Skip to content

Dimensionless Quantities

The quantities we discussed so far always had some specific type and physical dimension. However, this is not always the case. While performing various computations, we sometimes end up with so-called "dimensionless" quantities, which ISO defines as quantities of dimension one:

ISO/IEC Guide 99

  • Quantity for which all the exponents of the factors corresponding to the base quantities in its quantity dimension are zero.
  • 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.

Dividing two quantities of the same kind

Dividing two quantities of the same kind always results in a quantity of dimension one. However, depending on what type of quantities we divide or what their units are, we may end up with slightly different results.


In mp-units, dividing two quantities of the same dimension always results in a quantity with the dimension being dimension_one. This is often different for other physical units libraries, which may return a raw representation type for such cases. A raw value is also always returned from the division of two std::chrono::duration objects.

To read more about the reasoning for this design decision, please check our FAQ.

Dividing quantities of the same type

First, let's analyze what happens if we divide two quantities of the same type:

constexpr QuantityOf<dimensionless> auto q = isq::height(200 * m) / isq::height(50 * m);

In such a case, we end up with a dimensionless quantity that has the following properties:

static_assert(q.quantity_spec == dimensionless);
static_assert(q.dimension == dimension_one);
static_assert(q.unit == one);

In case we would like to print its value, we would see a raw value of 4 in the output with no unit being printed.

Dividing quantities of different types

Now let's see what happens if we divide quantities of the same dimension and unit but which have different quantity types:

constexpr QuantityOf<dimensionless> auto q = isq::work(200 * J) / isq::heat(50 * J);

Again we end up with dimension_one and one, but this time:

static_assert(q.quantity_spec == isq::work / isq::heat);

As shown above, the result is not of a dimensionless type anymore. Instead, we get a quantity type derived from the performed quantity equation. According to the ISQ, work divided by heat is the recipe for the thermodynamic efficiency quantity, thus:

static_assert(implicitly_convertible(q.quantity_spec, isq::efficiency_thermodynamics));


The quantity of isq::efficiency_thermodynamics is of a kind dimensionless, so it is implicitly convertible to dimensionless and satisfies the QuantityOf<dimensionless> concept.

Dividing quantities of different units

Now, let's see what happens when we divide two quantities of the same type but different units:

constexpr QuantityOf<dimensionless> auto q = isq::height(4 * km) / isq::height(2 * m);

This time, we still get a quantity of the dimensionless type with a dimension_one as its dimension. However, the resulting unit is not one anymore:

static_assert(q.unit == mag_power<10, 3> * one);

In case we would print the text output of this quantity, we would not see a raw value of 2000, but 2 km/m.

First, it may look surprising, but this is consistent with dividing quantities of different dimensions. For example, if we divide 4 * km / 2 * s, we do not expect km to be "expanded" to m before the division, right? We would expect the result of 2 km/s, which is exactly what we get when we divide quantities of the same kind.

This is a compelling feature that allows us to express huge or tiny ratios without the need for big and expensive representation types. With this, we can easily define things like a Hubble's constant that uses a unit that is proportional to the ratio of kilometers per megaparsecs, which are both units of length:

inline constexpr struct hubble_constant final :
    named_unit<{u8"H₀", "H_0"}, mag_ratio<701, 10> * si::kilo<si::metre> / si::second / si::mega<parsec>> {} hubble_constant;

Counts of things

Another important use case for dimensionless quantities is to provide strong types for counts of things. For example:

  • ISO-80000-3 provides a rotation quantity defined as the number of revolutions,
  • IEC-80000-6 provides a number of turns in a winding quantity,
  • IEC-80000-13 provides a Hamming distance quantity defined as the number of digit positions in which the corresponding digits of two words of the same length are different.

Thanks to assigning strong names to such quantities, later on, they can be explicitly used as arguments in the quantity equations of other quantities deriving from them.

Predefined units of the dimensionless quantity

As we observed above, the most common unit for dimensionless quantities is one. It has the ratio of 1 and does not output any textual symbol.

Important: one is an identity

A unit one is special in the entire type system of units as it is considered to be an identity operand in the unit expression templates. This means that, for example:

static_assert(one * one == one);
static_assert(one * si::metre == si::metre);
static_assert(si::metre / si::metre == one);

The same is also true for dimension_one and dimensionless in the domains of dimensions and quantity specifications.

