# 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](../../appendix/glossary.md#dimensionless-quantity): !!! quote "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](../../appendix/glossary.md#dimensionless-quantity). However, depending on what type of quantities we divide or what their units are, we may end up with slightly different results. !!! note 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](../../getting_started/faq.md#why-a-dimensionless-quantity-is-not-just-a-fundamental-arithmetic-type). ### Dividing quantities of the same type First, let's analyze what happens if we divide two quantities of the same type: ```cpp constexpr QuantityOf 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: ```cpp 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: ```cpp constexpr QuantityOf auto q = isq::work(200 * J) / isq::heat(50 * J); ``` Again we end up with `dimension_one` and `one`, but this time: ```cpp 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](../../appendix/glossary.md#quantity-equation). According to the [ISQ](../../appendix/glossary.md#isq), _work_ divided by _heat_ is the recipe for the _thermodynamic efficiency_ quantity, thus: ```cpp static_assert(implicitly_convertible(q.quantity_spec, isq::efficiency_thermodynamics)); ``` !!! note The quantity of `isq::efficiency_thermodynamics` is of a kind `dimensionless`, so it is implicitly convertible to `dimensionless` and satisfies the `QuantityOf` concept. ### Dividing quantities of different units Now, let's see what happens when we divide two quantities of the same type but different units: ```cpp constexpr QuantityOf 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: ```cpp 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_](https://en.wikipedia.org/wiki/Hubble%27s_law#Dimensionless_Hubble_constant) that uses a unit that is proportional to the ratio of kilometers per megaparsecs, which are both units of _length_: ```cpp inline constexpr struct hubble_constant final : named_unit<{u8"H₀", "H_0"}, mag_ratio<701, 10> * si::kilo / si::second / si::mega> {} 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](../../appendix/glossary.md#quantity-equation) 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 "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](interface_introduction.md#identities). This means that, for example: ```cpp 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: ```cpp 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; ``` ### Superpowers of the unit `one` Quantities of the unit `one` are the only ones that are implicitly convertible from a raw value and explicitly convertible to it. This property also expands to usual arithmetic operators. Thanks to the above, we can type: ```cpp quantity inc(quantity q) { return q + 1; } void legacy(double) { /* ... */ } if (auto q = inc(42); q != 0) legacy(static_cast(q)); ``` ## Angular quantities Special, often controversial, examples of dimensionless quantities are an _angular measure_ and _solid angular measure_ quantities that are defined in the [ISQ](../../appendix/glossary.md#isq) to be the result of a division of $arc\; length / radius$ and $area / radius^2$ respectively. Moreover, [ISQ](../../appendix/glossary.md#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](../../appendix/glossary.md#kind) dimensionless. On the other hand, [ISQ](../../appendix/glossary.md#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](../../appendix/glossary.md#kind) by themselves. !!! note 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](../systems/strong_angular_system.md). ### 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: ```cpp 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]}", glide_ratio.in(one)); std::println("Glide angle:"); std::println(" - {::N[.4f]}", glide_angle.in(rad)); std::println(" - {::N[.2f]}", glide_angle.in(deg)); std::println(" - {::N[.2f]}", glide_angle.in(grad)); ``` The above program prints: ```text Glide ratio: 48.0 Glide angle: - 0.0208 rad - 1.19° - 1.33ᵍ ``` !!! note In the production code the above `speed` and `rate_of_climb` quantities should probably be modelled as [separate typed quantities of the same kind](systems_of_quantities.md#system-of-quantities-is-not-only-about-kinds). ## 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](../../appendix/glossary.md#quantity-hierarchy) that contains more than one [quantity kind](../../appendix/glossary.md#kind) in its tree: ```mermaid flowchart TD dimensionless["dimensionless\n[one]"] 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: === "C++23" ```cpp inline constexpr struct angular_measure final : quantity_spec {} angular_measure; inline constexpr struct solid_angular_measure final : quantity_spec(radius), is_kind> {} solid_angular_measure; inline constexpr struct storage_capacity final : quantity_spec {} storage_capacity; ``` === "C++20" ```cpp inline constexpr struct angular_measure final : quantity_spec {} angular_measure; inline constexpr struct solid_angular_measure final : quantity_spec(radius), is_kind> {} solid_angular_measure; inline constexpr struct storage_capacity final : quantity_spec {} storage_capacity; ``` === "Portable" ```cpp 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: ```cpp inline constexpr struct radian final : named_unit<"rad", metre / metre, kind_of> {} radian; inline constexpr struct steradian final : named_unit<"sr", square(metre) / square(metre), kind_of> {} steradian; inline constexpr struct bit final : named_unit<"bit", one, kind_of> {} bit; ``` but still allow the usage of `one` and its scaled versions for such quantities.