Units #

Math.js supports units. Units can be used to do calculations and to perform conversions.

Usage #

Units can be created using the function math.unit. This function accepts either a single string argument containing a value and unit, or two arguments, the first being a numeric value and the second a string containing a unit. Most units support prefixes like k or kilo, and many units have both a full name and an abbreviation. The returned object is a Unit.


math.unit(value: number, valuelessUnit: string) : Unit
math.unit(value: number, valuelessUnit: Unit) : Unit
math.unit(unit: string) : Unit
math.unit(unit: Unit) : Unit

Example usage:

const a = math.unit(45, 'cm')             // Unit 450 mm
const b = math.unit('0.1 kilogram')       // Unit 100 gram
const c = math.unit('2 inch')             // Unit 2 inch
const d = math.unit('90 km/h')            // Unit 90 km/h
const e = math.unit('101325 kg/(m s^2)')  // Unit 101325 kg / (m s^2)

const f = c.to('cm')                      // Unit 5.08 cm
b.toNumber('gram')                        // Number 100
math.number(b, 'gram')                    // Number 100

c.equals(a)                               // false
c.equals(f)                               // true
c.equalBase(a)                            // true
c.equalBase(b)                            // false

f.toString()                              // String "5.08 cm"

const kph = math.unit('km/h')             // valueless Unit km/h
const mps = math.unit('m/s')              // valueless Unit m/s
const speed = math.unit(36, kph)          // Unit 36 km/h
speed.toNumber(mps)                       // Number 10

Use care when creating a unit with multiple terms in the denominator. Implicit multiplication has the same operator precedence as explicit multiplication and division, which means these three expressions are identical:

// These three are identical
const correct1 = math.unit('8.314 m^3 Pa / mol / K')           // Unit 8.314 (m^3 Pa) / (mol K)
const correct2 = math.unit('8.314 (m^3 Pa) / (mol K)')         // Unit 8.314 (m^3 Pa) / (mol K)
const correct3 = math.unit('8.314 (m^3 * Pa) / (mol * K)')     // Unit 8.314 (m^3 Pa) / (mol K)

But this expression, which omits the second / between mol and K, results in the wrong value:

// Missing the second '/' between 'mol' and 'K'
const incorrect = math.unit('8.314 m^3 Pa / mol K')            // Unit 8.314 (m^3 Pa K) / mol

The function math.unit has its own small parser. This parser differs a bit from the expression parser math.evaluate, and returns the expected result in this case:

// using math.evaluate instead of math.unit
const correct4 = math.evaluate('8.314 (m^3 * Pa) / (mol * K)') // Unit 8.314 (m^3 Pa) / (mol K)

In summary: be careful with implicit multiplication. In case of doubt, always use an explicit * and parenthesis.

Calculations #

The operations that support units are add, subtract, multiply, divide, pow, abs, sqrt, square, cube, and sign. Trigonometric functions like cos are also supported when the argument is an angle.

const a = math.unit(45, 'cm')       // Unit 450 mm
const b = math.unit('0.1m')         // Unit 100 mm
math.add(a, b)                      // Unit 0.55 m
math.multiply(b, 2)                 // Unit 200 mm

const c = math.unit(45, 'deg')      // Unit 45 deg
math.cos(c)                         // Number 0.7071067811865476

// Kinetic energy of average sedan on highway
const d = math.unit('80 mi/h')      // Unit 80 mi/h
const e = math.unit('2 tonne')      // Unit 2 tonne
const f = math.multiply(0.5, math.multipy(math.pow(d, 2), e))
                                    // 1.2790064742399996 MJ

Operations with arrays are supported too:

// Force on a charged particle moving through a magnetic field
const B = math.evaluate('[1, 0, 0] T')         // [1 T, 0 T, 0 T]
const v = math.evaluate('[0, 1, 0] m/s')       // [0 m / s, 1 m / s, 0 m / s]
const q = math.evaluate('1 C')                 // 1 C

const F = math.multiply(q, math.cross(v, B))   // [0 N, 0 N, -1 N]

