Units and Quantities

The Compute Engine provides a unit system for representing and computing with physical quantities. Units are organized around SI base dimensions with support for derived, prefixed, and non-SI units.

Read the Units and Quantities Guide for a tutorial introduction with examples

Quantity

Quantity(magnitude, unit)

A value paired with a physical unit.

The magnitude is any numeric expression. The unit is a unit symbol (string) or a compound unit expression built from Multiply, Divide, and Power.

9.8\,\mathrm{m/s^{2}}

$$$9.8\,\mathrm{m/s^{2}}$$

["Quantity", 3.5, "m"]
// A simple quantity: 3.5 meters

["Quantity", 9.8, ["Divide", "m", ["Power", "s", 2]]]
// A compound unit: 9.8 m/s^2

["Quantity", 100, "km"]
// Prefixed unit: 100 kilometers

A DSL string shorthand is accepted for compound units and canonicalized to the structured form:

["Quantity", 9.8, "m/s^2"]
// → ["Quantity", 9.8, ["Divide", "m", ["Power", "s", 2]]]

["Quantity", 1, "kg/(m*s^2)"]
// → ["Quantity", 1, ["Divide", "kg", ["Multiply", "m", ["Power", "s", 2]]]]

Arithmetic on quantities:

Add and Subtract require compatible dimensions and express the result in the unit with the largest scale factor:

["Add", ["Quantity", 12, "cm"], ["Quantity", 1, "m"]]
// → ["Quantity", 1.12, "m"]

Multiply and Divide combine units:

["Multiply", ["Quantity", 5, "m"], ["Quantity", 3, "s"]]
// → ["Quantity", 15, ["Multiply", "m", "s"]]

Power raises the unit to the exponent:

["Power", ["Quantity", 3, "m"], 2]
// → ["Quantity", 9, ["Power", "m", 2]]

QuantityMagnitude

QuantityMagnitude(quantity)

Extract the numeric magnitude from a Quantity expression.

["QuantityMagnitude", ["Quantity", 3.5, "m"]]
// → 3.5

QuantityUnit

QuantityUnit(quantity)

Extract the unit expression from a Quantity.

["QuantityUnit", ["Quantity", 3.5, "m"]]
// → "m"

["QuantityUnit", ["Quantity", 9.8, ["Divide", "m", ["Power", "s", 2]]]]
// → ["Divide", "m", ["Power", "s", 2]]

UnitConvert

UnitConvert(quantity, target-unit)

Convert a quantity to a different compatible unit. The target can be a simple unit symbol or a compound unit expression.

Returns the converted Quantity, or an Error if the units are incompatible.

Supports affine temperature conversions (degC, degF, K).

["UnitConvert", ["Quantity", 1500, "m"], "km"]
// → ["Quantity", 1.5, "km"]

["UnitConvert", ["Quantity", 180, "deg"], "rad"]
// → ["Quantity", 3.14159..., "rad"]

["UnitConvert",
  ["Quantity", 36, ["Divide", "km", "h"]],
  ["Divide", "m", "s"]]
// → ["Quantity", 10, ["Divide", "m", "s"]]

["UnitConvert", ["Quantity", 100, "degC"], "degF"]
// → ["Quantity", 212, "degF"]

["UnitConvert", ["Quantity", 5, "m"], "s"]
// → Error (incompatible dimensions)

UnitSimplify

UnitSimplify(quantity)

Reduce a compound unit to a named derived SI unit if one matches the dimension and has scale factor 1.

["UnitSimplify",
  ["Quantity", 100, ["Multiply", "kg", "m", ["Power", "s", -2]]]]
// → ["Quantity", 100, "N"]

If no simpler named form exists, the quantity is returned unchanged.

IsCompatibleUnit

IsCompatibleUnit(unit-a, unit-b)

Test whether two units have the same dimension vector. Returns True or False. Both simple unit symbols and compound unit expressions are accepted.

["IsCompatibleUnit", "m", "km"]
// → True

["IsCompatibleUnit", "m", "s"]
// → False

["IsCompatibleUnit", "J", "eV"]
// → True

["IsCompatibleUnit", ["Divide", "m", "s"], ["Divide", "km", "h"]]
// → True

UnitDimension

UnitDimension(unit)

Return the 7-element dimension vector of a unit. The vector encodes the exponents of the SI base dimensions: [length, mass, time, current, temperature, amount, luminosity].

Both simple unit symbols and compound unit expressions are accepted.

["UnitDimension", "m"]
// → [1, 0, 0, 0, 0, 0, 0]

["UnitDimension", "N"]
// → [1, 1, -2, 0, 0, 0, 0]

["UnitDimension", "V"]
// → [2, 1, -3, -1, 0, 0, 0]

["UnitDimension", ["Divide", "m", ["Power", "s", 2]]]
// → [1, 0, -2, 0, 0, 0, 0]

Physics Constants

The physics library provides physical constants as Quantity expressions. These constants use the 2018 CODATA recommended values.

Symbol	Value	Unit	Description
SpeedOfLight	299792458	m/s	Speed of light in vacuum
PlanckConstant	6.62607015e-34	J s	Planck constant
Mu0	1.25663706212e-6	N/A^2	Vacuum permeability
StandardGravity	9.80665	m/s^2	Standard acceleration due to gravity
ElementaryCharge	1.602176634e-19	C	Elementary charge
BoltzmannConstant	1.380649e-23	J/K	Boltzmann constant
AvogadroConstant	6.02214076e23	mol^-1	Avogadro constant
VacuumPermittivity	8.8541878128e-12	F/m	Vacuum permittivity
GravitationalConstant	6.67430e-11	m^3/(kg s^2)	Gravitational constant
StefanBoltzmannConstant	5.670374419e-8	W/(m^2 K^4)	Stefan-Boltzmann constant
GasConstant	8.314462618	J/(mol K)	Molar gas constant

\mu_0

$$$\mu_0$$

The vacuum permeability constant Mu0 can be entered in LaTeX as \mu_0.

