Systems of Quantities¶

Most physical units libraries focus on modeling one or more systems of units. However an equally (or more) important abstraction is the system of quantities.

Info

mp-units is likely the first Open Source library (in any language) that models the ISQ with the full ISO 80000 definition set. Feedback is welcome.

Dimension is not enough to describe a quantity¶

Most libraries understand dimensions, yet a dimension alone does not fully describe a quantity. Consider:

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

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

This interface is ambiguous. Many strongly typed libraries cannot do better

Another common question: how to differentiate work and torque? They share a dimension yet differ semantically.

A similar issue is related to figuring out what should be the result of:

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

where:

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

All have the same dimension \(\mathsf{T}^{-1}\), but adding or comparing them is meaningless.

Consider fuel consumption (fuel volume divided by distance, e.g. 6.7 l/km) vs an area. Both have dimension \(\mathsf{L}^{2}\) yet adding them is nonsensical and should fail.

Important

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

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

These issues require 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

ISO 80000 answers the earlier questions: two quantities cannot be added, subtracted, or compared unless they are of the same kind. Thus frequency, activity, and modulation rate are incompatible.

System of quantities is not only about kinds¶

ISO 80000 specifies hundreds of quantities in many kinds; kinds often contain multiple quantities forming a hierarchy.

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

flowchart TD
    length["<b>length</b><br>[m]"]
    length --- width["<b>width</b> | <b>breadth</b>"]
    length --- altitude["<b>altitude | <b>depth</b></b>†"]
    altitude --- height["<b>height</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>"]

V2 Workaround: Reversed Hierarchy for Signed Coordinates

† In the ISO 80000 model, height is the parent with altitude and depth as children. However, mp-units V2 temporarily reverses this: altitude and depth are children of length, while height (explicitly tagged non_negative) is a child of altitude.

Rationale: altitude and depth represent signed vertical coordinates (positions relative to a reference plane), while height is an unsigned magnitude. The reversed hierarchy enables implicit height → altitude conversions in affine space operations (e.g., mean_sea_level + height_value works without explicit casts).

This workaround will be removed in V3 when the point_for<> mechanism becomes available, allowing the proper ISO hierarchy with altitude/depth as point_for<height> types.

Special is_non_negative() overloads ensure altitude and depth remain unbounded despite inheriting from non-negative length.

Each quantity above expresses some kind of length and can be measured with si::metre. Each has different semantics and sometimes a distinct representation (e.g. position_vector and displacement are vector quantities).

The hierarchy guides valid arithmetic and conversion rules for quantities of the same kind.

Defining quantities¶

All quantity information resides in quantity_spec. To define a quantity inherit a strong type from a suitable instantiation.

Tip

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.

See more in the C++ compiler support chapter.

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

C++23C++20Portable

inline constexpr struct length final : quantity_spec<dim_length> {} length;
inline constexpr struct width final : quantity_spec<length> {} width;
inline constexpr auto breadth = width;
// V2 workaround: altitude/depth as children of length, height as child of altitude
inline constexpr struct altitude final : quantity_spec<length> {} altitude;
inline constexpr auto depth = altitude;
inline constexpr struct height final : quantity_spec<altitude, non_negative> {} height;
inline constexpr struct thickness final : quantity_spec<width> {} thickness;
inline constexpr struct diameter final : quantity_spec<width> {} diameter;
inline constexpr struct radius final : quantity_spec<width> {} radius;
inline constexpr struct radius_of_curvature final : quantity_spec<radius> {} radius_of_curvature;
inline constexpr struct path_length final : quantity_spec<length> {} path_length;
inline constexpr auto arc_length = path_length;
inline constexpr struct distance final : quantity_spec<path_length> {} distance;
inline constexpr struct radial_distance final : quantity_spec<distance> {} radial_distance;
inline constexpr struct wavelength final : quantity_spec<length> {} wavelength;
inline constexpr struct displacement final : quantity_spec<length, quantity_character::vector> {} displacement;
inline constexpr struct position_vector final : quantity_spec<displacement> {} position_vector;