All arithmetic operators act on the value of the unit as it is represented in SI units. This may lead to surprising behavior when working with temperature scales like celsius (or degC) and fahrenheit (or degF). In general, you should avoid calculations using celsius and fahrenheit. Rather, use kelvin (or K) and rankine (or degR) instead. This example highlights some problems when using celsius and fahrenheit in calculations:

const T_14F = math.unit('14 degF')            // Unit 14 degF (263.15 K)
const T_28F = math.multiply(T_14F, 2)         // Unit 28 degF (270.93 K), not 526.3 K

const Tnegative = math.unit(-13, 'degF')      // Unit -13 degF (248.15 K)
const Tpositive = math.abs(Tnegative)         // Unit -13 degF (248.15 K), not 13 degF

The expression parser supports units too. This is described in the section about units on the page Syntax.

User-Defined Units #

You can add your own units to Math.js using the math.createUnit function. The following example defines a new unit furlong, then uses the user-defined unit in a calculation:

math.createUnit('furlong', '220 yards')
math.evaluate('1 mile to furlong')            // 8 furlong

If you cannot express the new unit in terms of any existing unit, then the second argument can be omitted. In this case, a new base unit is created:

// A 'foo' cannot be expressed in terms of any other unit.
math.evaluate('8 foo * 4 feet')               // 32 foo feet

The second argument to createUnit can also be a configuration object consisting of the following properties:

An optional options object can also be supplied as the last argument to createUnits. Currently only the override option is supported:

// Redefine the mile (would not be the first time in history)
math.createUnit('mile', '1609.347218694 m', {override: true})

Base units created without specifying a definition cannot be overridden.

Create several units at once #

Multiple units can defined using a single call to createUnit by passing an object map as the first argument, where each key in the object is the name of a new unit and the value is either a string defining the unit, or an object with the configuration properties listed above. If the value is an empty string or an object lacking a definition property, a new base unit is created.

For example:

math.createUnit( {
  foo: {
    prefixes: 'long',
    baseName: 'essence-of-foo'
  bar: '40 foo',
  baz: {
    definition: '1 bar/hour',
    prefixes: 'long'
  override: true
math.evaluate('50000 kilofoo/s')  // 4.5 gigabaz

Return Value #

createUnit returns the created unit, or, when multiple units are created, the last unit created. Since createUnit is also compatible with the expression parser, this allows you to do things like this:

math.evaluate('45 mile/hour to createUnit("knot", "0.514444m/s")')
// 39.103964668651976 knot

Support of custom characters in unit names #

Per default, the name of a new unit:

It is possible to allow the usage of special characters (such as Greek alphabet, cyrillic alphabet, any Unicode symbols, etc.) by overriding the Unit.isValidAlpha static method. For example:

const isAlphaOriginal = math.Unit.isValidAlpha
const isGreekLowercaseChar = function (c) {
  const charCode = c.charCodeAt(0)
  return charCode > 944 && charCode < 970
math.Unit.isValidAlpha = function (c) {
  return isAlphaOriginal(c) || isGreekLowercaseChar(c)

math.createUnit('θ', '1 rad')
math.evaluate('1θ + 3 deg').toNumber('deg') // 60.29577951308232

Numeric type of the value of a unit #

The built-in units are always created with a value being a number. To turn the value into for example a BigNumber or Fraction, you can convert the value using the function math.fraction and math.bignumber:

math.unit(math.fraction(10), 'inch').toNumeric('cm')  // Fraction 127/5
math.fraction(math.unit(10, 'inch')).toNumeric('cm')  // Fraction 127/5

math.bignumber(math.unit(10, 'inch')).toNumeric('cm') // BigNumber 25.4
math.unit(math.bignumber(10), 'inch').toNumeric('cm') // BigNumber 25.4

When using the expression parser, it is possible to configure numeric values to be parsed as Fraction or BigNumber:

math.config({ number: 'Fraction' })
math.evaluate('10 inch').toNumeric('cm') // Fraction 127/5


A Unit object contains the following functions:

unit.clone() #

Clone the unit, returns a new unit with the same parameters.

unit.equalBase(unit) #

Test whether a unit has the same base as an other unit: length, mass, etc.

unit.equals(unit) #

Test whether a unit equals an other unit. Units are equal when they have the same base and same value when normalized to SI units.

