Unit one is intrusive

Abstract

The SI brochure’s treatment of quantities that it regards as dimensionless, with the associated unit one, requires certain physical quantities to be regarded as simply numbers. The resulting formal system erases the nature of these quantities and excludes them from important benefits that quantity calculus provides over numerical value calculations, namely, that accidental confusion of different units and different kinds of quantities is sometimes prevented. I propose a better treatment that entails removing from the SI brochure those prescriptions that conflict with common practices in the treatment of dimensionless quantities, especially the definition and use of non-SI dimensionless units that are distinguished by kind.

Full text

Metrologia
Metrologia 61 (2024) 033002 (5pp) https://doi.org/10.1088/1681-7575/ad4bea
Letter to the Editor
Unit one is intrusive
David Flater1
Information Technology Laboratory, National Institute of Standards and Technology, Gaithersburg, MD,
United States of America
E-mail: david.ater@nist.gov
Received 4 March 2024, revised 7 May 2024
Accepted for publication 15 May 2024
Published 24 May 2024
Abstract
The SI brochure’s treatment of quantities that it regards as dimensionless, with the associated
unit one, requires certain physical quantities to be regarded as simply numbers. The resulting
formal system erases the nature of these quantities and excludes them from important benets
that quantity calculus provides over numerical value calculations, namely, that accidental
confusion of different units and different kinds of quantities is sometimes prevented. I propose a
better treatment that entails removing from the SI brochure those prescriptions that conict with
common practices in the treatment of dimensionless quantities, especially the denition and use
of non-SI dimensionless units that are distinguished by kind.
Keywords: unit one, dimensionless quantity, SI
1. Introduction
A quantity in the International System of Units (SI) can be
stated as the product of a numerical value and a unit of meas-
urement. The unit is derived from the seven SI base quant-
ities, which measure seven different physical dimensions.
However, many kinds of quantities—all ratios of two quantit-
ies of the same kind, all counted quantities (numbers of entit-
ies or events), all ‘characteristic numbers’, and all products
where the dimensions cancel out—have no extent in any of
those dimensions and are thus called ‘dimensionless quantit-
ies’ (among other names).
The unit that the SI brochure [1] identies for dimension-
less quantities is the special unit one, which behaves as a mul-
tiplicative identity element. One might assume that unit one
is an algebraic formality with no operational impact, but this
1Ofcial contribution of the National Institute of Standards and Technology
(NIST); not subject to copyright in the United States. The opinions, recom-
mendations, ndings, and conclusions in this publication do not necessarily
reect the views or policies of NIST or the United States Government.
Original Content from this work may be used under the
terms of the Creative Commons Attribution 4.0 licence. Any
further distribution of this work must maintain attribution to the author(s) and
the title of the work, journal citation and DOI.
is not the case. Unit one is intrusive; i.e. its presence as the
unit associated with a quantity is unwelcome and inconveni-
ent. By invisibly occupying the place where a meaningful unit
could otherwise go, it prevents us from treating counted quant-
ities as anything more than plain numbers and deprives us of
advantages that quantity calculus has over purely numerical
calculations (unless we disregard the prescriptions of the SI
brochure).
The following sections provide the details supporting this
claim, a review of related work in the literature, an assessment,
and nally an outline of changes to the SI brochure that would
mitigate the problems.
2. Unit one in SI
Editions 1 through 6 of the SI brochure did not call unit one
by name but included a note on the subject of ‘dimension-
less quantities’ or ‘quantities expressed as pure numbers’ that
changed little from 1970 to 1991. The note did not specically
address counted quantities. The following is the English text
from the 6th edition: ‘Certain so-called dimensionless quant-
ities, as for example refractive index, relative permeability,
or friction factor, are dened as the ratio of two comparable
quantities. Such quantities have a dimensional product—or
dimension—equal to 1 and are therefore expressed by pure
1 © 2024 The Author(s). Published on behalf of BIPM by IOP Publishing Ltd

Metrologia 61 (2024) 033002 Letter to the Editor
numbers. The coherent SI unit is then the ratio of two identical
SI units and may be expressed by the number 1.’ Notice that
the nal sentence does not name the unit one; it only mentions
the number 1 as an expression, i.e. a unit symbol.
