Units of Measure: Concepts & Vocabulary

Quantities of different dimensions multiply freely — 1 m × 1 s is an unremarkable metre-second — yet they do not add: every formalization in the theory subtree and every system in this survey treats 1 m + 1 s as a defect, while disagreeing profoundly about what kind of defect it is. Why is the product total and the sum partial? That asymmetry is the spine of the whole catalog: each theory page answers a shared protocol question about it, each system page documents how a checker enforces it, and the capstone comparison reconciles the answers. This page fixes the vocabulary those pages lean on — quantity and quantity value, kind of quantity, dimension, unit, system of quantities vs system of units, base / derived / coherent / off-system units, quantity of dimension one, and the recurring affine and logarithmic edge cases — as the metrology primaries define them, and records where the mathematical literature and the surveyed libraries part ways with metrology (and where metrology parts ways with itself).

NOTE

The primaries. The definitional authority throughout is the VIM — the International Vocabulary of Metrology, 3rd edition (JCGM 200:2012, identical in content to ISO/IEC Guide 99), quoted by clause number below. The SI Brochure, 9th edition (BIPM 2019, v4.01), supplies the SI-specific definitions (coherence, the dimensional product, rad/sr, Np/B/dB). ISO 80000-1:2022 — the ISQ's own standard — is paywalled with no legitimate open copy, so wherever ISQ specifics matter this survey grounds them in the SI Brochure and in NIST's guide to the SI (SP 811, 2008 edition), the open secondaries; claims that only ISO 80000-1 itself could settle are flagged. The machine-facing angle comes from the UCUM specification (case-sensitive unit codes for electronic interchange) and the QUDT ontology (units as RDF data; repo pinned locally at bb9e04d).

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One prohibition, six accounts

What, exactly, is wrong with 1 m + 1 s? The survey's sources give answers that are genuinely incompatible — not paraphrases of one answer — and the disagreement is the single most useful orientation device for a reader of this tree:

• Inexpressible — Whitney's measurement models are disjoint carriers equipped only with physically warranted operations, so m + l is not an error the axioms rule out; it is a string with no denotation. A heterogeneous equation is not false but unformulable. (And yet Whitney's own counting example computes 6(2 bl + 3 ck) = 12 bl + 18 ck distributively, without ever naming the structure that sum lives in.)
• Ill-typed but semantically defined — and not invariant — in Kennedy's type system the mixed sum is a static type error, but the erased program underneath it is an ordinary rational addition that never gets stuck. What fails is equivariance: the sum's value depends on the arbitrary choice of units, whereas multiplication commutes with every rescaling. "Meaningless" is rendered as "defined but not invariant".
• Total, but exiled from every weight space — in the graded reading (tensor of lines, torsor / scaling torus) the ambient algebra happily contains 1 m + 1 s as a "hybrid" element lying in no weight space; what physical laws demand is homogeneity (equivariance under the scaling action), and Tao's convex-hull criterion measures exactly how little law-like content the hybrid part carries.
• Moved up a level, into the vector space — Hart keeps scalar cross-type addition undefined by definition, but makes the heterogeneous aggregate legitimate as a dimensioned vector: a tuple with one component per type, i.e. a point of the product (equivalently the direct sum) of the fibers. The formal sums Whitney computed with but never housed are, up to notation, Hart's vectors — the comparison works through that identification.
• Raised as an error at evaluation — the runtime systems define the sum and make it fail: Pint raises DimensionalityError at the moment two incompatible quantities actually meet; Unitful resolves the check per JIT specialization, compiling a mismatch into an unconditional throw.
• Not even checked until asked — MATLAB's symunit lets 1*u.m + 1*u.s flow through arithmetic as inert symbolic factors; only an explicit checkUnits call reports (as a logical, not an error) that the expression is inconsistent (wolfram-matlab).

No winner is crowned here. The full reconciliation — including Bridgman's demonstration that the prohibition is a conditional theorem rather than an axiom, and the Lean mechanization's total-but-unknowable rendering via Classical.epsilon — lives in the comparison's seven-readings ledger; the checkable type-system renderings are catalogued in the mechanisms bridge.

