# Glossary¶

## ISO definitions¶

Note

The ISO terms provided below are only a few of many defined in the ISO/IEC Guide 99.

quantity
• Property of a phenomenon, body, or substance, where the property has a magnitude that can be expressed by means of a number and a reference.
• A reference can be a measurement unit, a measurement procedure, a reference material, or a combination of such.
• A quantity as defined here is a scalar. However, a vector or a tensor, the components of which are quantities, is also considered to be a quantity.
• The concept ’quantity’ may be generically divided into, e.g. ‘physical quantity’, ‘chemical quantity’, and ‘biological quantity’, or ‘base quantity’ and ‘derived quantity’.
• Examples of quantities are: length, radius, wavelength, energy, electric charge, etc.
kind of quantity, kind
• Aspect common to mutually comparable quantities.
• The division of the concept ‘quantity’ into several kinds is to some extent arbitrary, for example:
• the quantities diameter, circumference, and wavelength are generally considered to be quantities of the same kind, namely, of the kind of quantity called length,
• the quantities heat, kinetic energy, and potential energy are generally considered to be quantities of the same kind, namely of the kind of quantity called energy.
• 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.
• For 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.
system of quantities
base quantity
• Quantity in a conventionally chosen subset of a given system of quantities, where no quantity in the subset can be expressed in terms of the others.
• Base quantities are referred to as being mutually independent since a base quantity cannot be expressed as a product of powers of the other base quantities.
• ‘Number of entities’ can be regarded as a base quantity in any system of quantities.
derived quantity
International System of Quantities, ISQ
quantity dimension, dimension of a 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.
• e.g. in the ISQ, the quantity dimension of force is denoted by $$\textsf{dim }F = \mathsf{LMT}^{–2}$$.
• A power of a factor is the factor raised to an exponent. Each factor is the dimension of a base quantity.
• In deriving the dimension of a quantity, no account is taken of its scalar, vector, or tensor character.
• In a given system of quantities:
• quantities of the same kind have the same quantity dimension,
• quantities of different quantity dimensions are always of different kinds,
• quantities having the same quantity dimension are not necessarily of the same kind.
• Symbols representing the dimensions of the base quantities in the ISQ are:

Base quantity Symbol for dimension
length $$\mathsf{L}$$
mass $$\mathsf{M}$$
time $$\mathsf{T}$$
electric current $$\mathsf{I}$$
thermodynamic temperature $$\mathsf{Θ}$$
amount of substance $$\mathsf{N}$$
luminous intensity $$\mathsf{J}$$

Thus, the dimension of a quantity $$Q$$ is denoted by $$\textsf{dim }Q = \mathsf{L}^α\mathsf{M}^β\mathsf{T}^γ\mathsf{I}^δ\mathsf{Θ}^ε\mathsf{N}^ζ\mathsf{J}^η$$ where the exponents, named dimensional exponents, are positive, negative, or zero.