In the 7th edition (1998), the previous note expanded into a
section (section 2.2.3) that explicitly named one as a coher-
ent SI unit and listed three categories of quantities that are
referred to it: ratios of two quantities of the same kind, ‘charac-
teristic numbers’, and ‘numbers which represent a count’. ‘All
of these quantities are described as being dimensionless, or of
dimension one, and have the coherent SI unit 1. Their values
are simply expressed as numbers and, in general, the unit 1 is
not explicitly shown. In a few cases, however, a special name
is given to this unit, mainly to avoid confusion between some
compound derived units. This is the case for the radian, stera-
dian, and neper.’ Unit one was referenced explicitly in tables 2
and 3 and in table footnotes.
The 8th edition (2006) discussed unit one in three separate
sections, elaborating the narrative of the 7th edition.
•Section 1.3 addressed unit one in the context of dimen-
sional products. For derived quantities where the dimen-
sional exponents are all zero, ‘The coherent derived unit...
is always the number one, 1’. Counted quantities ‘cannot be
described in terms of the seven base quantities of the SI at
all’ but ‘are also usually regarded as dimensionless quantit-
ies, or quantities of dimension one, with the unit one, 1.’
•Section 2.2.3 said that one is both a base unit and a derived
unit. ‘Certain quantities are dened as the ratio of two quant-
ities of the same kind. ...There are also some quantities that
are dened as a more complex product of simpler quantit-
ies in such a way that the product is dimensionless. ...For all
these cases the unit may be considered as the number one,
which is a dimensionless derived unit.’ But counted quant-
ities ‘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 quant-
ities, the unit one may instead be regarded as a further base
unit.’
•Section 5.3.7 added a prohibition on writing unit one expli-
citly except where a ‘special name’ has been accepted for it
(i.e. units for angles and logarithmic ratio quantities): ‘The
unit symbol 1 or unit name ‘one’ are not explicitly shown,
nor are special symbols or names given to the unit one,
apart from a few exceptions as follows.’ It also discussed
the meaning of percent, parts per million, and the like.
The 9th edition (2019) retained essentially the same role for
unit one but signicantly changed its denition. Text address-
ing whether unit one is base or derived was removed as was
text explicitly naming one as an SI unit. Instead, section
2.3.3 said ‘The unit one is the neutral element of any sys-
tem of units—necessary and present automatically. There is no
requirement to introduce it formally by decision. Therefore, a
formal traceability to the SI can be established through appro-
priate, validated measurement procedures.’
Section 2.3.3 avoided directly describing angles or counted
quantities as ‘just numbers’. Nevertheless, counted quantities
were said to be ‘just numbers’ in section 5.4.7. Section 5.4.7
also introduced the option to use unsimplied unit expressions
such as m/m to express ratios of two quantities of the same
kind.
The text of the 9th edition has been updated several
times. This letter refers specically to version 2.01, updated
December 2022 [2].
3. Unit one in the literature
In 1822, Fourier introduced the concept of physical dimen-
sion, thereby creating the distinction between dimensional and
dimensionless quantities [3, art. 160].
In 1933, Busemann introduced a classication system for
physical quantities in which dimensionless quantities were
grouped into Class 0 (‘0. Klasse’) [4]. Class 0 quantities were
those that he regarded as being measurable without rst estab-
lishing an arbitrary scale. For counted quantities, the unit of
counting implicitly established the scale; for angles, the cycle
lled the same role. He concluded that Class 0 quantities can
be described as pure numbers.
In 1955, unit one appeared as ‘die Einheit ‘Eins” in a
book by Stille [5, I.8]. After introducing the number one as
an effective unit, Stille immediately discussed the practice of
adding designations to various quantities that are nominally
expressed in unit one to distinguish them from one another.
Stille regarded these designations as arithmetically equivalent
to one, foreshadowing the SI brochure’s description of them
as ‘special names’ for the unit one. He included a version of
the widely-quoted equation where a dimensional product is
apparently equated to the number 1 or to a dimension that the
number 1 is meant to represent: Dim[Y] = ∏n
i=1A0
i=1. He
noted also that the term Dimension Zero (‘Dimension Null’)
was already in use.
In 1981, Reichardt rejected the position that numerical
quantities (‘Zahlengrößen’), elsewhere called dimensionless
quantities, are ‘just numbers’ [6]:
Numerical quantities are measurable properties
of objects or processes, that is to say phys-
ical quantities, not ‘just numbers’. ...Replacing
a numerical quantity by a number is only
permissible when numerically evaluating an
equation connecting numerical quantities. It
amounts to renouncing the qualitative proper-
ties of the particular quantity. It is not pos-
sible to deduce from mathematical derivations
whether it is possible to assign qualitative prop-
erties to the resulting values. For this one
has to resort to the physical considerations
which gave rise to formulating and solving the
equations in the rst place. Therefore all sym-
bols representing numerical quantities are con-
tainers for pure quantities, not for numbers.