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Quantity — and quantity value

The VIM's opening definition, the one every other clause builds on:

"quantity — property of a phenomenon, body, or substance, where the property has a magnitude that can be expressed as a number and a reference" — VIM 1.1

Three of its notes carry weight for this survey. Note 2: "A reference can be a measurement unit, a measurement procedure, a reference material, or a combination of such" — metrology's quantity concept is deliberately wider than unit-referenced measurement (Rockwell hardness is a quantity whose reference is a procedure; UCUM's arbitrary units are the interchange rendering of that width, and no surveyed type system models it). Note 5: "A quantity as defined here is a scalar" — vectors and tensors are quantities only componentwise, which is exactly the stance Hart attacks as under-ambitious. And the VIM keeps a separate concept for the number-and-reference pair:

"quantity value — number and reference together expressing magnitude of a quantity" — VIM 1.19

That quantity/quantity-value distinction is the cleanest lens on the formalizations' deepest split: Whitney and the tensor-of-lines school axiomatize VIM's quantity — the property itself, prior to any number — while Buckingham and Bridgman "deliberately conflate a quantity with its numerical measure" and Hart's (f, g) pairs formalize VIM's quantity value (a number with a group-element reference). Kennedy's quantities are a third thing: bare rationals at run time whose reference lives only in the type, erased before evaluation. When a theory page says "quantity", it matters which of these three it means; the pages say so, and this glossary is the place the words are held apart.

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Unit — and the unit–quantity circle

"measurement unit — real scalar quantity, defined and adopted by convention, with which any other quantity of the same kind can be compared to express the ratio of the two quantities as a number" — VIM 1.9

Two structural facts hide in this sentence. First, the definition is circular with 1.1 by design: a unit is itself a quantity, while a quantity's magnitude is expressed by reference to a unit (1.1 Note 2). The VIM is a concept system, not an axiomatization, and tolerates the circle; the formalizations each break it at a chosen point, and where they break it is what distinguishes them:

Where the circle is cut	Unit becomes…	Page
Quantities primitive	any element "kept fixed for a period" — pure bookkeeping, no algebraic privilege	Whitney
Carriers (1-D lines) primitive	a (positive) basis vector — JMV: "a semi-basis … is called a unit"	tensor
The group action primitive	a torsor point; a whole system of units = a section u : D → R, never canonical	torsor
Unit syntax primitive	a unit variable — base units and polymorphism distinguished only as free vs bound occurrences	Kennedy
Neither — the pair is primitive	no formal object at all (a recorded silence: writing (f, g) has already chosen a unit per dimension)	Hart
Operational (interchange)	a code with conversion data — UCUM's (r, û) magnitude-and-vector pair, QUDT's conversionMultiplier	below

Second, the ratio phrasing — "any other quantity of the same kind" — makes kind prior to unit in the metrology concept order: comparability comes first, units presuppose it. Almost every surveyed type system inverts this, deriving comparability from unit (or dimension) equality; the consequences are the kind cluster's story below.

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Dimension

"quantity dimension — expression of the dependence of a quantity on the base quantities of a system of quantities as a product of powers of factors corresponding to the base quantities, omitting any numerical factor" — VIM 1.7

Metrology's dimension is a piece of notation — an expression like dim F = LMT⁻², derived from the quantity's defining equations and read modulo the "omitting any numerical factor" convention. The SI Brochure fixes the SI's instance of it:

"In general the dimension of any quantity Q is written in the form of a dimensional product, dim Q = T^α L^β M^γ I^δ Θ^ε N^ζ J^η where the exponents α, β, γ, δ, ε, ζ and η, which are generally small integers, which can be positive, negative, or zero, are called the dimensional exponents." — SI Brochure §2.3.3

The mathematical literature reads the same object structurally: a dimension is an element of the free abelian group on the base dimensions (≅ ℤ⁷ for the ISQ), the dimensional product is the group operation, and "omitting any numerical factor" is the quotient that Buckingham's tradition undoes by conflating quantities with measures. The free-abelian-group page develops this reading; the mechanisms bridge catalogues its type-system encodings; the CI-verified quantity-zn-graded.d prototype implements it directly. Three frictions between the metrology text and the group-theoretic reading are worth pinning:

• The basis is conventional, and so is its order. VIM 1.7 Note 5 writes the product dim Q = L^α M^β T^γ I^δ Θ^ε N^ζ J^η; the SI Brochure's own §2.3.3 writes T^α L^β M^γ …; QUDT's vector labels run A…E…L…I…M…H…T…D (below). The two flagship metrology primaries disagree on display order because the group has no canonical ordered basis — a triviality in the algebra that becomes load-bearing in positional encodings (dimensioned's tarr! arrays, and every diagnostics spine that leaks index positions). Base change — e.g. {L, T, M} → {L, T, F} — is a GL(n, ℤ) action the VIM has no vocabulary for at all (fag).
• "Generally small integers" is not a commitment. The VIM's very own Example 3 computes a fractional exponent — a pendulum analysis ending in dim C(g) = L^(−1/2) T — two paragraphs below a definition whose SI rendering says "generally small integers". Whether exponents live in ℤ, ℚ, or ℝ is precisely the axis the formalizations never settled and practice quietly resolved toward ℚ (comparison, Part I & III; quantity-rational-exponents.d).
• Dimension is not the whole identity. VIM 1.7 Note 4 states it as a three-bullet asymmetry: same kind ⟹ same dimension; different dimension ⟹ different kind; same dimension ⇏ same kind. Every formalization whose dimension is the quantity's whole identity collapses the third bullet — the kind cluster next.

For quantities whose exponents are all zero, the VIM keeps a dedicated term — "quantity of dimension one" (VIM 1.8, with "dimensionless quantity" retained "for historical reasons") — whose surprisingly sharp edges are deferred to the edge cases below.

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Kind of quantity

The one metrology concept most surveyed type systems drop — and the survey's widest theory/practice gap (comparison § kinds):

"kind of quantity, kind — aspect common to mutually comparable quantities" — VIM 1.2

The VIM immediately concedes the concept's softness, in exactly these words:

"NOTE 1 The division of ‘quantity’ according to ‘kind of quantity’ is to some extent arbitrary." — VIM 1.2, Note 1

and then states the relationship to dimension that no free-abelian-group encoding can express:

"NOTE 2 Quantities of the same kind within a given system of quantities have the same quantity dimension. However, quantities of the same dimension are not necessarily of the same kind. — EXAMPLE The quantities moment of force and energy are, by convention, not regarded as being of the same kind, although they have the same dimension. Similarly for heat capacity and entropy, as well as for number of entities, relative permeability, and mass fraction." — VIM 1.2, Note 2

Kind reaches into the unit system itself: VIM 1.9 Note 2 records that "in some cases special measurement unit names are restricted to be used with quantities of a specific kind only. For example, the measurement unit ‘second to the power minus one’ (1/s) is called hertz (Hz) when used for frequencies and becquerel (Bq) when used for activities of radionuclides" — and the SI Brochure's Table 4 turns the restriction into normative language ("The hertz shall only be used for periodic phenomena and the becquerel shall only be used for stochastic processes…"). The Brochure is explicit that some of these distinctions exist for safety:

"The special names becquerel, gray and sievert were specifically introduced because of the dangers to human health that might arise from mistakes involving the units reciprocal second and joule per kilogram, in case the latter units were incorrectly taken to identify the different quantities involved." — SI Brochure §2.3.4

NIST SP 811 adds the complementary rule — kind information belongs to the quantity, never to the unit: "it is incorrect to attach letters or other symbols to the unit in order to provide information about the quantity or its conditions of measurement. Instead, the letters or other symbols should be attached to the quantity" (Vmax = 1000 V, not V = 1000 Vmax; SP 811 §7.4).

