# 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.

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.

### Dividing quantities of the same type¶

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

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:

Again we end up with `dimension_one`

and `one`

, but this time:

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:

Note

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:

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:

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;
```

### Superpowers of the unit `one`

¶

Quantities of the unit `one`

are the only ones that are:

- implicitly constructible from the raw value,
- explicitly convertible to a raw value,
- comparable to a raw value.

```
quantity<one> inc(quantity<one> q) { return q + 1; }
void legacy(double) { /* ... */ }
if (auto q = inc(42); q != 0)
legacy(static_cast<int>(q));
```

This property also expands to usual arithmetic operators.

Note

Those rules do not apply to all the dimensionless quantities. It would be unsafe and misleading
to allow such operations on units with a magnitude different than `1`

(e.g., `percent`

or
`radian`

).

## 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.

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.

### 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]}", 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:

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.

## 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 quantity hierarchy contains more than one quantity kind and more than one unit in its tree:

```
flowchart TD
dimensionless["<b>dimensionless</b><br>[one]"]
dimensionless --- rotation["<b>rotation</b>"]
dimensionless --- thermodynamic_efficiency["<b>thermodynamic_efficiency</b><br><i>(work / heat)</i>"]
dimensionless --- angular_measure["<b>angular_measure</b><br><i>(arc_length / radius)</i><br>[rad]"]
angular_measure --- rotational_displacement["<b>rotational_displacement</b><br><i>(path_length / radius)</i>"]
angular_measure --- phase_angle["<b>phase_angle</b>"]
dimensionless --- solid_angular_measure["<b>solid_angular_measure</b><br><i>(area / pow<2>(radius))</i><br>[sr]"]
dimensionless --- drag_factor["<b>drag_factor</b><br><i>(drag_force / (mass_density * pow<2>(speed) * area))</i>"]
dimensionless --- storage_capacity["<b>storage_capacity</b><br>[bit]"] --- equivalent_binary_storage_capacity["<b>equivalent_binary_storage_capacity</b>"]
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;
```

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.