This is the only way in which the combina-
tions which led to the formulation and sub-
sequent treatment of the equations connecting
2

Metrologia 61 (2024) 033002 Letter to the Editor
the quantities adequately mirror the physical
connections.
While recognizing that the names and symbols were a mat-
ter of convention and unimportant from a mathematical per-
spective, Reichardt used Dimension 1 as the working name for
the dimension of numerical quantities and reported that East
German standards had used 1 as the symbol for the corres-
ponding unit.
In 1995, J de Boer wrote, ‘All the so-called ‘dimension-
less quantities’ belong to one class of quantities of the same
kind; they belong to the equivalence class of ‘dimensionless’
quantities’ [7, section A3.4]. ‘These dimensionless quantities
are pure numbers’ [7, section 2.1.4].
Some pages later in the same journal issue, Mills reviewed
‘arguments for regarding the number 1 as a unit of the SI’
[8]. Mills addressed both counted quantities and ratios of two
quantities of the same kind while de Boer addressed only the
ratios.
Also in 1995, Giacomo criticized the term ‘dimension one’,
pointing out that the analogy with x0=1 where xis a real num-
ber is unjustied for dimensions [9].
In 1998, Quinn and Mills recommended adoption of the
special name uno (symbol U) for unit one to enable SI pre-
xes to be used instead of percent, parts per million, and
suchlike, noting that it had been discussed in the Consultative
Committee for Units (CCU) [10]. The article drew a comment
from White and Nicholas saying that the problem ‘is funda-
mentally one of a lack of education, and not of a aw in the
SI’ [11].
In 2001, Mills, Taylor, and Thor looked at the denitions of
the units radian, neper, bel, and decibel [12]. They made the
following general comments on units for dimensionless quant-
ities: ‘It is clear that as the value of a dimensionless quantity
is always simply a number, the coherent unit... is always the
number one, symbol 1. It is none the less important to establish
the equation dening the quantity concerned... in order that the
meaning of the coherent unit one can be correctly interpreted.
...For all of them the coherent unit is one. However the sig-
nicance of this unit is different for different quantities.’ The
article drew a response from Emerson saying, ‘The concept of
units for dimensionless quantities, whether they be the number
one or other numbers, lacks convincing supporting logic’ and
‘Angle should instead be adopted formally as a base quantity
with a denition that is consistent with the general, universal
view of lexicographers and of the majority who understand
and use the term in its classical sense’ [13].
In 2002, Valdés opposed expansion of the treatment of
dimensionless quantities in SI: ‘Adopting special names for
other dimensionless quantities, even another general special
name for the number one, will add more confusion to that
already existing’ [14].
In 2004, Dybkaer reviewed the previous history of unit
one and the uno and asserted “Number of entities’ is dimen-
sionally independent of the current base quantities and should
take its rightful place among them’ [15]. He reported that
the International Committee for Weights and Measures had
decided not to act on the CCU’s recommendation of the uno.
Later in 2004, Emerson criticized the concept of unit one,
concluding ‘If countable quantities, each with its own unit, are
also recognized as base quantities, nothing remains of the case
for one as a practical unit’ [16]. He rejected de Boer’s assertion
(quoted above) in a footnote:
A unit of a quantity is necessarily a quantity
of the same kind. Every kind of dimensionless
quantity, were it to have a unit, would have
to have a unit of its own kind, even if it bore
the same name as the putative units of all other
kinds of dimensionless quantities, notably one.
Dimensionless quantities of all kinds are evalu-
ated as numbers and are at the same time quant-
ities of different kinds. The magnitude of a rel-
ative area, for example, cannot be meaningfully
compared with that of a Reynolds number.
In 2010, Johansson proposed a parametric unit one—
essentially, one of something that must be specied [17].
In 2013, Mitrokin introduced ‘[1]’ (the number 1 in square
brackets, not bibliographic reference 1!) as a symbol for the
‘dimensional 1’ to distinguish it from the number 1 [18].
In 2015, for counted quantities, Brown and Brewer pro-
posed explicitly writing 1 as a unit instead of omitting it,
together with a full description of what is being counted [19].
Around the same time, Mohr and Phillips argued that radi-
ans and steradians should not be algebraically eliminated and
advocated the introduction of counting units that allow clearer
denition of both becquerel and amounts of substances [20].