inline constexpr struct length final : quantity_spec<length, dim_length> {} length;
inline constexpr struct width final : quantity_spec<width, length> {} width;
inline constexpr auto breadth = width;
// V2 workaround: altitude/depth as children of length, height as child of altitude
inline constexpr struct altitude final : quantity_spec<altitude, length> {} altitude;
inline constexpr auto depth = altitude;
inline constexpr struct height final : quantity_spec<height, altitude, non_negative> {} height;
inline constexpr struct thickness final : quantity_spec<thickness, width> {} thickness;
inline constexpr struct diameter final : quantity_spec<diameter, width> {} diameter;
inline constexpr struct radius final : quantity_spec<radius, width> {} radius;
inline constexpr struct radius_of_curvature final : quantity_spec<radius_of_curvature, radius> {} radius_of_curvature;
inline constexpr struct path_length final : quantity_spec<path_length, length> {} path_length;
inline constexpr auto arc_length = path_length;
inline constexpr struct distance final : quantity_spec<distance, path_length> {} distance;
inline constexpr struct radial_distance final : quantity_spec<radial_distance, distance> {} radial_distance;
inline constexpr struct wavelength final : quantity_spec<wavelength, length> {} wavelength;
inline constexpr struct displacement final : quantity_spec<displacement, length, quantity_character::vector> {} displacement;
inline constexpr struct position_vector final : quantity_spec<position_vector, displacement> {} position_vector;

QUANTITY_SPEC(length, dim_length);
QUANTITY_SPEC(width, length);
inline constexpr auto breadth = width;
// V2 workaround: altitude/depth as children of length, height as child of altitude
QUANTITY_SPEC(altitude, length);
inline constexpr auto depth = altitude;
QUANTITY_SPEC(height, altitude, non_negative);
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(displacement, length, quantity_character::vector);
QUANTITY_SPEC(position_vector, displacement);

Note

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 states that width and height are quantities of the same kind; therefore 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 node in a hierarchy tree of the same kind. For example:

static_assert(get_common_quantity_spec(isq::width, isq::height) == isq::length);
static_assert(get_common_quantity_spec(isq::thickness, isq::radius) == isq::width);
static_assert(get_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.

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::width));
static_assert(implicitly_convertible(isq::radius, isq::length));

Implicit conversions are allowed on copy-initialization:

void foo(quantity<isq::length[m]> q);

quantity<isq::width[m]> q1 = 42 * m;
quantity<isq::length[m]> q2 = q1;  // implicit quantity conversion
foo(q1);                           // implicit quantity conversion

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::width, isq::radius));
static_assert(!implicitly_convertible(isq::length, isq::radius));
static_assert(explicitly_convertible(isq::length, isq::width));
static_assert(explicitly_convertible(isq::width, isq::radius));
static_assert(explicitly_convertible(isq::length, isq::radius));

Explicit conversions are forced by passing the quantity to a call operator of a quantity_spec type or by calling quantity's explicit constructor:

void foo(quantity<isq::height[m]> q);

quantity<isq::length[m]> q1 = 42 * m;
quantity<isq::height[m]> q2 = isq::height(q1);  // explicit quantity conversion
quantity<isq::height[m]> q3(q1);                // direct initialization
foo(isq::height(q1));                           // explicit quantity conversion

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));

Explicit casts are forced with a dedicated quantity_cast function:

void foo(quantity<isq::height[m]> q);

quantity<isq::width[m]> q1 = 42 * m;
quantity<isq::height[m]> q2 = quantity_cast<isq::height>(q1);  // explicit quantity cast
foo(quantity_cast<isq::height>(q1));                           // explicit quantity cast

No conversion

time has nothing in common with length

static_assert(!implicitly_convertible(isq::duration, isq::length));
static_assert(!explicitly_convertible(isq::duration, isq::length));
static_assert(!castable(isq::duration, isq::length));