How the rest of the survey treats kind:

• The theory corpus drops it, knowingly. Where dimension is a group element, the group element is the quantity's entire identity: torque ≡ energy, Hz ≡ Bq. Kennedy names the torque/energy problem and provides no mechanism; Jonsson turns the collapse into a definition (same kind iff commensurable); the only productive theoretical mechanism — Tao's structure-group enlargement — still never separates scalar same-dimension pairs like Hz/Bq (comparison § kinds).
• Most type systems inherit the collapse. 5.0<Hz> + 3.0<Bq> type-checks in F# — the stdlib's own SI.fs defines both as second^-1 — and the same holds across GNAT, dimensional, dimensioned, Pint, Unitful, Lean, and the D artifacts (d-quantities).
• The engineering resurrections are mutually incompatible.uom's flat Kind tags separate Hz from Bq but reset to the default kind under ×/÷; Boost.Units and Au mint extra base dimensions (radian/steradian; Angle, Information), which splits torque from energy but can never split anything sharing a genuine dimension; mp-units rebuilds the ISQ itself as a quantity_spec hierarchy with kind algebra — and its documentation grounds the design by quoting this very VIM text, arbitrariness caveat included.
• The ontology puts kind at the centre. In QUDT a unit must name its quantity kind and the dimension vector hangs off the kind — the exact inversion of the dimension-only type systems.

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System of quantities vs system of units

Metrology maintains two parallel systems, related but never identified:

"system of quantities — set of quantities together with a set of non-contradictory equations relating those quantities" — VIM 1.3

"system of units — set of base units and derived units, together with their multiples and submultiples, defined in accordance with given rules, for a given system of quantities" — VIM 1.13

The ISQ (VIM 1.6) is a system of quantities — seven base quantities and the equations of physics; the SI (VIM 1.16) is a system of units based on the ISQ. Each side has its own base/derived split. A base quantity is a member of "a conventionally chosen subset … where no subset quantity can be expressed in terms of the others" (VIM 1.4) — the note makes the independence multiplicative ("cannot be expressed as a product of powers of the other base quantities"), which the free-abelian-group page reads as linear independence in the exponent lattice, and whose failure modes (is the chosen basis actually of full rank for the problem at hand?) are the π-theorem's bread and butter. A derived quantity is "defined in terms of the base quantities" (VIM 1.5); a base unit is "adopted by convention for a base quantity" (VIM 1.10 — with Note 1's "in each coherent system of units, there is only one base unit for each base quantity", i.e. the torsor page's section u : D → R picks exactly one point per fiber); a derived unit is simply "a measurement unit for a derived quantity" (VIM 1.11).

Most surveyed libraries collapse the quantity level into the unit level — their only notion of "what this value is" is its unit (or its dimension vector). The exceptions prove the distinction is implementable: mp-units models the ISQ as a first-class quantity_spec tree above its units layer, and QUDT stores quantity kinds and units as separate node types. Meanwhile the formalizations mostly axiomatize the quantity side and let units be derived décor — the exact opposite collapse (comparison, Part I).

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Coherent, off-system, and accepted units

"coherent derived unit — derived unit that, for a given system of quantities and for a chosen set of base units, is a product of powers of base units with no other proportionality factor than one" — VIM 1.12

The SI Brochure explains what the property buys:

"The word 'coherent' here means that equations between the numerical values of quantities take exactly the same form as the equations between the quantities themselves." — SI Brochure §2.3.4

Coherence is relative twice over (VIM 1.12 Note 2: "only with respect to a particular system of quantities and a given set of base units"; Note 3's example: cm/s is coherent in CGS, not in the SI), and it is fragile: prefixes destroy it ("when prefixes are used with SI units, the resulting units are no longer coherent, because the prefix introduces a numerical factor other than one" — §2.3.4), with the kilogram as the standing historical joke ("the kilogram is the only coherent SI unit that includes a prefix in its name and symbol", §3). In the torsor page's vocabulary this all becomes one sentence: a system of units is a section u : D → R of the dimension fibration, and coherence is the demand that the section be multiplicative — u(d·e) = u(d)·u(e) with factor one — which is why "numerical-value equations keep the form of quantity equations" and why normalize-to-base libraries (uom, quantities) can do all arithmetic in one trivialization and convert only at the boundaries (comparison, Part III).

