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Systems of Quantities

The physical units libraries on the market typically only scope on modeling one or more systems of units. However, this is not the only system kind to model. Another, and maybe even more important, system kind is a system of quantities.


Please note that the mp-units is probably the first library on the Open Source market (in any programming language) that models the ISQ with all its definitions provided in ISO 80000. Please provide feedback if something looks odd or could be improved.

Dimension is not enough to describe a quantity

Most of the products on the market are aware of physical dimensions. However, a dimension is not enough to describe a quantity. For example, let's see the following implementation:

class Box {
  area base_;
  length height_;
  Box(length l, length w, length h) : base_(l * w), height_(h) {}
  // ...

Box my_box(2 * m, 3 * m, 1 * m);

How do you like such an interface? It turns out that in most existing strongly-typed libraries this is often the best we can do 🥴

Another typical question many users ask is how to deal with energy and moment of force. Both of those have the same dimension but are different quantities.

Another question is what should be the result of:

auto res = 1 * Hz + 1 * Bq + 1 * Bd;


  • Hz (hertz) - unit of frequency
  • Bq (becquerel) - unit of activity
  • Bd (baud) - unit of modulation rate

All of those quantities have the same dimension, namely \(\mathsf{T}^{-1}\), but probably it is not wise to allow adding, subtracting, or comparing them, as they describe vastly different physical properties.


More than one quantity may be defined for the same dimension:

  • quantities of different kinds (i.e. frequency, modulation rate, activity, ...)
  • quantities of the same kind (i.e. length, width, altitude, distance, radius, wavelength, position vector, ...)

It turns out that the above issues can't be solved correctly without proper modeling of a system of quantities.

Quantities of the same kind

ISO 80000-1

  • Quantities may be grouped together into categories of quantities that are mutually comparable
  • Mutually comparable quantities are called quantities of the same kind
  • Two or more quantities cannot be added or subtracted unless they belong to the same category of mutually comparable quantities
  • Quantities of the same kind within a given system of quantities have the same quantity dimension
  • Quantities of the same dimension are not necessarily of the same kind

The above quotes from ISO 80000 answer to all the issues above. Two quantities can't be added, subtracted, or compared unless they belong to the same kind. As frequency, activity, and modulation rate are different kinds, the expression provided above should not compile.

System of quantities is not only about kinds

ISO 80000 specify hundreds of different quantities. There are plenty of different kinds provided and often each kind contains more than one quantity. In fact, it turns out that such quantities form a hierarchy of quantities of the same kind.

For example, here are all quantities of the kind length provided in the ISO 80000:

flowchart TD
    length --- width[width, breadth]
    length --- height[height, depth, altitude]
    width --- thickness
    width --- diameter
    width --- radius
    length --- path_length
    path_length --- distance
    distance --- radial_distance
    length --- wavelength
    length --- position_vector["position_vector\n{vector}"]
    length --- displacement["displacement\n{vector}"]
    radius --- radius_of_curvature

Each of the above quantities expresses some kind of length, and each can be measured with si::metre. However, each of them has different properties, usage, and sometimes even a different representation type (notice that position_vector and displacement are vector quantities).

Analyzing such a hierarchy can help us in defining arithmetics and conversion rules.

Defining quantities

In the mp-units library all the information about the quantity is provided with the quantity_spec class template. In order to define a specific quantity a user should inherit a strong type from such an instantiation.


Quantity specification definitions benefit from an explicit object parameter added in C++23 to remove the need for CRTP idiom, which significantly simplifies the code. However, as C++23 is far from being mainstream today, a portability macro QUANTITY_SPEC() is provided and used consistently through the library to allow the code to compile with C++20 compilers, thanks to the CRTP usage under the hood.

For example, here is how the above quantity kind tree can be modeled in the library:

inline constexpr struct length : quantity_spec<dim_length> {} length;
inline constexpr struct width : quantity_spec<length> {} width;
inline constexpr auto breadth = width;
inline constexpr struct height : quantity_spec<length> {} height;
inline constexpr auto depth = height;
inline constexpr auto altitude = height;
inline constexpr struct thickness : quantity_spec<width> {} thickness;
inline constexpr struct diameter : quantity_spec<width> {} diameter;
inline constexpr struct radius : quantity_spec<width> {} radius;
inline constexpr struct radius_of_curvature : quantity_spec<radius> {} radius_of_curvature;
inline constexpr struct path_length : quantity_spec<length> {} path_length;
inline constexpr auto arc_length = path_length;
inline constexpr struct distance : quantity_spec<path_length> {} distance;
inline constexpr struct radial_distance : quantity_spec<distance> {} radial_distance;
inline constexpr struct wavelength : quantity_spec<length> {} wavelength;
inline constexpr struct position_vector : quantity_spec<length, quantity_character::vector> {} position_vector;
inline constexpr struct displacement : quantity_spec<length, quantity_character::vector> {} displacement;
inline constexpr struct length : quantity_spec<length, dim_length> {} length;
inline constexpr struct width : quantity_spec<width, length> {} width;
inline constexpr auto breadth = width;
inline constexpr struct height : quantity_spec<height, length> {} height;
inline constexpr auto depth = height;
inline constexpr auto altitude = height;
inline constexpr struct thickness : quantity_spec<thickness, width> {} thickness;
inline constexpr struct diameter : quantity_spec<diameter, width> {} diameter;
inline constexpr struct radius : quantity_spec<radius, width> {} radius;
inline constexpr struct radius_of_curvature : quantity_spec<radius_of_curvature, radius> {} radius_of_curvature;
inline constexpr struct path_length : quantity_spec<path_length, length> {} path_length;
inline constexpr auto arc_length = path_length;
inline constexpr struct distance : quantity_spec<distance, path_length> {} distance;
inline constexpr struct radial_distance : quantity_spec<radial_distance, distance> {} radial_distance;
inline constexpr struct wavelength : quantity_spec<wavelength, length> {} wavelength;
inline constexpr struct position_vector : quantity_spec<position_vector, length, quantity_character::vector> {} position_vector;
inline constexpr struct displacement : quantity_spec<displacement, length, quantity_character::vector> {} displacement;
QUANTITY_SPEC(length, dim_length);
QUANTITY_SPEC(width, length);
inline constexpr auto breadth = width;
QUANTITY_SPEC(height, length);
inline constexpr auto depth = height;
inline constexpr auto altitude = height;
QUANTITY_SPEC(thickness, width);
QUANTITY_SPEC(diameter, width);
QUANTITY_SPEC(radius, width);
QUANTITY_SPEC(radius_of_curvature, radius);
QUANTITY_SPEC(path_length, length);
inline constexpr auto arc_length = path_length;
QUANTITY_SPEC(distance, path_length);
QUANTITY_SPEC(radial_distance, distance);
QUANTITY_SPEC(wavelength, length);
QUANTITY_SPEC(position_vector, length, quantity_character::vector);
QUANTITY_SPEC(displacement, length, quantity_character::vector);


More information on how to define a system of quantities can be found in the "International System of Quantities (ISQ)" chapter.

Comparing, adding, and subtracting quantities

ISO 80000 explicitly states that width and height are quantities of the same kind, and as such they:

  • are mutually comparable
  • can be added and subtracted

If we take the above for granted, the only reasonable result of 1 * width + 1 * height is 2 * length, where the result of length is known as a common quantity type. A result of such an equation is always the first common branch in a hierarchy tree of the same kind. For example:

static_assert(common_quantity_spec(isq::width, isq::height) == isq::length);
static_assert(common_quantity_spec(isq::thickness, isq::radius) == isq::width);
static_assert(common_quantity_spec(isq::distance, isq::path_length) == isq::path_length);

Converting between quantities

Based on the same hierarchy of quantities of kind length, we can define quantity conversion rules.

  1. Implicit conversions

    • every width is a length
    • every radius is a width
    static_assert(implicitly_convertible(isq::width, isq::length));
    static_assert(implicitly_convertible(isq::radius, isq::length));
    static_assert(implicitly_convertible(isq::radius, isq::width));
  2. Explicit conversions

    • not every length is a width
    • not every width is a radius
    static_assert(!implicitly_convertible(isq::length, isq::width));
    static_assert(!implicitly_convertible(isq::length, isq::radius));
    static_assert(!implicitly_convertible(isq::width, isq::radius));
    static_assert(explicitly_convertible(isq::length, isq::width));
    static_assert(explicitly_convertible(isq::length, isq::radius));
    static_assert(explicitly_convertible(isq::width, isq::radius));
  3. Explicit casts

    • height is not a width
    • both height and width are quantities of kind length
    static_assert(!implicitly_convertible(isq::height, isq::width));
    static_assert(!explicitly_convertible(isq::height, isq::width));
    static_assert(castable(isq::height, isq::width));
  4. No conversion

    • time has nothing in common with length
    static_assert(!implicitly_convertible(isq::time, isq::length));
    static_assert(!explicitly_convertible(isq::time, isq::length));
    static_assert(!castable(isq::time, isq::length));

Hierarchies of derived quantities

Derived quantity equations often do not automatically form a hierarchy tree. This is why it is sometimes not obvious what such a tree should look like. Also, ISO explicitly states:

ISO/IEC Guide 99

The division of ‘quantity’ according to ‘kind of quantity’ is, to some extent, arbitrary.