Even the explicit casts will not force such a conversion:

void foo(quantity<isq::length[m]>);

quantity<isq::length[m]> q1 = 42 * s;    // Compile-time error
foo(quantity_cast<isq::length>(42 * s)); // Compile-time error

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["<b>energy</b><br><i>(mass * length<sup>2</sup> / time<sup>2</sup>)</i><br>[J]"]
    energy --- signal_energy_per_binary_digit["<b>signal_energy_per_binary_digit</b><br><i>(carrier_power * period_of_binary_digits)</i>"]
    energy --- mechanical_work["<b>mechanical_work</b><br><i>(force * displacement)</i>"]
    mechanical_work --- mechanical_energy["<b>mechanical_energy</b><br><i>(mass * length<sup>2</sup> / time<sup>2</sup>)</i>"]
    mechanical_energy --- potential_energy["<b>potential_energy</b>"]
    potential_energy --- gravitational_potential_energy["<b>gravitational_potential_energy</b><br><i>(mass * acceleration_of_free_fall * height)</i>"]
    potential_energy --- elastic_potential_energy["<b>elastic_potential_energy</b><br><i>(spring_constant * amount_of_compression<sup>2</sup>)</i>"]
    mechanical_energy --- kinetic_energy["<b>kinetic_energy</b><br><i>(mass * speed<sup>2</sup>)</i>"]
    energy --- radiant_energy["<b>radiant_energy</b>"]
    energy --- internal_energy["<b>internal_energy</b> | <b>thermodynamic_energy</b>"]
    internal_energy --- Helmholtz_energy["<b>Helmholtz_energy</b> | <b>Helmholtz_function</b>"]
    internal_energy --- enthalpy["<b>enthalpy</b>"]
    enthalpy --- Gibbs_energy["<b>Gibbs_energy</b> | <b>Gibbs_function</b>"]
    internal_energy --- heat["<b>heat</b> | <b>amount_of_heat</b>"]
    heat --- latent_heat["<b>latent_heat</b>"]
    energy --- active_energy["<b>active_energy</b><br><i>(instantaneous_power * time)</i>"]

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 that can be 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 from their trees 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::duration), isq::energy));
static_assert(implicitly_convertible(isq::mass * pow<2>(isq::height) / pow<2>(isq::duration), 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::duration),
                                      isq::mechanical_energy));
static_assert(explicitly_convertible(isq::mass * pow<2>(isq::length) / pow<2>(isq::duration),
                                     isq::mechanical_energy));

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

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>.

Important

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::duration>, kind_of<isq::length / isq::duration>>);

However, if at least one equation's operand is not a quantity 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::duration, kind_of<isq::length / isq::duration>>);
static_assert(same_type<kind_of<isq::length> / isq::duration, isq::length / isq::duration>);

Info

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. However, we can call get_kind(q) to obtain a kind of any quantity:

static_assert(get_kind(isq::width) == kind_of<isq::length>);

Creating distinct quantity kinds with `is_kind`¶

While dimension-based type safety prevents many errors, sometimes quantities share the same dimension but represent fundamentally incompatible physical concepts. The is_kind specifier allows creating distinct quantity types that cannot be mixed even though they share the same dimension and quantity hierarchy tree.

When to use `is_kind`?¶

Use is_kind to create distinct subkinds within an existing quantity hierarchy when:

Multiple incompatible concepts need to share the same parent quantity's properties (unit or quantity type)
These concepts cannot be meaningfully added or compared to each other without explicit conversion
They represent different reference frames or measurement contexts, but derive from the same physical basis

The key insight: use is_kind when quantities need to inherit from a parent (quantity type, unit) but must be isolated from each other.