Outside a system sit the off-system units — "measurement unit that does not belong to a given system of units" (VIM 1.15; its examples are the electronvolt and day/hour/minute) — and, cutting across that, the accepted ones: units "outside the SI but accepted for use with the SI" (VIM 1.11's km/h example). The 9th-edition Brochure's chapter 4 gathers these into Table 8 ("Non-SI units") with a shrug of realism: "It is recognized that some non-SI units are widely used and that this is expected to continue for many years." The practice side mirrors the taxonomy exactly: registry systems carry the whole Table-8 world as data (Pint's default_en.txt), unit-in-type systems mint off-system units with exact rational/symbolic conversion factors (Au's magnitude vectors, Unitful's Rational{Int} powers — the comparison's "exact conversion factors are kept exact" consensus), and minimal systems like F# ship no metrology at all — every unit is off-system because there is no system.

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UCUM: units as case-sensitive codes

The Unified Code for Units of Measure is what the unit concept looks like when the consumer is a wire protocol: "a code system intended to include all units of measures being contemporarily used in international science, engineering, and business … to facilitate unambiguous electronic communication of quantities together with their units" (UCUM, Introduction). Its angles, in survey terms:

• Case-sensitivity as a correctness feature. UCUM's predecessor standards (ISO 2955, ANSI X3.50) were case-insensitive and paid with name collisions — "cd means candela and centi-day and PEV means peta-volt and pico-electronvolt" (UCUM, Introduction). UCUM's primary symbols are case-sensitive (Cel is not cel); a separate case-insensitive variant exists for degraded channels and is declared "incompatible to the case sensitive symbols" (§3). The lesson generalizes: every registry-driven system in the survey (Pint, Wolfram) owns an equivalent symbol-disambiguation problem.
• The algebra is stated in the spec, not implied. UCUM defines equality = and commensurability ~ as equivalence relations (§17); units are the =-classes and "(U, ·) is an Abelian group" (§18); dimensions are the ~-classes (§19); any system is generated by "a finite set B of mutually independent base units", against which every unit is "a pair (r, û) of magnitude r and dimension û" (§20). This is the free-abelian-group picture published as an interchange spec — with the spec-level twist that conformance is semantic: full implementations "must compare unit expressions by their semantics, i.e. they must detect equivalence for different expressions with the same meaning" (§2).
• The metric predicate marks which atoms take prefixes (§11): all base units are metric, customary units never are, and — decisively — "a unit must be a quantity on a ratio scale in order to be metric".
• Special units quarantine the non-ratio scales. Interval- and logarithmic-scale symbols (Cel, [degF], B, Np, dB) "do not represent proper units as elements of the group (U, ·)"; each is a triple (u, fₛ, fₛ⁻¹) — a corresponding proper unit plus a conversion function pair — and "in theory, special units cannot take part in any algebraic operations" (§§21–22). UCUM here disagrees with the VIM, which counts the decibel among "special names" for units of dimension one (VIM 1.9 Note 3); the edge cases below inherit the dispute. The spec's editorial aside on the CCU's 1995 decision to allow prefixes on °C is the sharpest sentence in the corpus on the affine problem: "One wonders why the CGPM keeps the Celsius temperature in the SI as it is superfluous and in a unique way incoherent with the SI" (§22).
• Arbitrary units ([IU], procedure-defined assay units) are "not 'of any specific dimension' and are not 'commensurable with' any other unit" (§24) — the interchange rendering of VIM 1.1 Note 2's procedure-referenced quantities. Curly-brace annotations carry human context but "do not contribute to the semantics of the unit" — UCUM's mechanized form of SP 811 §7.4's no-information-on-the-unit rule.

