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International System of Quantities (ISQ)

The International System of Quantities (ISQ) is the system of quantities used as the basis for the International System of Units (SI) and other systems of units.

Info

For a general introduction to systems of quantities and their importance, see Systems of Quantities in the Framework Basics section.

The ISQ is defined across the ISO/IEC 80000 series, which currently includes parts 1-17 covering quantities and units in various domains.

mp-units provides comprehensive support for the ISQ, likely making it the first library (in any programming language) to model the complete ISO/IEC 80000 quantity specification.

Base Quantities and Dimensions

The ISQ defines seven base quantities, each with its own dimension. According to the SI Brochure:

SI Brochure (9th edition)

Physical quantities can be organized in a system of dimensions, where the system used is decided by convention. Each of the seven base quantities used in the SI is regarded as having its own dimension.

The mp-units library defines these base dimensions as follows:

namespace mp_units::isq {

inline constexpr struct dim_length : base_dimension<"L"> {} dim_length;
inline constexpr struct dim_mass : base_dimension<"M"> {} dim_mass;
inline constexpr struct dim_time : base_dimension<"T"> {} dim_time;
inline constexpr struct dim_electric_current : base_dimension<"I"> {} dim_electric_current;
inline constexpr struct dim_thermodynamic_temperature : base_dimension<symbol_text{u8"Θ", "O"}> {} dim_thermodynamic_temperature;
inline constexpr struct dim_amount_of_substance : base_dimension<"N"> {} dim_amount_of_substance;
inline constexpr struct dim_luminous_intensity : base_dimension<"J"> {} dim_luminous_intensity;

}
Base Quantity Dimension Symbol mp-units Identifier
length L isq::dim_length
mass M isq::dim_mass
time T isq::dim_time
electric current I isq::dim_electric_current
thermodynamic temperature Θ isq::dim_thermodynamic_temperature
amount of substance N isq::dim_amount_of_substance
luminous intensity J isq::dim_luminous_intensity

These dimension symbols are consistently defined by both the SI Brochure and ISO/IEC 80000.

Base and Derived Quantities

For each base dimension, the ISQ defines corresponding base quantities:

namespace mp_units::isq {

inline constexpr struct length : quantity_spec<dim_length, non_negative> {} length;
inline constexpr struct mass : quantity_spec<dim_mass, non_negative> {} mass;
inline constexpr struct time : quantity_spec<dim_time, non_negative> {} time;
inline constexpr struct electric_current : quantity_spec<dim_electric_current> {} electric_current;
inline constexpr struct thermodynamic_temperature : quantity_spec<dim_thermodynamic_temperature, non_negative> {} thermodynamic_temperature;
inline constexpr struct amount_of_substance : quantity_spec<dim_amount_of_substance, non_negative> {} amount_of_substance;
inline constexpr struct luminous_intensity : quantity_spec<dim_luminous_intensity, non_negative> {} luminous_intensity;

}

Derived Quantities

The SI Brochure states:

SI Brochure (9th edition)

Since the number of quantities is without limit, it is not possible to provide a complete list of derived quantities and derived units.

However, ISO/IEC 80000 defines hundreds of standardized quantities across many domains. The mp-units library provides definitions for all quantities specified in the ISO/IEC 80000 series, organized by part:

  • Part 3: Space and time (e.g., velocity, acceleration, area, volume)
  • Part 4: Mechanics (e.g., force, energy, power, pressure)
  • Part 5: Thermodynamics (e.g., entropy, heat_capacity)
  • Part 6: Electromagnetism (e.g., voltage, resistance, capacitance)
  • Part 7: Light and radiation (partial support)
  • Part 8: Acoustics (TBD)
  • Part 9: Physical chemistry and molecular physics (TBD)
  • Part 10: Atomic and nuclear physics (TBD)
  • Part 11: Characteristic numbers (TBD)
  • Part 12: Condensed matter physics (TBD)
  • Part 13: Information science and technology (e.g., traffic_intensity, storage_capacity, transfer_rate)
  • Part 14: Telebiometrics related to human physiology (TBD)

Example: Derived Quantities in Mechanics

namespace mp_units::isq {

// Kinematics
inline constexpr struct speed : quantity_spec<length / duration> {} speed;
inline constexpr struct velocity : quantity_spec<speed, displacement / duration> {} velocity;
inline constexpr struct acceleration : quantity_spec<velocity / duration> {} acceleration;

// Dynamics
inline constexpr struct force : quantity_spec<mass * acceleration> {} force;
inline constexpr struct pressure : quantity_spec<force / area, quantity_tensor_order::scalar> {} pressure;
inline constexpr struct energy : quantity_spec<mass* pow<2>(length) / pow<2>(time)> {} energy;
inline constexpr struct power : quantity_spec<mass* pow<2>(length) / pow<3>(time)> {} power;

// Many more...