Common examples of subkinds within existing trees include:

Angular measure (rad), solid angular measure (sr), storage capacity (bit) — subkind of dimensionless
Fluid head and water head in hydraulic engineering — subkinds of height (dimension of length)

Defining a distinct kind¶

Important

The is_kind specifier creates subkinds within an existing quantity hierarchy tree, not independent trees. This allows the subkind to inherit properties from its parent:

Unit of measure: fluid head and water head inherit metre from height; angular measure inherits one from dimensionless
Quantity type: Subkinds inherit their parent's quantity type, which is crucial when they appear in derived quantities involving this quantity (e.g., sampling rate, tempo can use Hz because they properly model the dimensionless component divided by duration)

For quantities that should be completely independent (different dimension trees), define separate root quantities instead (e.g., frequency and activity are independent roots, not subkinds).

To create a distinct quantity kind as a subkind, add the is_kind specifier to the quantity_spec definition:

C++23C++20Portable

inline constexpr struct fluid_head final : quantity_spec<isq::height, is_kind> {} fluid_head;
inline constexpr struct water_head final : quantity_spec<isq::height, is_kind> {} water_head;
// Both inherit metre as unit and length as dimension from isq::height

inline constexpr struct fluid_head final : quantity_spec<fluid_head, isq::height, is_kind> {} fluid_head;
inline constexpr struct water_head final : quantity_spec<water_head, isq::height, is_kind> {} water_head;
// Both inherit metre as unit and length as dimension from isq::height

QUANTITY_SPEC(fluid_head, isq::height, is_kind);
QUANTITY_SPEC(water_head, isq::height, is_kind);
// Both inherit metre as unit and length as dimension from isq::height

Both fluid_head and water_head are subkinds of height (inheriting its dimension of length and unit of metre), but marking them with is_kind makes them distinct incompatible kinds that require explicit conversion.

Behavior of `is_kind` quantities¶

Quantities marked with is_kind behave differently from regular hierarchy members:

Cannot be implicitly converted to each other:

static_assert(!implicitly_convertible(fluid_head, water_head));
static_assert(!explicitly_convertible(fluid_head, water_head));
static_assert(!castable(fluid_head, water_head));

Cannot be added or compared directly:

quantity h_fluid = fluid_head(2 * m);
quantity h_water = water_head(10 * m);

// auto sum = h_fluid + h_water;  // Compile-time error!
// bool cmp = h_fluid < h_water;  // Compile-time error!

Require explicit conversion to base quantity:

To perform generic operations or conversions between kinds, explicit conversion to the base quantity is required:

// Convert to base quantity explicitly
quantity h1 = isq::height(h_fluid);  // explicit conversion required
quantity h2 = isq::height(h_water);  // explicit conversion required

// Now generic operations are possible
quantity sum = h1 + h2;  // OK: both are isq::height

Warning

Implicit conversion from is_kind quantities to their base is not allowed:

quantity<isq::height[m]> h = h_fluid;  // Compile-time error!

Can be used with kind_of:

Unlike regular hierarchy members, is_kind quantities can be used with kind_of:

static_assert(get_kind(fluid_head) == kind_of<fluid_head>);
static_assert(get_kind(water_head) == kind_of<water_head>);
static_assert(get_kind(isq::height) == kind_of<isq::length>);
// static_assert(get_kind(isq::height) == kind_of<isq::height>);  // Compile-time error!

// Both are kinds of height, but different kinds
static_assert(get_kind(fluid_head) != get_kind(water_head));
static_assert(get_kind(fluid_head) != get_kind(isq::height));

Implementing physics-based conversions¶

When quantities are distinct kinds, domain-specific conversion functions should be provided to perform the correct physics-based transformations (if applicable):

// Define specific gravity as dimensionless
inline constexpr struct specific_gravity final : quantity_spec<dimensionless> {} specific_gravity;

// Physics: H_water = H_fluid * SG
constexpr QuantityOf<water_head> auto to_water_head(QuantityOf<fluid_head> auto h_fluid,
                                                    QuantityOf<specific_gravity> auto sg)
{
  return water_head(isq::height(h_fluid) * sg);
}

