Quantitative
When working with physical quantities, such as lengths, masses or temperatures,
it can be easy to mix up quantities with different units, especially if we
represent all quantities with Double
s, which is often necessary for
performance.
Quantitative represents physical quantities with a generic Quantity
type, an
opaque alias of Double
, which statically encodes the value’s units in its
type parameter. This provides all the desirable homogeneity constraints when
combining quantities, with the performance of Double
s, and without
compromising on intuitive syntax for arithmetic operations.
Quantities can be multiplied and divided arbitrarily, with new units computed by the compiler, and checked for consistency in additions and subtractions.
All Quantitative terms and types are defined in the quantitative
package,
import
quantitative
.
*
and exported to the soundness
package:
import
soundness
.
*
Quantity
types
Physical quantities can be represented by different Quantity
types, with an appropriate parameter that encodes
the value’s units. We can create a quantity by multiplying an existing Double
(or any numeric type) by some
unit value, such as Metre
or Joule
, which are just Quantity
values equal to 1.0
of the appropriate unit.
For example:
val
distance
=
58.3
*
Metre
The types of these values will be inferred. The value distance
will get the type Quantity[Metres[1]]
, since
its value is a number of metres (raised to the power 1
).
In general, types representing units are written in the plural (for example, Metres
, Feet
, Candelas
), with
a bias for distinction when the singular name is often used in the plural; for example, the type is Kelvins
even though “Kelvins” and “Kelvin” are both commonly used for plural values. Unit instances are always named in
the singular.
We can compute an area
value by squaring the distance,
val
area
=
distance
*
distance
which should have units of square metres (m²
). Quantitative represents this as the type, Quantity[Metres[2]]
; the
2
singleton literal value represents the metres being squared. Likewise, a volume would have the parameter
Metres[3]
.
Representation and displaying
Each quantity, regardless of its units, is represented in the JVM as a Double
using an opaque type
alias.
The precise types, representing units, are known statically, but are erased by runtime. Hence, all
dimensionality checking takes place at compiletime, after which, operations on Quantity
s will be
operations on Double
s, and will achieve similar performance.
The raw Double
value of a Quantity
can always be obtained with Quantity#value
Due to this representation, the toString
method on Quantity
s is the same as Double
s toString
,
so the toString
representations will show just the raw numerical value, without any units. In
general, toString
should not be used. A gossamer.Show
instance is provided to produce
humanreadable Text
values, so calling show
on a Quantity
will produce much better output.
Derived units
We can also define:
val
energy
=
Joule
*
28000
The type of the energy
value could have been defined as Quantity[Joule[1]]
, but 1 J is equivalent to 1
kg⋅m²⋅s¯², and it’s more useful for the type to reflect a product of thes more basic units (even though we
can still use the Joule
value to construct it).
Metres, seconds and kilograms are all SI base units. Kilograms are a little different, since nominally, a kilogram is one thousand grams (while a gram is not an SI base unit), and this has a small implication on the way we construct such units.
Quantitative provides general syntax for metric naming conventions, allowing prefixes such as Nano
or Mega
to be applied to existing unit values to specify the appropriate scale to the value. Hence, a kilogram value
is written, Kilo(Gram)
. But since the SI base unit is the kilogram, this and any other multiple of Gram
,
such as Micro(Gram)
, will use the type Kilogram
, or more precisely, Kilogram[1]
.
Therefore, the type of energy
is Quantity[Grams[1] & Metres[2] & Second[2]]
, using a combination of three
base units raised to different powers. They are combined into an intersection type with the &
type operator,
which provides the useful property that the order of the intersection is unimportant;
Second[2] & Metres[2] & Grams[1]
is an identical type, much as kg m²s¯² and s¯²m²kg are identical
units.
Just as we could construct an area by multiplying two lengths, we can compute a new value with appropriate units
by combining, say, area
and energy
,
val
volume
=
distance
*
distance
*
distance
val
energyDensity
=
energy
/
volume
and its type will be inferred with the parameter Kilogram[1] & Metres[1] & Second[2]
.
If we had instead calculated energy/area
, whose units do not include metres, the type parameter would be just
Kilogram[1] & Second[2]
; the redundant Metres[0]
would be automatically removed from the conjunction.
We can go further. For example, the ”SUVAT” equations of motion can be safely implemented as methods, and their dimensionality will be checked at compiletime. For example, the equation,
$s=ut+\frac{1}{2}a{t}^{2}$
calculating a distance (s
) from an initial velocity (u
), acceleration (a
) and time (t
) can be
implemented using Quantitative Quantity
s with,
type
Velocity
=
Quantity
[
Metres
[
1
]
&
Seconds
[