The article drew a comment from Leonard alleging a num-
ber of ‘errors’ [21], in response to which Mohr and Phillips
emphasized that they were proposing to change the denitions
that Leonard was citing [22].
Also in 2015, Krystek distinguished the multiplicative neut-
ral element in dimensional algebra from the number 1 (cf.
Reichardt) and reiterated Giacomo’s criticism of the term
‘dimension one’: ‘the symbol ‘1’ denotes the number ‘one’
and ‘x0=1’ is only valid, if xactually represents a number.
A dimensional product is however clearly not a number’ [23].
He proposed referring to dimension number with symbol Z
and the coherent unit 1 instead of saying ‘dimension one’ or
‘dimensionless’ and described the values of the quantity num-
ber as pure numbers. ‘The quantity dimension Z is not a base
dimension, like [the seven enumerated in SI], but rather the
neutral element needed in any system of quantity dimensions.’
In 2016, Mills argued that the radian and the cycle should
be treated as units of dimension angle, thereby clarifying
that certain pairs of denitions refer to the same quantity
expressed with different units [24]. Discussion of proposals to
add dimension angle continued as a separate thread from the
treatment of dimensionless quantities and has remained active
[25–31].
In 2017, Flater proposed to use dimensionless units like
those introduced by Mohr and Phillips to discern different
kinds of dimensionless quantities in quantity calculus [32]. He
arranged dimensionless units in a type system [33] to support
generalization so that, for example, a number of neutrons and
a number of protons could be added, resulting in a number of
3

Metrologia 61 (2024) 033002 Letter to the Editor
nucleons. Unit one remained as the maximally general (and
thus completely uninformative) ‘top type’.
In 2021, Brown reviewed the situation with counted quant-
ities and unit one, expanding on the issues of traceability and
realisation of units [34].
Finally, in 2023, Flater described counts as a particular case
of quantal quantities, which are constrained to exist in integ-
ral multiples of a quantum, and argued that the unit used to
express a count need not be the same as the quantum (the single
entity or event being counted) [35]. The principle of invariance
of a quantity to a change of unit then applied to counted quant-
ities and counting units the same as it does to other physical
quantities and units.
4. Assessment
4.1. Dimensions not included in SI
Unit one is indeed recognized as a unit in the 9th edition of the
SI brochure [2]. But unlike the base and derived SI units that
are listed normally, unit one is a ghostlike entity that mani-
fests only on occasion. It is not included in the denition of
the SI (section 2.2). It is said to have arisen spontaneously
as an entailment of dimensional algebra rather than through
a formal decision to adopt it as other units were adopted
(section 2.3.3). The reader is prohibited from writing it expli-
citly in a quantitative expression (section 5.4.7). Nevertheless,
it plays a unique role in establishing formal traceability to
the SI for counted quantities (section 2.3.3)—albeit a vacu-
ous traceability that carries no burden of calibration. For what
is presented as an indispensable unit, one receives a confusing
treatment.
The importance of treating numerical physical quantities
as more than just numbers was fully explained by Reichardt
(see the block quote in section 3). When the structure real-
ized by the SI system and quantity calculus is different than
the inherent structure of the material world, we have to con-
clude that the SI structure is either incomplete or wrong. Yet
when dimensional analysis suggests that counted quantities,
angles, and ratios of two quantities of the same kind are all
dimensionless, some evidently conclude that it must be so
and bravely toss them into the set of real numbers. Section
2.3.3: ‘Such quantities are simply numbers.’ Section 5.4.7:
‘Quantities related to counting... are just numbers.’
The resulting system erases the nature of numerical phys-
ical quantities and facilitates two kinds of errors. First, one can
make the errors that in every other dimension are recognized
as errors of unit conversion; e.g. one can produce an erroneous
factor of 2πfrom the confusion of radians with cycles or a
factor of eight from the confusion of bits with bytes. Second,
one can add, subtract, or convert among quantities of differ-
ent kinds; e.g. one can add an angle to an amount of data. The
risk of confusing different kinds of quantities that happen to
be expressed in the same units is not limited to dimension-
less quantities, but it is worse because of the sheer number of
different kinds that all must refer to the unit one. A simple
quantity calculator that implements the system as given will
realize de Boer’s assertion ‘All the so-called ‘dimensionless
quantities’ belong to one class of quantities of the same kind’
literally and freely convert any dimensionless quantity to any
other.