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QUDT: units as an ontology

QUDT ("Quantities, Units, Dimensions and Types") is the semantic-web codification: everything the static libraries put into types is here RDF data — classes, properties, and SHACL constraints in a public repo (pinned locally at bb9e04d). Its ontological commitments answer this page's vocabulary questions in an instructively different order:

• Kind is mandatory; dimension hangs off the kind. A qudt:Unit must carry at least one qudt:hasQuantityKind (owl:minCardinality 1 in the schema, sh:minCount 1 in the SHACL shapes), and a qudt:QuantityKind — "any observable property that can be measured and quantified numerically … Less familiar examples include currency, interest rate, price to earning ratio, and information capacity" (SCHEMA_QUDT.ttl) — must carry exactly one qudt:hasDimensionVector. Where the dimension-only type systems make the exponent vector the identity and drop kind, QUDT makes kind the identity and demotes the vector to one of its properties.
• The Hz/Bq split is data. Via unitForQuantityKind, unit:HZ points at quantitykind:Frequency and unit:BQ at quantitykind:Activity, while both carry the same dimension vector qkdv:A0E0L0I0M0H0T-1D0. The becquerel's own description note reads: "both the becquerel and the hertz are basically defined as one event per second, yet they measure different things" (VOCAB_QUDT-UNITS-ALL.ttl). Likewise one vector node, qkdv:A0E0L2I0M1H0T-2D0, lists both quantitykind:Energy and quantitykind:Torque as reference kinds — the VIM 1.2 Note 2 example encoded as a many-kinds-per-vector graph shape.
• An eight-slot dimension vector. QUDT's vectors carry the ISQ seven plus a D slot: quantitykind:Dimensionless — and unit:RAD — get …T0D1, not the all-zero vector. Dimensionless-ness is a coordinate, not the group identity: an engineering move (it keeps dimensionless kinds from colliding with the empty product) that no formalization in the theory subtree endorses, and a concrete instance of the angle problem being patched at the data layer.
• Conversion is two numbers. qudt:conversionMultiplier and — for the affine cases — qudt:conversionOffset (unit:DEG_C carries conversionOffset 273.15) put the point/difference distinction of the affine edge case into plain properties, with none of the operational safety of a point/difference type split: what a consumer does with the offset is its own affair.
• The bridge to UCUM is a property. A unit may carry at most one qudt:ucumCode — the ontology and the wire code deliberately cross-reference rather than duplicate.

QUDT is thus the run-time-registry pole of the mechanisms spectrum taken to its limit: a units "library" with no checker at all, only queryable structure — and, precisely because nothing must type-check, the richest kind vocabulary in the survey.

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The recurring edge cases

Three quantity shapes recur throughout the survey wherever a page has an "Expressiveness edges" section; they are defined once here, as the canonical link target. What unites them: each is a place where the plain multiply-freely/add-within-a-fiber picture — the consensus core of both metrology and the formalizations — stops describing the physics people actually record.

Affine quantities: points vs differences

Celsius temperature is the SI's own affine quantity: not a magnitude but a displacement from a conventional origin,

"it remains common practice to express a thermodynamic temperature, symbol T, in terms of its difference from the reference temperature T₀ = 273.15 K … This difference is called the Celsius temperature, symbol t, which is defined by the quantity equation t = T − T₀." — SI Brochure §2.3.1

with the degree Celsius "by definition equal in magnitude to the unit kelvin" — and the Brochure concedes the algebraic consequence in one line: "The unit degree Celsius is only coherent when expressing temperature differences" (§2.3.4). Points (temperatures, timestamps, positions, voltages) support point − point → difference and point ± difference → point, but not point + point, not scalar × point, and their zero is a convention, not an identity. The theory home is the torsor page: each fiber's points form a torsor under its additive group — a second, different torsor from the multiplicative torsor of unit choices — and Whitney had the structure in 1968 (his Part I §6 time-point space T* with a biray T of translations). UCUM quarantines the same cases as special units on interval scales; QUDT stores them as a conversionOffset.

Practice converged independently on the point/difference split wherever temperature is handled correctly — Pint's auto-generated delta_ units and OffsetUnitCalculusError, Unitful's @affineunit + AffineError (for any dimension, not just temperature), mp-units's quantity_point with typed origins, Au's QuantityPoint, D's 2011 std.units AffineUnit (d-quantities), C++'s own std::chrono time_point/duration pair (boost-units) — while the systems that skip it exhibit the two standard failure modes: a linear-unit costume (Celsius_Temperature as a Kelvin vector with a °C display symbol in GNAT; enum celsius = kelvin in quantities — d-quantities) or a documented trap (MATLAB's 0*u.Celsius collapsing to a dimensionless 0 — wolfram-matlab). The convergence and its exceptions are tabulated in comparison § affine.