quantity of dimension one, dimensionless quantity
• quantity for which all the exponents of the factors corresponding to the base quantities in its quantity dimension are zero.
• The term “dimensionless quantity” is commonly used and is kept here for historical reasons. It stems from the fact that all exponents are zero in the symbolic representation of the dimension for such quantities. The term “quantity of dimension one” reflects the convention in which the symbolic representation of the dimension for such quantities is the symbol $$1$$.
• The measurement units and values of quantities of dimension one are numbers, but such quantities convey more information than a number.
• Some quantities of dimension one are defined as the ratios of two quantities of the same kind.
• Numbers of entities are quantities of dimension one.
measurement unit, unit of 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.
• Measurement units are designated by conventionally assigned names and symbols.
• Measurement units of quantities of the same quantity dimension may be designated by the same name and symbol even when the quantities are not of the same kind.
• For example, joule per kelvin and J/K are respectively the name and symbol of both a measurement unit of heat capacity and a measurement unit of entropy, which are generally not considered to be quantities of the same kind.
• However, in some cases special measurement unit names are restricted to be used with quantities of specific kind only.
• For example, the measurement unit ‘second to the power minus one’ ($$\mathsf{1/s}$$) is called hertz ($$\mathsf{Hz}$$) when used for frequencies and becquerel ($$\mathsf{Bq}$$) when used for activities of radionuclides. As another example, the joule ($$\mathsf{J}$$) is used as a unit of energy, but never as a unit of moment of force, e.g. the newton metre ($$\mathsf{N·m}$$).
• Measurement units of quantities of dimension one are numbers. In some cases, these measurement units are given special names, e.g. radian, steradian, and decibel, or are expressed by quotients such as millimole per mole equal to $$10^{−3}$$ and microgram per kilogram equal to $$10^{−9}$$.
base unit
derived unit
• Measurement unit for a derived quantity.
• For example, the metre per second, symbol m/s, and the centimetre per second, symbol cm/s, are derived units of speed in the SI. The kilometre per hour, symbol km/h, is a measurement unit of speed outside the SI but accepted for use with the SI. The knot, equal to one nautical mile per hour, is a measurement unit of speed outside the SI.
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.
• A power of a base unit is the base unit raised to an exponent.
• Coherence can be determined only with respect to a particular system of quantities and a given set of base units.
• For example, if the metre, the second, and the mole are base units, the metre per second is the coherent derived unit of velocity when velocity is defined by the quantity equation $$v = \mathsf{d}r/\mathsf{d}t$$, and the mole per cubic metre is the coherent derived unit of amount-of-substance concentration when amount-of-substance concentration is defined by the quantity equation $$c = n/V$$. The kilometre per hour and the knot, given as examples of derived units, are not coherent derived units in such a system of quantities.
• A derived unit can be coherent with respect to one system of quantities but not to another.
• For example, the centimetre per second is the coherent derived unit of speed in a CGS system of units but is not a coherent derived unit in the SI.
• The coherent derived unit for every derived quantity of dimension one in a given system of units is the number one, symbol $$1$$. The name and symbol of the measurement unit one are generally not indicated.
system of units
coherent system of units
off-system measurement unit, off-system unit
• Measurement unit that does not belong to a given system of units.
• For example, the electronvolt (about $$1.602\;18 × 10^{–19}\;\mathsf{J}$$) is an off-system measurement unit of energy with respect to the SI. Day, hour, minute are off-system measurement units of time with respect to the SI.
International System of Units, SI
• System of units, based on the International System of Quantities, their names and symbols, including a series of prefixes and their names and symbols, together with rules for their use, adopted by the General Conference on Weights and Measures (CGPM).
quantity value, value of a quantity, value
• Number and reference together expressing magnitude of a quantity.
• For example, length of a given rod: $$5.34\;\mathsf{m}$$ or $$534\;\mathsf{cm}$$.
• The number can be complex.
• A quantity value can be presented in more than one way.
• In the case of vector or tensor quantities, each component has a quantity value.
• For example, force acting on a given particle, e.g. in Cartesian components $$(F_x; F_y; F_z) = (−31.5; 43.2; 17.0)\;\mathsf{N}$$.
numerical quantity value, numerical value of a quantity, numerical value
• Number in the expression of a quantity value, other than any number serving as the reference
• For example, in an amount-of-substance fraction equal to $$3\;\mathsf{mmol/mol}$$, the numerical quantity value is $$3$$ and the unit is $$\mathsf{mmol/mol}$$. The unit $$\mathsf{mmol/mol}$$ is numerically equal to $$0.001$$, but this number $$0.001$$ is not part of the numerical quantity value, which remains $$3$$.
quantity equation
unit equation
• Mathematical relation between base units, coher­ent derived units or other measurement units.
• For example, $$\mathsf{J} := \mathsf{kg}\:\mathsf{m}^2/\mathsf{s}^2$$, where, $$\mathsf{J}$$, $$\mathsf{kg}$$, $$\mathsf{m}$$, and $$\mathsf{s}$$ are the symbols for the joule, kilogram, metre, and second, respectively. (The symbol $$:=$$ denotes “is by definition equal to” as given in the ISO 80000 and IEC 80000 series.). $$1\;\mathsf{km/h} = (1/3.6)\;\mathsf{m/s}$$.
numerical value equation, numerical quantity value equation
• Mathematical relation between numerical quantity values, based on a given quantity equation and specified measurement units.
• For example, in the quantity equation for kinetic energy of a particle, $$T = (1/2) mv^2$$, if $$m = 2\;\mathsf{kg}$$ and $$v = 3\;\mathsf{m/s}$$, then $${T} = (1/2)\:×\:2\:×\:3^2$$ is a numerical value equation giving the numerical value $$9$$ of $$T$$ in joules.

## Other definitions¶

Info

The below terms extend the official ISO glossary and are commonly referred to by the mp-units library.

base dimension
derived dimension
dimension equation
quantity kind hierarchy, quantity hierarchy
• Quantities of the same kind form a hierarchy that determines their:
• convertibility (e.g. every width is a length, but width should not be convertible to height)
• common quantity type (e.g. width + height -> length)
quantity character, character of a quantity, character
• Scalars, vectors and tensors are mathematical objects that can be used to denote certain physical quantities and their values. They are as such independent of the particular choice of a coordinate system, whereas each scalar component of a vector or a tensor and each component vector and component tensor depend on that choice.
• A vector is a tensor of the first order and a scalar is a tensor of order zero.
• For vectors and tensors, the components are quantities that can be expressed as a product of a number and a unit.
• Vectors and tensors can also be expressed as a numerical value vector or tensor, respectively, multiplied by a unit.
• Quantities of different characters support different set of operations.
• For example, a quantity can be multiplied by another one only if any of them has scalar character. Vectors and tensors can't be multiplied or divided, but they support additional operations like dot and cross products, which are not available for scalars.
• The term ’character’ was borrowed from the below quote:

ISO 80000-1_2009

In deriving the dimension of a quantity, no account is taken of its scalar, vector, or tensor character.

quantity specification, quantity_spec
unit with an associated quantity, associated unit
quantity reference, reference
• According to its definition, quantity can be expressed by means of a number and a reference
• In the mp-units library, a reference describes all the required meta-information associated with a specific quantity (quantity specification and unit).
canonical representation of a unit, canonical unit
• A canonical representation of a unit consists of:
• a reference unit being the result of extraction of all the intermediate derived units,
• a magnitude being a product of all the prefixes and magnitudes of extracted scaled units.
• All units having the same canonical unit are deemed equal.
• All units having the same reference unit are convertible (their magnitude may differ and is used during conversion).
reference unit
absolute quantity point origin, absolute point origin
• An explicit point on an axis of values of a specific quantity type that serves as an absolute reference point for all quantity points which definitions are (explicitly or implicitly) based on it.
• For example, mean sea level is commonly used as an absolute reference point to measure altitudes.
relative quantity point origin, relative point origin
• An explicit, known at compile-time, point on an axis of values of a specific quantity type serving as a reference for other quantities.
• For example, an ice point is a quantity point with a value of $$273.15\;\mathsf{K}$$ that is used as the zero point of a degree Celsius scale.
quantity point origin, point origin
quantity point, absolute quantity
• An absolute quantity with respect to an origin.
• For example, timestamp (as opposed to duration), altitude (as opposed to height), absolute temperature (as opposed to temperature difference).