\varepsilon_0

$$$\varepsilon_0$$

The vacuum permittivity constant VacuumPermittivity can be entered in LaTeX as \varepsilon_0.

Supported Units

SI Base Units

Symbol	Quantity	Name
m	Length	meter
kg	Mass	kilogram
s	Time	second
A	Electric current	ampere
K	Temperature	kelvin
mol	Amount of substance	mole
cd	Luminous intensity	candela

Named Derived SI Units

Symbol	Quantity	Name	In base units
Hz	Frequency	hertz	s^-1
N	Force	newton	kg m s^-2
Pa	Pressure	pascal	kg m^-1 s^-2
J	Energy	joule	kg m^2 s^-2
W	Power	watt	kg m^2 s^-3
C	Electric charge	coulomb	A s
V	Voltage	volt	kg m^2 s^-3 A^-1
F	Capacitance	farad	kg^-1 m^-2 s^4 A^2
ohm	Resistance	ohm	kg m^2 s^-3 A^-2
S	Conductance	siemens	kg^-1 m^-2 s^3 A^2
Wb	Magnetic flux	weber	kg m^2 s^-2 A^-1
T	Magnetic flux density	tesla	kg s^-2 A^-1
H	Inductance	henry	kg m^2 s^-2 A^-2
lm	Luminous flux	lumen	cd
lx	Illuminance	lux	cd m^-2
Bq	Radioactivity	becquerel	s^-1
Gy	Absorbed dose	gray	m^2 s^-2
Sv	Equivalent dose	sievert	m^2 s^-2
kat	Catalytic activity	katal	mol s^-1

SI Prefixes

All SI prefixes are supported on base and named derived units:

Prefix	Symbol	Factor
quetta	Q	10^30
ronna	R	10^27
yotta	Y	10^24
zetta	Z	10^21
exa	E	10^18
peta	P	10^15
tera	T	10^12
giga	G	10^9
mega	M	10^6
kilo	k	10^3
hecto	h	10^2
deca	da	10^1

Prefix	Symbol	Factor
deci	d	10^-1
centi	c	10^-2
milli	m	10^-3
micro	u	10^-6
nano	n	10^-9
pico	p	10^-12
femto	f	10^-15
atto	a	10^-18
zepto	z	10^-21
yocto	y	10^-24
ronto	r	10^-27
quecto	q	10^-30

Note

Prefixes are applicable to all base and derived units except kg. Use g as the base for mass prefixing: mg, ug, etc.

Non-SI Units

Symbol	Quantity	SI Equivalent
min	Time	60 s
h	Time	3600 s
d	Time	86400 s
ha	Area	10^4 m^2
L	Volume	10^-3 m^3
t	Mass	10^3 kg
eV	Energy	1.602176634e-19 J
Da	Mass	1.66053906660e-27 kg
au	Length	1.495978707e11 m
in	Length	0.0254 m
ft	Length	0.3048 m
mi	Length	1609.344 m
lb	Mass	0.45359237 kg
oz	Mass	0.028349523125 kg
gal	Volume	3.785411784e-3 m^3
atm	Pressure	101325 Pa
bar	Pressure	10^5 Pa
cal	Energy	4.184 J
kWh	Energy	3.6e6 J
degC	Temperature	K - 273.15
degF	Temperature	(K - 459.67) x 5/9

Note

degC and degF use affine conversions (offset + scale), not just linear scaling. UnitConvert handles these correctly.

Angular Units

Symbol	Name	In radians
rad	radian	1
deg	degree	pi/180
grad	gradian	pi/200
turn	turn	2 pi
arcmin	arcminute	pi/10800
arcsec	arcsecond	pi/648000

Angular units are dimensionless. Trigonometric functions (Sin, Cos, Tan, etc.) automatically convert Quantity arguments with angular units to radians.

Dimensionless Units

Symbol	Scale factor
percent	0.01
ppm	10^-6
dB	1 (nominal)
Np	1 (nominal)

Note

Logarithmic units (dB, Np) are parsed and stored but conversion between logarithmic and linear scales is not supported in v1.

LaTeX Parsing

Supported Input Forms

LaTeX	MathJSON
5\,\mathrm{m}	["Quantity", 5, "m"]
3\,\text{kg}	["Quantity", 3, "kg"]
9.8\,\mathrm{m/s^{2}}	["Quantity", 9.8, ["Divide", "m", ["Power", "s", 2]]]
\qty{12}{cm}	["Quantity", 12, "cm"]
\SI{5}{kg}	["Quantity", 5, "kg"]
\unit{m/s}	["Divide", "m", "s"]
\si{MHz}	"MHz"
9°30'15"	["Add", ["Quantity", 9, "deg"], ["Quantity", 30, "arcmin"], ["Quantity", 15, "arcsec"]]
9°30'	["Add", ["Quantity", 9, "deg"], ["Quantity", 30, "arcmin"]]

Unit Parsing Rules

Within \mathrm{...} or \text{...}, the parser interprets:

• / as division
• ^ and ^{...} as exponents
• \cdot as multiplication
• Juxtaposed symbols as individual units or prefixed units