}

Quantity Hierarchies

As described in Systems of Quantities, the ISQ organizes quantities of the same kind into hierarchies. For example, quantities of kind length:

flowchart TD
    length["<b>length</b><br>[m]"]
    length --- width["<b>width</b> | <b>breadth</b>"]
    length --- height["<b>height</b> | <b>depth</b> | <b>altitude</b>"]
    width --- thickness["<b>thickness</b>"]
    width --- diameter["<b>diameter</b>"]
    width --- radius["<b>radius</b>"]
    length --- path_length["<b>path_length</b>"]
    path_length --- distance["<b>distance</b>"]
    distance --- radial_distance["<b>radial_distance</b>"]
    length --- wavelength["<b>wavelength</b>"]
    length --- displacement["<b>displacement</b><br>{vector}"]
    displacement --- position_vector["<b>position_vector</b>"]
    radius --- radius_of_curvature["<b>radius_of_curvature</b>"]

In code:

inline constexpr struct length : quantity_spec<dim_length, non_negative> {} 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 displacement : quantity_spec<length, quantity_tensor_order::vector> {} displacement;
inline constexpr struct position_vector : quantity_spec<displacement> {} position_vector;

This hierarchy enables strong type safety:

using namespace mp_units::si::unit_symbols;

quantity<isq::height[m]> tower_height = 42 * m;
quantity<isq::distance[m]> walking_distance = 500 * m;

// quantity<isq::width[m]> w = tower_height;  // Compile-time error!
quantity<isq::length[m]> some_length = tower_height;  // OK: height is-a length

Usage with Units

ISQ quantities are typically paired with SI units (or other compatible unit systems):

using namespace mp_units;
using namespace mp_units::si::unit_symbols;

// Using the ISQ-SI pairing
quantity<isq::length[m]> distance = 100 * m;
quantity<isq::speed[m / s]> velocity = distance / (10 * s);
quantity<isq::force[N]> force = isq::mass(5 * kg) * isq::acceleration(9.81 * m / s2);

The library automatically ensures dimensional consistency and provides meaningful error messages when incompatible operations are attempted.

Using QuantityOf Concept

The QuantityOf concept allows constraining function parameters to accept quantities of a specific kind, leveraging the hierarchy for type safety:

using namespace mp_units;
using namespace mp_units::si::unit_symbols;

// Function accepting any length quantity
void process_length(QuantityOf<isq::length> auto length)
{
  std::cout << "Length: " << length << "\n";
}

// Function specifically for height measurements
void set_height(QuantityOf<isq::height> auto h)
{
  std::cout << "Height set to: " << h << "\n";
}

// Function for distances only
double unsafe_travel_time_in_s(QuantityOf<isq::distance> auto d, QuantityOf<isq::speed> auto v)
{
  return (d / v).numerical_value_in(s);
}

// Usage
quantity tower = isq::height(42 * m);
quantity road = isq::distance(500 * m);
quantity river = 10 * isq::width[m];

process_length(tower);   // OK: height is-a length
process_length(road);    // OK: distance is-a length
process_length(river);   // OK: width is-a length

set_height(tower);       // OK: height matches exactly
// set_height(road);     // Compile-time error: distance is not a height
// set_height(river);    // Compile-time error: width is not a height

double time = unsafe_travel_time_in_s(road, 50. * km / h);  // OK
// unsafe_travel_time_in_s(river, 50. * km / h);  // Compile-time error: width is not a distance

This provides powerful compile-time checking that enforces the semantic meaning of quantities, not just their dimensions. A width cannot be used where a height is required, even though both are lengths.

Photometric Conditions

ISO 80000-7 defines every luminous quantity for several photometric conditions: photopic vision (cone cells, daylight), scotopic vision (rod cells, night), and mesopic vision (both, twilight). Each condition weights the radiometric spectrum with a different spectral luminous efficiency function, so a photopic and a scotopic value of the same source differ by a spectrum-dependent factor, while both are expressed in the same units (lm, cd, lx). A calculation that mixes values of two conditions is wrong by a factor that depends on the light source, and because the units match, no dimensional analysis can catch it.

This is why the library models the condition as a template argument on the luminous quantities and keeps every condition in its own quantity hierarchy. Following the standard, an unspecified condition means photopic vision, so the plain names (isq::luminous_flux, si::lumen) are the photopic entities and existing code works unchanged:

quantity daylight = 1000. * si::lumen;
quantity night = 250. * si::lumen_of<isq::scotopic_vision>;

// auto sum = daylight + night;        // Compile-time error: different quantity kinds
quantity sp_ratio = night / daylight;  // OK: the S/P ratio, 0.25 lm/lm

Quantities of different conditions can never be added, compared, or converted to each other (there is no spectrum-independent conversion factor), while their products and quotients stay well-formed. The set of conditions is open: users may derive their own tags (e.g. the CIE S 026 α-opic weightings) from isq::photometric_condition_base.

A condition argument on a unit does not create a different unit

SI defines exactly one candela, lumen, and lux, and the standard distinguishes the conditions by the quantity symbols, never by the units. si::lumen_of<C> is the SI lumen: it prints as lm, its conversion factor to si::lumen is exactly one, and its photopic instantiation is si::lumen itself. The condition argument only states which quantity kind the unit is associated with. This association is what preserves quantity-kind safety in simple mode, where the unit is the only carrier of the quantity semantics:

quantity night = 250. * si::lumen_of<isq::scotopic_vision>;  // no quantity spec spelled

In a quotient of two conditions the lumens deliberately do not cancel: the lm/lm output records that quantities of two different kinds were divided, the same way the SI Brochure keeps unit ratios like mg/kg on dimensionless quantities to convey information about the quantities involved. When a pure number is wanted, the reduction is an explicit .in(one).

The condition safety applies to all quantities with a dimension. The dimensionless luminous ratios (luminous reflectance, luminous transmittance, luminance factor) remain distinct quantity specs per condition with no implicit conversion between them, but they add and compare through dimensionless like every other efficiency or factor in the library. Absorbance (items 7-32.1 and 7-32.2) is a logarithmic quantity and is not provided yet, and the colorimetric items 7-26 to 7-28 (tristimulus values, colour-matching functions, chromaticity coordinates) are triples of tabulated functions rather than scalar quantities of a single kind, so they are not modelled as quantity specs.

See Working with Photometric Conditions for the recipes, including bridging between conditions with a known S/P ratio.

References