. Torque expressed in J/rad cannot then accidentally reduce to energy as it can when rad is just a “special name for the number 1” [2, Table 3, footnote b]. However, the reduction that is possible in SI can be obtained by generalizing rad up to the unit 1, which discards the “per what” information contained in the J/rad expression, leaving only J/1 = J.

In the proposed model, the practice of using the types of entities counted as if they were units is reconciled with SI by making all “counting units” specializations of the SI unit 1. Consequently, the hertz that is interpreted as clockticks per second and the becquerel that is interpreted as nuclear decays per second will be distinguishable, even though both of them reduce to the SI definition of s⁻¹. Moreover, the latter interpretation of hertz will be distinguishable from the one that is 2π rad/s.

Type systems formalize the concept of generalization in such a way that if a thing is an instance of a given type, then it is also an instance of every supertype of that type, up to and including the ‘top type.’ It is valid to use the subtype in a context where an instance of one of its supertypes is called for, but in doing so, one discards the details that distinguished it as an instance of the subtype.

Applying this concept to units of measurement, the context of use that determines the generalization/specialization relationships is the algebraic addition or subtraction of quantities. Unit A is a subtype of unit B only if any sum or difference with a quantity expressed in unit B would remain valid if a quantity expressed in unit A were substituted. For example, a calculation that involves a count of atoms, with no further constraint on the atoms, remains valid regardless whether those atoms happen to be hydrogen. Contrariwise, if a calculation specifically requires a count of hydrogen atoms, it would not do to provide a count of some other kind of atoms. The counting unit for hydrogen atoms, therefore, is a subtype of the more general counting unit for atoms.

Addition or subtraction of quantities is logically defined whenever their units share a common supertype (or one is a supertype of the other), but the result is a quantity that is expressed in that more general unit. That result may or may not be acceptable, meaningful, or useful in a particular context, depending on what was specified or assumed about the quantity expected (e.g., whether it was supposed to be a count of atoms generically or hydrogen atoms specifically). Referring to a type system thus enables a more exact determination of compatibility than what the yes-or-no test of dimensional homogeneity can provide.

The top levels of a type system that interprets dimensionless units, as well as numerical factors that have been “unitized” by convention, are shown in Figs. 1 and 2. All identified types are merely prominent examples chosen from a set that can be extended without limit. The system must be extensible, i.e., it must allow users to add new types, since it is plainly impossible to catalog every possible counting unit. The following subsections provide further detail.

5. Comparison with previous proposals

The controversy over quantities of dimension one has revealed that different people subscribe to different and conflicting mental models of how quantity calculus should work. As Guggenheim observed, if a reader disputes the philosophical commitments of a proposal, then it is profitless to go into technical detail [4]. The following comparison of different proposals consequently focuses on identifying the philosophical commitments that may be decisive.

The type system proposed here makes a distinction between quantities that are measured in some ‘dimensionless unit’ and pure numbers that have no units at all. Stated another way, it makes a distinction between the unit 1 and the number 1. This allows a formal method to prevent the mishaps that have occurred when the former have been reduced to the latter, without introducing a new dimension each time. It also formalizes the type hierarchy or lattice to allow the units for sums and differences of different subtypes to be methodically determined.

Brown and Brewer [21] proposed retaining the explicit 1 in unit expressions as an indicator of counted quantities. Their proposed convention suggests that some difference exists between the unit of counting, which should be retained for clarity, and the plain number, which is safe to eliminate. That much is philosophically consistent with the type system introduced here. However, they rejected the proliferation of different counting units that the type system allows, asserting instead that all counts are quantities of the same kind, and did not propose any new convention for dimensionless quantities other than counts.

Like the type system, Johansson’s proposal of parametric units [9] provides a formal way to distinguish multiple specializations of a unit. The “qualitative parameter” can be thought of as indicating a distinct subtype. However, a parametric unit without some value specified for the parameter is not itself a fully-defined unit, so generalization is impossible. Johansson discusses the hierarchy or lattice of counting units, but only to conclude that the base unit 1 is not useful. With respect to ratios of two quantities of the same kind, Johansson’s solution in general is to reject their algebraic simplification quite broadly (e.g., “the dimension of rainfall is volume per area, not length”), while in the specific case of the radian he appears to support adding a dimension for angle. The type system allows significant ratios to be given names and treated as distinct specializations of unit 1, but there is no philosophical objection to simplifying away those that carry no significant information.