unit.format([options]) #

Get a string representation of the unit. The function will determine the best fitting prefix for the unit. See the Format page for available options.

unit.fromJSON(json) #

Revive a unit from a JSON object. Accepts An object {mathjs: 'Unit', value: number, unit: string, fixPrefix: boolean}, where the property mathjs and fixPrefix are optional. Used when deserializing a unit, see Serialization.

unit.splitUnit(parts) #

Split a unit into the specified parts. For example:

const u = math.unit(1, 'm')
u.splitUnit(['ft', 'in'])    // 3 feet,3.3700787401574765 inch

unit.to(unitName) #

Convert the unit to a specific unit name. Returns a clone of the unit with a fixed prefix and unit.

unit.toJSON() #

Returns a JSON representation of the unit, with signature {mathjs: 'Unit', value: number, unit: string, fixPrefix: boolean}. Used when serializing a unit, see Serialization.

unit.toNumber(unitName) #

Get the value of a unit when converted to the specified unit (a unit with optional prefix but without value). The type of the returned value is always number.

unit.toNumeric(unitName) #

Get the value of a unit when converted to the specified unit (a unit with optional prefix but without value). The type of the returned value depends on how the unit was created and can be number, Fraction, or BigNumber.

unit.toSI() #

Returns a clone of a unit represented in SI units. Works with units with or without a value.

unit.toString() #

Get a string representation of the unit. The function will determine the best fitting prefix for the unit.

unit.valType() #

Get the string name of the current type of the value of this Unit object, e.g. ‘number’, ‘BigNumber’, etc.

Unit reference #

This section lists all available units, prefixes, and physical constants. These can be used via the Unit object, or via math.evaluate().

Reference #

Math.js comes with the following built-in units.

Base Unit
Length meter (m), inch (in), foot (ft), yard (yd), mile (mi), link (li), rod (rd), chain (ch), angstrom, mil
Surface area m2, sqin, sqft, sqyd, sqmi, sqrd, sqch, sqmil, acre, hectare
Volume m3, litre (l, L, lt, liter), cc, cuin, cuft, cuyd, teaspoon, tablespoon
Liquid volume minim, fluiddram (fldr), fluidounce (floz), gill (gi), cup (cp), pint (pt), quart (qt), gallon (gal), beerbarrel (bbl), oilbarrel (obl), hogshead, drop (gtt)
Angles rad (radian), deg (degree), grad (gradian), cycle, arcsec (arcsecond), arcmin (arcminute)
Time second (s, secs, seconds), minute (min, mins, minutes), hour (h, hr, hrs, hours), day (days), week (weeks), month (months), year (years), decade (decades), century (centuries), millennium (millennia)
Frequency hertz (Hz)
Mass gram(g), tonne, ton, grain (gr), dram (dr), ounce (oz), poundmass (lbm, lb, lbs), hundredweight (cwt), stick, stone
Electric current ampere (A)
Temperature kelvin (K), celsius (degC), fahrenheit (degF), rankine (degR)
Amount of substance mole (mol)
Luminous intensity candela (cd)
Force newton (N), dyne (dyn), poundforce (lbf), kip
Energy joule (J), erg, Wh, BTU, electronvolt (eV)
Power watt (W), hp
Pressure Pa, psi, atm, torr, bar, mmHg, mmH2O, cmH2O
Electricity and magnetism ampere (A), coulomb (C), watt (W), volt (V), ohm, farad (F), weber (Wb), tesla (T), henry (H), siemens (S), electronvolt (eV)
Binary bits (b), bytes (B)

Note: all time units are based on the Julian year, with one month being 1/12th of a Julian year, a year being one Julian year, a decade being 10 Julian years, a century being 100, and a millennium being 1000.

Note that all relevant units can also be written in plural form, for example 5 meters instead of 5 meter or 10 seconds instead of 10 second.

Surface and volume units can alternatively be expressed in terms of length units raised to a power, for example 100 in^2 instead of 100 sqin.

Prefixes #

The following decimal prefixes are available.