Besides the unit one, there are a few other scaled units predefined in the library for usage with dimensionless quantities:

inline constexpr struct percent final : named_unit<"%", mag_ratio<1, 100> * one> {} percent;
inline constexpr struct per_mille final : named_unit<{u8"‰", "%o"}, mag_ratio<1, 1000> * one> {} per_mille;
inline constexpr struct parts_per_million final : named_unit<"ppm", mag_ratio<1, 1'000'000> * one> {} parts_per_million;
inline constexpr auto ppm = parts_per_million;

Angular quantities

Special, often controversial, examples of dimensionless quantities are an angular measure and solid angular measure quantities that are defined in the ISQ to be the result of a division of \(arc\; length / radius\) and \(area / radius^2\) respectively. Moreover, ISQ also explicitly states that both can be expressed in the unit one. This means that both angular measure and solid angular measure should be of a kind dimensionless.

On the other hand, ISQ also specifies that a unit radian can be used for angular measure, and a unit steradian can be used for solid angular measure. Those should not be mixed or used to express other types of dimensionless quantities. This means that both angular measure and solid angular measure should also be quantity kinds by themselves.


Many people claim that angle being a dimensionless quantity is a bad idea. There are proposals submitted to make an angle a base quantity and rad to become a base unit. More on this topic can be found in the "Strong Angular System" chapter.

Radians and degrees support

Thanks to the usage of magnitudes the library provides efficient strong types for all angular types. This means that with the built-in support for magnitudes of \(\pi\) we can provide accurate conversions between radians and degrees. The library also provides common trigonometric functions for angular quantities:

using namespace mp_units::si::unit_symbols;
using mp_units::angular::unit_symbols::rad;
using mp_units::angular::unit_symbols::deg;
using mp_units::angular::unit_symbols::grad;

quantity speed = 110 * km / h;
quantity rate_of_climb = -0.63657 * m / s;
quantity glide_ratio = speed / -rate_of_climb;
quantity glide_angle = angular::asin(1 / glide_ratio);

std::println("Glide ratio: {::N[.1f]}",;
std::println("Glide angle:");
std::println(" - {::N[.4f]}",;
std::println(" - {::N[.2f]}",;
std::println(" - {::N[.2f]}",;

The above program prints:

Glide ratio: 48.0
Glide angle:
 - 0.0208 rad
 - 1.19°
 - 1.33ᵍ


In the production code the above speed and rate_of_climb quantities should probably be modelled as separate typed quantities of the same kind.

Nested quantity kinds

Angular quantities are not the only ones with such a "strange" behavior. Another but a similar case is a storage capacity quantity specified in IEC-80000-13 that again allows expressing it in both one and bit units.

Those cases make dimensionless quantities an exceptional tree in the library. This is the only quantity hierarchy that contains more than one quantity kind in its tree:

flowchart TD
    dimensionless --- rotation
    dimensionless --- efficiency
    dimensionless --- angular_measure["angular_measure\n[rad]"]
    angular_measure --- rotational_displacement
    angular_measure --- phase_angle
    dimensionless --- solid_angular_measure["solid_angular_measure\n[sr]"]
    dimensionless --- drag_factor
    dimensionless --- storage_capacity["storage_capacity\n[bit]"] --- equivalent_binary_storage_capacity
    dimensionless --- ...

To provide such support in the library, we provided an is_kind specifier that can be appended to the quantity specification:

inline constexpr struct angular_measure final : quantity_spec<dimensionless, arc_length / radius, is_kind> {} angular_measure;
inline constexpr struct solid_angular_measure final : quantity_spec<dimensionless, area / pow<2>(radius), is_kind> {} solid_angular_measure;
inline constexpr struct storage_capacity final : quantity_spec<dimensionless, is_kind> {} storage_capacity;
inline constexpr struct angular_measure final : quantity_spec<angular_measure, dimensionless, arc_length / radius, is_kind> {} angular_measure;
inline constexpr struct solid_angular_measure final : quantity_spec<solid_angular_measure, dimensionless, area / pow<2>(radius), is_kind> {} solid_angular_measure;
inline constexpr struct storage_capacity final : quantity_spec<storage_capacity, dimensionless, is_kind> {} storage_capacity;
QUANTITY_SPEC(angular_measure, dimensionless, arc_length / radius, is_kind);
QUANTITY_SPEC(solid_angular_measure, dimensionless, area / pow<2>(radius), is_kind);
QUANTITY_SPEC(storage_capacity, dimensionless, is_kind);

With the above, we can constrain radian, steradian, and bit to be allowed for usage with specific quantity kinds only:

inline constexpr struct radian final : named_unit<"rad", metre / metre, kind_of<isq::angular_measure>> {} radian;
inline constexpr struct steradian final : named_unit<"sr", square(metre) / square(metre), kind_of<isq::solid_angular_measure>> {} steradian;
inline constexpr struct bit final : named_unit<"bit", one, kind_of<storage_capacity>> {} bit;

but still allow the usage of one and its scaled versions for such quantities.