Logarithmic quantities: Np, B, dB

The SI Brochure itself ships three logarithmic units — in Table 8 ("Non-SI units"), with a footnote where their conversion value should be:

"The neper, bel and decibel are used to express values of specified logarithmic ratio quantities. When using these units, it is important that the quantity be specified, and that any reference value used be specified. … The statement LX = m dB = (m/10) B (where m is a number) is interpreted to mean that m = 10 lg(X/X₀)." — SI Brochure, Table 8, note (m)

A decibel value is thus parameterized twice — by the quantity it describes and by a reference value X₀ — and the primaries disagree on what the thing even is: the VIM counts the decibel among the special names for units of dimension one (VIM 1.9 Note 3), while UCUM expels it from the unit group as a special unit on a logarithmic scale (§21). The theory subtree has no account at all — the torsor page records the fourfold silence of its sources — and practice is nearly as thin: Pint (documented Beta) and Unitful (@logscale; dB, Np, referenced levels like dBm) are the only shipped implementations in the survey, with mp-units carrying an in-source TODO ("how to support those? // neper // bel // decibel") and the rest silent (comparison § affine/log).

Angles and quantities of dimension one

"quantity of dimension one — quantity for which all the exponents of the factors corresponding to the base quantities in its quantity dimension are zero" — VIM 1.8

In the group reading this is the identity element, and everything at it is invariant under all rescalings — which is exactly why it is a dumping ground: length ratios, amounts-of-substance fractions, counts of entities, Mach numbers, and angles all land on the same group element, and Kennedy's Π-theorem-as-type-isomorphism says every unit-polymorphic program factors through it. The SI's treatment of angles is an explicit, documented compromise:

"For reasons of history and convention, plane and solid angles are treated within the SI as quantities with the unit one." — SI Brochure §2.3.3

with the kind problem conceded in Table 4's footnote on the radian: rad = m/m "is not intrinsic and may be misleading since angle is not the same kind of quantity as other length ratios" (note (b); mutatis mutandis for the steradian, note (c)). So rad and sr are dimension-one units with kind-restricted use — the same shape as Hz/Bq, one level down. The systems split three ways: follow the SI and pay the documented 2π trap between frequency and angular frequency (Pint); promote angle to a base dimension and leave the SI (Boost.Units's nine-base-unit system, Au's Angle); or tag kinds above an SI-conformant dimension (uom, mp-units). QUDT gives unit:RAD the D¹ slot of its eight-component vector plus an angle quantity kind — both workarounds at once. No option is cost-free; the trade-offs are the comparison's angle-policy row.