A logical separation of counted quantities from others can arise through an extended version of Stevens’ scale theory that includes the absolute scale (defined as the most restrictive scale where the only admissible transformation is identity) [24,25]. Scale theory is well-known and thus useful for communication, but there are significant disagreements over the scope of its applicability. Ref. [26] argues, “scale types are not fundamental attributes of the data, but rather, derive from both how the data were measured and what we conclude from the data.” It would follow that a unit of measurement does not uniquely determine the scale of a measurement result, and it would be misleading to refer to scale theory within a type system for units. Nevertheless, scale theory may characterize some of the grievances with SI’s treatment of quantities of dimension 1 in a manner that is more intuitive to some readers. Approaching the issues from the perspective of scale theory rather than units, and furthermore introducing cyclical scales as found necessary by Chrisman and others [27], one would arrive at something similar to the conceptual analysis of quantities of dimension 1 that was done by Josef Kogan [23, Section 1.3].

6. Reduction to SI units

Reduction to SI units will be necessary, at least as a finishing step for publications, as long as the current set of rules for SI remains in force. This section describes the reduction process.

First, any ‘non-coherent’ units that are defined as multiples of others must be replaced with the corresponding coherent units. For counting units, this means reducing to the unit of the single, indivisible counted entity. If using a definition of cycle that is 2π rad, multiply the value by 2π and change the unit from cycle to rad. If using a definition of byte that is 8 bits, multiply the value by 8 and change the unit from byte to bit. Without this step, one risks introducing the multiplier as an erroneous factor when cycles, radians, bits, and bytes all reduce to the unit 1.

Then, every quantity of dimension 1 that is not an accepted “special name for the unit 1” or dimensionless derived unit must be replaced with the SI unit 1, or, optionally, with an unsimplified units fraction for ratios of two quantities of the same kind.

Additional descriptive text should now be added to replace the information that was discarded from the unit expressions. ‘Before’ and ‘after’ examples of this textual transformation are already supplied in NIST SP 811 [13, Section 7.5]. It is likely that the 9th edition of the SI brochure and the corresponding update of SP 811 will provide more examples and discussion for both counted quantities and ratios.

Finally, the expression can be simplified as needed and the quantity description, unit expression, exponents, and prefixes can be readjusted for aesthetics.

7. Limitations

Introducing a type system of units for dimensionless quantities as described in Section 4 enables stronger integrity checking to prevent some notorious mistakes, but it is not a complete solution; rather, it is a compromise. For rigorous integrity checking, one must extend the implementation of quantity calculus to explicitly track kinds of quantities in addition to units. For example, in addition to performing the algebra on units to determine that $x\frac{\text{J}}{\text{rad}}\times y\text{rad}=xy\text{J}$, the formalism would explicitly represent that torque multiplied by angular displacement yields work.

To model all kinds of quantities and their functional interactions is a more ambitious undertaking than simply implementing a type system for units. When extending a type system for units, it is only necessary to identify the supertypes of a type being added; but when extending a system of quantity kinds, one must also identify meaningful functional interactions that the new kind has with previously-defined kinds. For example, consider the functional interactions that the kinds distance and amount of rainfall would have with fuel consumption. Distance and amount of rainfall may both be stated in meters, but while fuel consumption multiplied by distance yields amount of fuel, fuel consumption multiplied by amount of rainfall is an error.

Identifying the interactions for every possible combination of kinds of quantities known to science seems over-ambitious. Catalogs of kinds do already exist within ISO/IEC 80000 [28] and International Union of Pure and Applied Chemistry (IUPAC) recommendations [29], but these references only identify the most important kinds without identifying their interactions.

8. Conclusion

Utilizing a type system for ‘dimensionless quantities’ with quantity calculus will strengthen the integrity of calculations. Many errors that presently can be caught only through informal reasoning about the quantity kinds described by informative text will instead be caught by the formalism.

Eventually, the method of utilizing a type system with quantity calculus could be standardized at the CIPM level. The CCU would not need to occupy itself with defining counting units beyond the minimum set identified in Ref. [14]. Fleshing out large type systems with names and symbols for every countable thing of interest could be left as exercises for the various sciences to do for themselves in the appropriate standards-developing organizations.