Given the structural difference between a model with a
xed set of dimensions and a world that gives rise to addi-
tional dimensions as science progresses, what must yield is
the assumption that the set of dimensions enumerated in SI is
complete. Quantities that SI does not distinguish are not neces-
sarily the same. The enumeration in the SI brochure of certain
dimensions ought not be construed to create difculties for the
use of others in the various domains of science.
4.2. Counting units
The problem is exacerbated by rules that effectively ban the
appearance of counting units (section 5.4.7). The SI brochure
does not allow one to say ‘The throughput is 1 Mb/s’; one is
required to shift the unit of data into the description of the
quantity and say something equivalent to ‘The bit-throughput
is 106/s’. As Mohr and Phillips observed, ‘one cannot do cal-
culations or conversions with phrases that precede the quantity
in question’ [22, section 3]; for those working with data, com-
pliance comes at that cost. In practice, computer scientists use
bit and byte as units to avoid confusion and enable the expec-
ted calculations; compliance with section 5.4.7 would confuse
their colleagues and increase the risk of bit–byte conversion
errors.
With counting having been used as a measurement method
for mass, length, and volume since prehistoric times (e.g.
grains of wheat or barley), it is no coincidence that amounts
and counts have the same form of expression: a number fol-
lowed by a unit. When one is specied as the unit associated
with a count, it displaces a more informative counting unit that
in common usage would be supplied (e.g. 10 passengers, 23
kg/passenger). To a wide audience, ‘23 kg’ and ‘23 kg/pas-
senger’ express different quantities. Suppressing the counting
unit invites mistakes to be made.
5. Outline of changes to the SI brochure
Proposed is (1) to recognize in the SI brochure that numer-
ical physical quantities are not simply numbers, and (2) to
remove from the SI brochure those prescriptions that con-
ict with common practices in the treatment of dimensionless
quantities, especially the denition and use of dimensionless
units that are distinguished by kind. The corresponding opera-
tional model for quantity calculus has already been described
[32,35].
In the 9th edition of the SI brochure [2], the relevant pro-
posed changes are:
•Section 2.3.3: remove the assertion that any quantity that
is dened as the ratio of two quantities of the same kind
is simply a number, reduce the emphasis on unit one, and
address traceability for dimensionless ratios and counts in
a more candid way. For dimensionless ratios, we know that
4

Metrologia 61 (2024) 033002 Letter to the Editor
the numerator and denominator will have meaningful trace-
ability if they are considered separately. For counts, we
know that some denition for the identication of the entit-
ies or events counted must exist, ideally in a standard that is
‘downstream’ [34] of the SI brochure.
•Section 4: acknowledge the role of ‘downstream’ stand-
ards for other units and remove the implication that any
units not listed in the brochure are ‘unacceptable’ (cf. [36,
section 7.5], ‘quantities must be dened so that they can be
expressed solely in acceptable units’). The normative goal
is to proscribe the use of non-SI units for the dimensions
where SI units have been dened; it cannot be to prohibit
the denition of units for dimensions that SI does not con-
tain, for which unit one is a meaningless placeholder.
•Section 5.4.2: adjust rules about unit symbols and ‘extra
information’ so that they cannot be construed to prohibit the
use of dimensionless units beyond unit one (cf. [36, section
7.5], ‘Unacceptability of mixing information with units’).
•Section 5.4.7: remove the assertion that counted quantities
are just numbers and allow the use of SI prexes with count-
ing units other than unit one.
•Section 5.4.8: revise the treatment of angles, in particular the
presentation of the equation 1 rad =1, to catch up with the
characterization of the ‘radian convention’ that has appeared
in the literature [37].
Those changes would reduce the difculty of making do
without an SI dimension. Whether angles, data, or any other
kind of quantity should be allocated an SI dimension should
be regarded as a completely separate question.
ORCID iD
David Flater https://orcid.org/0000-0002-2546-2530
References
[1] BIPM 1970–2022 The International System of Units (SI)
every edn (available at: https://www.bipm.org/en/
publications/si-brochure/)
[2] BIPM 2022 The International System of Units (SI) 9th edn
version 2.01 (available at: https://web.archive.org/web/
20221219120033/https://www.bipm.org/documents/20126/
41483022/SI-Brochure-9.pdf)
[3] Fourier J 1822 Théorie Analytique de la Chaleur (Firmin
Didot) (available at: https://archive.org/details/
thorieanalytiq00four)
[4] Busemann A 1933 Die Temperatur im Rahmen der
Ähnlichkeitsbetrachtungen Z. Tech. Phys. 14 131–5
[5] Stille U 1955 Messen und Rechnen in der Physik 2nd edn
(Springer) (https://doi.org/10.1007/978-3-322-98458-6)
[6] Reichardt W 1981 Zahlengrößen—dimensionslose Größen
oder Größen der Dimension 1? Acta Acust. United Acust.