1
]
]
type
Time
=
Quantity
[
Seconds
[
1
]
]
type
Acceleration
=
Quantity
[
Metres
[
1
]
&
Seconds
[

2
]
]
type
Distance
=
Quantity
[
Metres
[
1
]
]
def
s
(
u
:
Velocity
,
t
:
Time
,
a
:
Acceleration
)
:
Distance
=
u
*
t
+
0.5
*
a
*
t
*
t
or more verbosely,
def
distance
(
velocity0
:
Quantity
[
Metres
[
1
]
&
Seconds
[

1
]
]
,
time
:
Quantity
[
Seconds
[
1
]
]
,
acceleration
:
Quantity
[
Metres
[
1
]
&
Seconds
[

2
]
]
)
:
Quantity
[
Metres
[
1
]
]
=
velocity0
*
time
+
0.5
*
acceleration
*
time
*
time
While the method arguments have more complex types, the expression, u*t + 0.5*a*t*t
, is checked for
dimensional consistency. If we had written t + 0.5*a*t*t
or u*t + 0.5*a*a*t
instead, these would
have produced errors at compiletime.
Combining mixed units
Kilograms, metres and seconds are units of in the mass, length and time dimensions, which are never interchangeable. Yet we sometimes need to work with different units of the same dimension, such as feet, metres, yards and miles as different (but interchangeable) units of length; or kilograms and pounds, as units of mass.
Each type representing units, such as Metres
or Kilograms
, must be a subtype of the Units
type,
which is parameterized with its power (with a singleton literal integer) and a dimension, i.e. another type
representing the nature of the measurement. For Metres
the dimension is Length
; for Kilograms
it is
Mass
; Candela
’s is Luminosity
.
Metres[PowerType]
is a subtype of Units[PowerType, Length]
, where PowerType
must be a singleton
integer type. More specifically, Metres[1]
would be a subtype of Units[1, Length]
.
Note that there are no special dimensions for compound units, like energy, since the time, length and mass
components of the units of an energy quantity will be associated with the Second
, Metres
and Kilogram
types respectively.
Encoding the dimension in the type makes it possible to freely mix different units of the same dimension.
It is possible to create new length or mass units, such as Inch
or Pound
, which share the Length
or Mass
dimensions. This allows them to be considered equivalent in some calculations, if a conversion coefficient is
available.
Quantitative defines a variety of imperial measurements, and will automatically convert units of the same dimension to the same units in multiplications and divisions. For example,
val
width
=
0.3
*
Metre
val
height
=
5
*
Inch
val
area2
=
width
*
height
will infer the type Quantity[Metres[2]]
for area
.
However, the conversion of one of the units from inches to metres was necessary only to avoid a mixture of
Inches
and Metres
in the resultant type, but the expression, height*height
would produce a value with the
units, Inches[2]
, performing no unnecessary conversions.
Conversions
Addition & subtraction
Addition and subtraction are possible between quantities which share the same dimension.
We can safely add an inch and a metre,
val
length
=
1
*
Inch
+
1
*
Metre
but we can’t subtract a second from a litre:
For the addition and subtraction of values with mixed units, the question arises of which units the result
should take. Quantitative will use the principal unit for the dimension, which is determined by the presence
of a unique contextual PrincipalUnit
instance, parameterized on Dimension
and Units
types.
In general, if the units for the same dimension don’t match between the operands, then the principal unit
will be used for both. This may mean that adding a foot to a mile produces a result measured in metres,
but a new PrincipalUnit[Length, Miles[1]]()
contextual value could always be provided inscope,
which will take precedence over the PrincipalUnit[Length, Metres[1]]
in scope.
Some additional contextual values may be required, though. See below for more information on conversions.
Inequality Comparisons
Likewise, we can compare units in like or mixed values with the four standard inequality operators
(<
, >
, <=
, >=
). These will return true
or false
if the operands have the same dimension,
even if they have different units, for example,
while incompatible units will result in a compile error.
Equality
Equality between different Quantity
values should be treated with care, since all such values are
represented as Double
s at runtime, and the JVM’s standard equality will not take units into
account. So, by default, 3*Foot == 3*Metre
will yield true
, since 3.0 == 3.0
!
This is highly undesirable, but luckily there’s a solution:
import
language
.
strictEquality
This turns on Scala’s strictequality feature, which forbids comparisons between any two types unless
a corresponding CanEqual[LeftOperandType, RightOperandType]
exists in scope for the appropriate
operand types. Quantitative provides just such an instance for Quantity
instances with the same units.
The runtime equality check, however, is performed in exactly the same way: by comparing two Double
s.
That is absolutely fine if we know the units are identical, but it does not allow equality comparisons
between Quantity
s of the same dimension and different units.
For this, there are two possibilities:

convert one of the
Quantity
s to the units of the other 
test
left <= right && left >= right
, which will only be true ifleft
equalsright
Conversion ratios
In order to automatically convert between two units, Quantitative needs to know the ratio between them.
This is provided with a contextual Ratio
value for the appropriate pair of units: one with the
power 1
and the other with the power 1
. The rate of conversion should be specified as a singleton
literal Double
as the second parameter. The given
may be erased
, if using Scala’s erased definitions.
For example,
which specifies that there are about 1016 kilograms in a ton, and will be used if Quantitative ever needs to convert between kilograms and tons.
By making the conversion rate a type (a singleton literal, specifically), its value is available at
compiletime, even while the given
is erased
. This has the further advantage that any calculations on
Quantity
s which need to use the conversion ratio in a calculation involving other constants will use
constant folding to automatically perform arithmetic operations on constants at compiletime, saving the
performance cost of doing these at runtime.
Explicit Conversions
To convert a quantity to different units, we can use the in
method, passing it an unapplied units type
constructor, such as Hour
or Furlong
. The significance of the type being “unapplied” is that a units type
constructor is typically applied to an integer singleton type, such as Metres[2]
representing square
metres. Each dimension in a quantity must have the same units, no matter what its power, so it doesn’t make
sense to specify that power when converting.
So, (10*Metre).in[Yards]
, would create a value representing approximately 10.94 yards, while,
(3*Foot * 1*Metre * 0.4*Centi(Metre)).in[Inches]
, would calculate a volume in cubic inches.
If a quantity includes units in multiple dimensions, these can be converted in steps, for example,
val
distance2
=
100
*
Metre
val
time
=
9.8
*
Second
val
speed
=
distance2
/
time
SI definitions
There are seven SI base dimensions, with corresponding units, which are defined by Quantitative:

Length
with units type,Metres
, and unit value,Metre

Mass
with units,Kilograms
, and unit value,Kilogram

Time
with units,Seconds
, and unit value,Second

Current
with units,Amperes
, and unit value,Ampere

Luminosity
with units,Candelas
, and unit value,Candela

AmountOfSubstance
with units,Moles
, and unit value,Mole

Temperature
with units,Kelvins
, and unit value,Kelvin
As well as these, the following SI derived unit values are defined in terms of the base units:

Hertz
, for measuring frequency, as one per second 
Newton
, for measuring force, as one metrekilogram per square second 
Pascal
, for measuring pressure, as one Newton per square metre 
Joule
, for measuring energy, as one Newtonmetre 
Watt
, for measuring power, as one Joule per second 
Coulomb
, for measuring electric charge, as one secondAmpere 
Volt
, for measuring electric potential, as one Watt per Ampere 
Farad
, for measuring electrical capacitance, as one Coulomb per Volt 
Ohm
, for measuring electrical resistance, as one Volt per Ampere 
Siemens
, for measuring electrical conductance, as one Ampere per Volt 
Weber
, for measuring magnetic flux, as one Voltsecond 
Tesla
, for measuring magnetic flux density, as one Weber per square metre 
Henry
, for measuring electrical inductance, as one Weber per Ampere 
Lux
, for measuring illuminance, as one Candela per square metre 
Becquerel
, for measuring radioactivity, as one per second 
Gray
, for measuring ionizing radiation dose, as one Joule per kilogram 
Sievert
, for measuring stochastic health risk of ionizing radiation, as one Joule per kilogram 
Katal
, for measuring catalytic activity, as one mole per second
Defining your own units
Quantitative provides implementations of a variety of useful (and some less useful) units from the metric system, CGS and imperial. It’s also very easy to define your own units.
Imagine we wanted to implement the FLOPS unit, for measuring the floatingpoint performance of a CPU: floatingpoint instructions per second.
Trivially, we could create a value,
val
SimpleFlop
=
1.0
/
Second
and use it in equations such as, 1000000*SimpleFlop * Minute
to yield an absolute number representing
the number of floatingpoint instructions that could (theoretically) be calculated in one minute by
a onemegaFLOP CPU.
But this definition is just a value, not a unit. We can tweak the definition slightly to,
val
Flop
=
MetricUnit
(
1.0
/
Second
)
and it becomes possible to use metric prefixes on the value. So we could rewrite the above expression
as, Mega(Flop) * Minute
.
Introducing new dimensions
The result is just a Double
, though, which is a little unsatisfactory, since it represents
something more specific: a number of instructions. To do better, we need to introduce a new
Dimension
, distinct from length, mass and other dimensions, and representing a CPU’s
performance,
trait
CpuPerformance
extends
Dimension
and create a Flops
type corresponding to this dimension:
import
rudiments
.
*
trait
Flops
[
PowerType
<:
Nat
]
extends
Units
[
PowerType
,
CpuPerformance
]
val
Flop
:
MetricUnit
[
Flops
[
1
]
]
=
MetricUnit
(
1
)
The type parameter, PowerType
, is a necessary part of this definition, and must be constrained on
the Nat
type defined in Rudiments, which is just an
alias for Int & Singleton
. If you are using Scala’s erased definitions, both CpuPerformance
and
Flops
may be made erased trait
s to reduce the bytecode size slightly.
With these definitions, we can now write Mega(Flop) * Minute
to get a result with the dimensions
”FLOPSseconds”, represented by the type, Quantity[Flops[1] & Seconds[1]]
.
If we want to show the FLOPS value as Text
, a symbolic name is required. This can be specified
with a contextual instance of UnitName[Flops[1]]
,
given
UnitName
[
Flops
[
1
]
]
=
(
)
=>
t
"FLOPS"
which will allow show
to be called on a quantity involving FLOPs.
Describing physical quantities
English provides many names for physical quantities, including the familiar base dimensions of length, mass, time and so on, as well as combinations of these, such as velocity, acceleration and electrical resistance.
Definitions of names for many of these physical quantities are already defined, and will appear in error messages when a mismatch occurs.
It is also possible to define your own, for example, here is the definition for “force”:
erased
given
DimensionName
[
Units
[
1
,
Mass
]
&
Units
[
1
,
Length
]
&
Units
[

2
,
Time
]
,
"force"
]
=
erasedValue
The singleton type "force"
is the provided name for any units corresponding to the dimensions,
mass×length×time¯².
Substituting simplified units
While the SI base units can be used to describe the units of most physical quantities, there often
exist simpler forms of their units. For example, the Joule, J
, is equal to kg⋅m²⋅s¯²
, and is
much easier to write.
By default, Quantitative will use the latter form, but it is possible to define alternative representations of units where these exist, and Quantitative will use these whenever a quantity is displayed. A contextual value can be defined, such as the following,
import
gossamer
.
t
given
SubstituteUnits
[
Kilograms
[
1
]
&
Metres
[
2
]
&
Seconds
[

2
]
]
(
t
"J"
)
and then a value such as, 2.8*Kilo(Joule)
will be rendered as 2800 J
instead of 2800 kg⋅m²⋅s¯²
.
Note that this only applies if the quantity’s units exactly match the type parameter of
SubstituteUnits
, and units such as Jouleseconds would still be displayed as kg⋅m²⋅s¯¹
.