The below presents some arbitrary hierarchy of derived quantities of kind energy:

flowchart TD
    energy["energy\n(mass * length^2 / time^2)"]
    energy --- mechanical_energy
    mechanical_energy --- potential_energy
    potential_energy --- gravitational_potential_energy["gravitational_potential_energy\n(mass / acceleration_of_free_fall / height)"]
    potential_energy --- elastic_potential_energy["elastic_potential_energy\n(spring_constant * amount_of_compression^2)"]
    mechanical_energy --- kinetic_energy["kinetic_energy\n(mass * speed^2)"]
    energy --- enthalpy
    enthalpy --- internal_energy[internal_energy, thermodynamic_energy]
    internal_energy --- Helmholtz_energy[Helmholtz_energy, Helmholtz_function]
    enthalpy --- Gibbs_energy[Gibbs_energy, Gibbs_function]
    energy --- active_energy

Notice, that even though all of those quantities have the same dimension and can be expressed in the same units, they have different quantity equations used to create them implicitly:

  • energy is the most generic one and thus can be created from base quantities of mass, length, and time. As those are also the roots of quantities of their kinds and all other quantities are implicitly convertible to them (we agreed on that "every width is a length" already), it means that an energy can be implicitly constructed from any quantity of mass, length, and time.

    static_assert(implicitly_convertible(isq::mass * pow<2>(isq::length) / pow<2>(isq::time), isq::energy));
    static_assert(implicitly_convertible(isq::mass * pow<2>(isq::height) / pow<2>(isq::time), isq::energy));
  • mechanical_energy is a more "specialized" quantity than energy (not every energy is a mechanical_energy). It is why an explicit cast is needed to convert from either energy or the results of its quantity equation.

    static_assert(!implicitly_convertible(isq::energy, isq::mechanical_energy));
    static_assert(explicitly_convertible(isq::energy, isq::mechanical_energy));
    static_assert(!implicitly_convertible(isq::mass * pow<2>(isq::length) / pow<2>(isq::time),
    static_assert(explicitly_convertible(isq::mass * pow<2>(isq::length) / pow<2>(isq::time),
  • gravitational_potential_energy is not only even more specialized one but additionally, it is special in a way that it provides its own "constrained" quantity equation. Maybe not every mass * pow<2>(length) / pow<2>(time) is a gravitational_potential_energy, but every mass * acceleration_of_free_fall * height is.

    static_assert(!implicitly_convertible(isq::energy, gravitational_potential_energy));
    static_assert(explicitly_convertible(isq::energy, gravitational_potential_energy));
    static_assert(!implicitly_convertible(isq::mass * pow<2>(isq::length) / pow<2>(isq::time),
    static_assert(explicitly_convertible(isq::mass * pow<2>(isq::length) / pow<2>(isq::time),
    static_assert(implicitly_convertible(isq::mass * isq::acceleration_of_free_fall * isq::height,

Modeling a quantity kind

In the physical units library, we also need an abstraction describing an entire family of quantities of the same kind. Such quantities have not only the same dimension but also can be expressed in the same units.

To annotate a quantity to represent its kind (and not just a hierarchy tree's root quantity) we introduced a kind_of<> specifier. For example, to express any quantity of length, we need to type kind_of<isq::length>.


isq::length and kind_of<isq::length> are two different things.

Such an entity behaves as any quantity of its kind. This means that it is implicitly convertible to any quantity in a tree.

static_assert(!implicitly_convertible(isq::length, isq::height));
static_assert(implicitly_convertible(kind_of<isq::length>, isq::height));

Additionally, the result of operations on quantity kinds is also a quantity kind:

static_assert(same_type<kind_of<isq::length> / kind_of<isq::time>, kind_of<isq::length / isq::time>>);

However, if at least one equation's operand is not a kind, the result becomes a "strong" quantity where all the kinds are converted to the hierarchy tree's root quantities:

static_assert(!same_type<kind_of<isq::length> / isq::time, kind_of<isq::length / isq::time>>);
static_assert(same_type<kind_of<isq::length> / isq::time, isq::length / isq::time>);


Only a root quantity from the hierarchy tree or the one marked with is_kind specifier in the quantity_spec definition can be put as a template parameter to the kind_of specifier. For example, kind_of<isq::width> will fail to compile.