// Physics: H_fluid = H_water / SG
constexpr QuantityOf<fluid_head> auto to_fluid_head(QuantityOf<water_head> auto h_water,
                                                    QuantityOf<specific_gravity> auto sg)
{
  return fluid_head(isq::height(h_water) / sg);
}

This pattern:

Makes conversions explicit and visible in the code
Encodes the physics (specific gravity conversion formula)
Provides type-safe boundaries via QuantityOf constraints
Documents the relationship between different quantity kinds

Guidelines for using `is_kind`¶

Use is_kind when:

Quantities share a parent but have fundamentally different physical meanings
Adding or comparing them is physically nonsensical (e.g., plane angles + solid angles, fluid head + water head)
You need compile-time prevention of a known category of errors
Conversions between kinds either don't exist (plane vs solid angles) or require domain-specific formulas (fluid head ↔ water head via specific gravity)

Don't use is_kind when:

Quantities are naturally part of the same hierarchy (use regular quantity_spec hierarchy)
Conversions are just unit changes (use regular unit conversions)
The distinction is purely semantic without different physics (document in comments instead)

Tip

For a complete practical example demonstrating how is_kind prevents catastrophic engineering errors in hydraulic systems, see Workshop: Preventing Confusion with Distinct Kinds.

Note

Special dimensionless quantity kinds like angular measure, solid angular measure, and storage capacity are discussed in detail in the Dimensionless Quantities chapter.

Non-negative quantities¶

Many physical quantities are inherently non-negative. For example, length, mass, and thermodynamic temperature cannot have negative values in their physical domains. The library models this with the non_negative property tag:

C++23C++20Portable

inline constexpr struct length final : quantity_spec<dim_length, non_negative> {} length;
inline constexpr struct mass final : quantity_spec<dim_mass, non_negative> {} mass;
inline constexpr struct duration final : quantity_spec<dim_time, non_negative> {} duration;
// electric_current is NOT non_negative (current can flow in either direction)
inline constexpr struct electric_current final : quantity_spec<dim_electric_current> {} electric_current;

inline constexpr struct length final : quantity_spec<length, dim_length, non_negative> {} length;
inline constexpr struct mass final : quantity_spec<mass, dim_mass, non_negative> {} mass;
inline constexpr struct duration final : quantity_spec<duration, dim_time, non_negative> {} duration;
// electric_current is NOT non_negative (current can flow in either direction)
inline constexpr struct electric_current final : quantity_spec<electric_current, dim_electric_current> {} electric_current;

QUANTITY_SPEC(length, dim_length, non_negative);
QUANTITY_SPEC(mass, dim_mass, non_negative);
QUANTITY_SPEC(duration, dim_time, non_negative);
// electric_current is NOT non_negative (current can flow in either direction)
QUANTITY_SPEC(electric_current, dim_electric_current);

Propagation through equations¶

The non_negative property automatically propagates through derived quantity equations. A derived quantity is non_negative when all factors in its defining equation are non_negative:

// area = length² → non_negative (length is non_negative)
static_assert(is_non_negative(isq::area));

// speed = length / duration → non_negative (both are non_negative)
static_assert(is_non_negative(isq::speed));

// energy = mass * length² / duration² → non_negative (all factors are non_negative)
static_assert(is_non_negative(isq::energy));

When any factor in the equation is not non_negative, the derived quantity is not either:

// electric_current is not non_negative, so:
// electric_charge = electric_current * duration → not non_negative
static_assert(!is_non_negative(isq::electric_charge));

Named real-scalar children inherit from parents¶

A named child quantity automatically inherits non_negative from its parent, as long as the child's character is real_scalar. This reflects the physical reality: every height IS a length, so if length is non-negative then every specific type of length must be too:

C++23C++20Portable

inline constexpr struct width final : quantity_spec<length> {} width;
inline constexpr struct height final : quantity_spec<length> {} height;
inline constexpr struct radius final : quantity_spec<width> {} radius;
static_assert(is_non_negative(width));
static_assert(is_non_negative(height));
static_assert(is_non_negative(radius));

inline constexpr struct width final : quantity_spec<width, length> {} width;
inline constexpr struct height final : quantity_spec<height, length> {} height;
inline constexpr struct radius final : quantity_spec<radius, width> {} radius;
static_assert(is_non_negative(width));
static_assert(is_non_negative(height));
static_assert(is_non_negative(radius));

QUANTITY_SPEC(width,  length);  // inherits non_negative from length
QUANTITY_SPEC(height, length);  // inherits non_negative from length
QUANTITY_SPEC(radius, width);   // inherits non_negative from width → from length
static_assert(is_non_negative(width));
static_assert(is_non_negative(height));
static_assert(is_non_negative(radius));

Important

Once non_negative is applied to a parent quantity, the constraint propagates unconditionally and irremovably to all real-scalar descendants. There is no mechanism for a child to opt out. This is physically sound: a more specific quantity cannot validly produce values outside the domain of its parent. No exception to this rule is known for the real-scalar quantities defined in the ISQ framework — every named child of a non-negative quantity is itself non-negative by physical necessity.

Quantities of vector, complex, or tensor character cannot be non-negative by definition (they are direction-sensitive or multi-component). Applying the non_negative tag to such a quantity produces a compile-time error:

C++23C++20Portable

// displacement is a child of length with vector character — correctly NOT non_negative
inline constexpr struct displacement final : quantity_spec<length, quantity_character::vector> {} displacement;
static_assert(!is_non_negative(displacement));

// compile-time error — non_negative is incompatible with vector character:
// inline constexpr struct bad final : quantity_spec<length, quantity_character::vector, non_negative> {} bad;  // ← error

// displacement is a child of length with vector character — correctly NOT non_negative
inline constexpr struct displacement final : quantity_spec<displacement, length, quantity_character::vector> {} displacement;
static_assert(!is_non_negative(displacement));

// compile-time error — non_negative is incompatible with vector character:
// inline constexpr struct bad final : quantity_spec<bad, length, quantity_character::vector, non_negative> {} bad;  // ← error

// displacement is a child of length with vector character — correctly NOT non_negative
QUANTITY_SPEC(displacement, length, quantity_character::vector);
static_assert(!is_non_negative(displacement));

// compile-time error — non_negative is incompatible with vector character:
// QUANTITY_SPEC(bad, length, quantity_character::vector, non_negative);  // ← error

Kinds are never non-negative

A kind_of<QS> represents the entire quantity tree rooted at QS — including vector quantities (e.g., displacement) and signed coordinates (e.g., altitude, depth). Therefore, kind_of<QS> is never non-negative, even when QS itself is tagged non_negative:

static_assert(is_non_negative(isq::length));           // ✓ tagged as non_negative
static_assert(!is_non_negative(kind_of<isq::length>)); // ✗ kind encompasses signed subtypes

This matters when using CTAD with bare SI units, as they deduce kind_of origins:

quantity_point generic{5.0 * m};  // origin = natural_point_origin<kind_of<isq::length>>
                                  // NOT auto-bounded (kind is not non-negative)

Always use an explicit quantity specification when you need non-negative guarantees.

Systems of Quantities¶

Dimension is not enough to describe a quantity¶

Quantities of the same kind¶

System of quantities is not only about kinds¶

Defining quantities¶

Comparing, adding, and subtracting quantities¶

Converting between quantities¶

Hierarchies of derived quantities¶

Modeling a quantity kind¶

Creating distinct quantity kinds with is_kind¶

When to use is_kind?¶

Defining a distinct kind¶

Behavior of is_kind quantities¶

Implementing physics-based conversions¶

Guidelines for using is_kind¶

Non-negative quantities¶

Propagation through equations¶

Named real-scalar children inherit from parents¶

Creating distinct quantity kinds with `is_kind`¶

When to use `is_kind`?¶

Behavior of `is_kind` quantities¶

Guidelines for using `is_kind`¶