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The vocabulary at a glance

Term	VIM/SI says	The mathematical formalizations say	Where they part ways
Quantity	property with "a magnitude that can be expressed as a number and a reference" (VIM 1.1)	element of a model (Whitney); of a 1-D line (tensor); a typed bare rational (Kennedy)	metrology's reference can be a procedure or material; every formalization assumes unit-referenced ratio scales
Quantity value	"number and reference together" (VIM 1.19), a concept distinct from the quantity	Hart's (f, g) pair; Buckingham's measure-numbers — the pair is the primitive	half the formalizations axiomatize the quantity, half the quantity value; the trivialization gap between them is the torsor page's theorem
Unit	"real scalar quantity, defined and adopted by convention" (VIM 1.9) — itself a quantity, circularly	basis vector, torsor point, unit variable, or nothing at all — each formalization cuts the 1.1↔1.9 circle elsewhere	metrology tolerates the circularity; the formalizations must not, and where they cut it is what distinguishes them
Dimension	an expression: "product of powers of factors … omitting any numerical factor" (VIM 1.7)	an element of the free abelian group ℤⁿ (or ℚⁿ) on the base dimensions (fag)	exponent ring (ℤ/ℚ/ℝ) unsettled — VIM's own Example 3 is fractional while the SI says "generally small integers"
Kind of quantity	"aspect common to mutually comparable quantities" (VIM 1.2) — partitioning "to some extent arbitrary"	mostly dropped: the group element is the whole identity; Jonsson defines kind as commensurability	the survey's widest gap: metrology restricts Hz/Bq by kind; only engineering overlays (mp-units, uom, QUDT) recover it
System of quantities	quantities + "non-contradictory equations relating those quantities" (VIM 1.3); the ISQ	usually implicit — the equations become the grading's algebra; only the basis survives	mp-units and QUDT model it explicitly; every other system collapses it into the unit system
System of units	base + derived units + multiples "for a given system of quantities" (VIM 1.13); the SI	a section u : D → R of the dimension fibration — one torsor point per fiber (torsor)	the section is never canonical; metrology's "adopted by convention" is the theory's non-canonicity, agreed on by both sides
Coherent unit	product of powers of base units "with no other proportionality factor than one" (VIM 1.12; SI §2.3.4)	the section being multiplicative — numerical-value equations keep the form of quantity equations	prefixes break coherence in metrology; in the algebra a prefix is just another scalar — coherence is invisible to the group structure
Off-system / accepted	"does not belong to a given system of units" (VIM 1.15); Table 8's tolerated non-SI units	no concept — all units are interchangeable trivializations	registries (Pint) encode Table 8 as data; typed systems either normalize it away or carry exact factors
Dimension one	all exponents zero (VIM 1.8); rad, sr there "for reasons of history and convention"	the group identity; the target of the Π-theorem's factorization (pi, kennedy)	SI's own footnotes concede angle is "not the same kind" as other ratios; the identity element can't say so
Affine quantity (°C)	t = T − T₀; °C "only coherent when expressing temperature differences" (SI §2.3.4)	an additive torsor per fiber — a second torsor the graded picture does not supply (torsor)	UCUM calls Celsius temperature "in a unique way incoherent with the SI"; practice re-invents point/difference splits seven times over
Logarithmic unit (dB)	a special name for a dimension-one unit (VIM 1.9 Note 3); Table 8 with a footnote for a value (SI)	no account — a recorded fourfold silence (torsor)	VIM counts dB a unit; UCUM expels it from the unit group; the theory corpus says nothing at all

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Sources

• JCGM 200:2012 — International vocabulary of metrology — Basic and general concepts and associated terms (VIM), 3rd edition (identical to ISO/IEC Guide 99). Clauses 1.1–1.19 quoted throughout. (PDF; local: jcgm-2012-vim-3rd-ed.pdf.)
• BIPM — The International System of Units (SI), SI Brochure, 9th edition (2019; update V4.01, June 2026). §§2.3.1–2.3.4 and Table 4, §3, chapter 4/Table 8 with note (m). (publication page; local: bipm-2019-si-brochure-9th-ed.pdf.)
• NIST SP 811 (2008 edition) — Guide for the Use of the International System of Units (SI); the survey's open stand-in, together with the SI Brochure, for the paywalled ISO 80000-1:2022 (catalog page). §7.4 quoted. (DOI; local: nist-2008-sp811-guide-si.pdf.)
• UCUM — The Unified Code for Units of Measure specification. §§1–4 (grammar, case sensitivity), §§16–20 (semantics: the unit group, commensurability, dimensions), §§21–24 (special and arbitrary units). (spec; local: ucum-spec.html.)
• QUDT — schema SCHEMA_QUDT.ttl (class definitions and SHACL cardinalities) and vocabularies VOCAB_QUDT-QUANTITY-KINDS-ALL.ttl, VOCAB_QUDT-UNITS-ALL.ttl, VOCAB_QUDT-DIMENSION-VECTORS.ttl (HZ/BQ, DEG_C, RAD, torque/energy). (qudt.org · repository; local clone pinned at bb9e04d, 2026-07-02.)
• In-tree pages carry the survey-side citations quoted here: the theory deep-dives (Whitney, Buckingham π, free abelian group, tensor of lines, torsor, Kennedy, Hart, mechanisms), the fourteen system pages, and the comparison capstone.