49 173–8 (available at: https://www.ingentaconnect.com/
content/dav/aaua/1981/00000049/00000003/art00003)
[7] de Boer J 1995 On the history of quantity calculus and the
international system Metrologia 31 405–29
[8] Mills I M 1995 Unity as a unit Metrologia 31 537–41
[9] Giacomo P 1995 Sans dimension ou de dimension un?
Metrologia 32 311–3
[10] Quinn T J and Mills I M 1998 The use and abuse of the terms
percent, parts per million and parts in 10nMetrologia
35 807–10
[11] White D R and Nicholas J V 2001 Comment on Quinn and
Mills’ proposal for the uno Metrologia 38 369–71
[12] Mills I M, Taylor B N and Thor A J 2001 Denitions of the
units radian, neper, bel and decibel Metrologia 38 353–61
[13] Emerson W H 2002 A reply to ‘Denitions of the units radian,
neper, bel and decibel’ by I M Mills etal Metrologia
39 105–9
[14] Valdés J 2002 The unit one, the neper, the bel and the future of
the SI Metrologia 39 543–9
[15] Dybkaer R 2004 Units for quantities of dimension one
Metrologia 41 69–73
[16] Emerson W H 2004 One as a ‘unit’ in expressing the
magnitudes of quantities Metrologia 41 L26–L28
[17] Johansson I 2010 Metrological thinking needs the notions of
parametric quantities, units and dimensions Metrologia
47 219–30
[18] Mitrokhin A N 2013 On the dimensionality of dimensionless
quantities (English translation) Meas. Tech. 56 17–24
[19] Richard J C B and Paul J B 2015 Proposals for nomenclature
to clarify the expression of units for counting quantities
Metrologia 52 L1–L3
[20] Mohr P J and Phillips W D 2015 Dimensionless units in the SI
Metrologia 52 40–47
[21] Leonard B P 2015 Comment on ‘Dimensionless units in the
SI’ Metrologia 52 613–6
[22] Mohr P J and Phillips W D 2015 Reply to comments on
‘Dimensionless units in the SI’ Metrologia 52 617–8
[23] Krystek M P 2015 The term ‘dimension’ in the international
system of units Metrologia 52 297–300
[24] Mills I 2016 On the units radian and cycle for the quantity
plane angle Metrologia 53 991–7
[25] Leonard B P 2021 Proposal for the dimensionally consistent
treatment of angle and solid angle by the International
System of Units (SI) Metrologia 58 052001
[26] Quincey P 2021 Angles in the SI: a detailed proposal for
solving the problem Metrologia 58 053002
[27] Leonard B P 2022 Comment on ‘Angles in the SI: a detailed
proposal for solving the problem’ Metrologia 59 038001
[28] Quincey P 2022 Reply to comment on ‘Angles in the SI: a
detailed proposal for solving the problem’ Metrologia
59 038002
[29] Mohr P J, Shirley E L, Phillips W D and Trott M 2022 On the
dimension of angles and their units Metrologia 59 053001
[30] Quincey P 2023 Comment on ‘On the dimension of angles and
their units’ Metrologia 60 048001
[31] Mohr P J, Shirley E, Phillips W D and Trott M 2023 Reply to
comment on ‘On the dimension of angles and their units’
Metrologia 60 068001
[32] Flater D 2017 Redressing grievances with the treatment of
dimensionless quantities in SI Measurement 109 105–10
[33] Coquand T 2022 Type theory Stanford Encyclopedia of
Philosophy ed E N Zalta and U Nodelman (Metaphysics
Research Lab, Stanford University) (available at: https://
plato.stanford.edu/entries/type-theory/)
[34] Brown R J C 2021 A metrological approach to quantities that
are counted and the unit one Metrologia 58 035014
[35] Flater D 2023 Dealing with counts and other quantal quantities
in quantity calculus Measurement 206 112226
[36] Thompson A and Taylor B N 2008 NIST SP 811 Guide for the
use of the International System of Units (SI) National
Institute of Standards and Technology (https://doi.org/
10.6028/NIST.SP.811e2008)
[37] Quincey P and Burrows K 2019 The role of unit systems in
expressing and testing the laws of nature Metrologia
56 065001
5