Name Abbreviation Value
deca da 1e1
hecto h 1e2
kilo k 1e3
mega M 1e6
giga G 1e9
tera T 1e12
peta P 1e15
exa E 1e18
zetta Z 1e21
yotta Y 1e24
ronna R 1e27
quetta Q 1e30
Name Abbreviation Value
deci d 1e-1
centi c 1e-2
milli m 1e-3
micro u 1e-6
nano n 1e-9
pico p 1e-12
femto f 1e-15
atto a 1e-18
zepto z 1e-21
yocto y 1e-24
ronto r 1e-27
quecto q 1e-30

The following binary prefixes are available. They can be used with units bits (b) and bytes (B).

Name Abbreviation Value
kibi Ki 1024
mebi Mi 1024^2
gibi Gi 1024^3
tebi Ti 1024^4
pebi Pi 1024^5
exi Ei 1024^6
zebi Zi 1024^7
yobi Yi 1024^8
Name Abbreviation Value
kilo k 1e3
mega M 1e6
giga G 1e9
tera T 1e12
peta P 1e15
exa E 1e18
zetta Z 1e21
yotta Y 1e24

Physical Constants #

Math.js includes the following physical constants. See Wikipedia for more information.

Universal constants #

Name Symbol Value Unit
speedOfLight c 299792458 m · s-1
gravitationConstant G 6.6738480e-11 m3 · kg-1 · s-2
planckConstant h 6.626069311e-34 J · s
reducedPlanckConstant h 1.05457172647e-34 J · s

Electromagnetic constants #

Name Symbol Value Unit
magneticConstant μ0 1.2566370614e-6 N · A-2
electricConstant ε0 8.854187817e-12 F · m-1
vacuumImpedance Z0 376.730313461 &ohm;
coulomb κ 8.9875517873681764e9 N · m2 · C-2
elementaryCharge e 1.60217656535e-19 C
bohrMagneton μB 9.2740096820e-24 J · T-1
conductanceQuantum G0 7.748091734625e-5 S
inverseConductanceQuantum G0-1 12906.403721742 &ohm;
magneticFluxQuantum f0 2.06783375846e-15 Wb
nuclearMagneton μN 5.0507835311e-27 J · T-1
klitzing RK 25812.807443484 &ohm;

Atomic and nuclear constants #

Name Symbol Value Unit
bohrRadius a0 5.291772109217e-11 m
classicalElectronRadius re 2.817940326727e-15 m
electronMass me 9.1093829140e-31 kg
fermiCoupling GF 1.1663645e-5 GeV-2
fineStructure α 7.297352569824e-3 -
hartreeEnergy Eh 4.3597443419e-18 J
protonMass mp 1.67262177774e-27 kg
deuteronMass md 3.3435830926e-27 kg
neutronMass mn 1.6749271613e-27 kg
quantumOfCirculation h / (2me) 3.636947552024e-4 m2 · s-1
rydberg R 10973731.56853955 m-1
thomsonCrossSection   6.65245873413e-29 m2
weakMixingAngle   0.222321 -
efimovFactor   22.7 -

Physico-chemical constants #

Name Symbol Value Unit
atomicMass mu 1.66053892173e-27 kg
avogadro NA 6.0221412927e23 mol-1
boltzmann k 1.380648813e-23 J · K-1
faraday F 96485.336521 C · mol-1
firstRadiation c1 3.7417715317e-16 W · m2
loschmidt n0 2.686780524e25 m-3
gasConstant R 8.314462175 J · K-1 · mol-1
molarPlanckConstant NA · h 3.990312717628e-10 J · s · mol-1
molarVolume Vm 2.241396820e-10 m3 · mol-1
sackurTetrode   -1.164870823 -
secondRadiation c2 1.438777013e-2 m · K
stefanBoltzmann σ 5.67037321e-8 W · m-2 · K-4
wienDisplacement b 2.897772126e-3 m · K

Note that the values of loschmidt and molarVolume are at T = 273.15 K and p = 101.325 kPa. The value of sackurTetrode is at T = 1 K and p = 101.325 kPa.

Adopted values #

Name Symbol Value Unit
molarMass Mu 1e-3 kg · mol-1
molarMassC12 M(12C) 1.2e-2 kg · mol-1
gravity gn 9.80665 m · s-2
atm atm 101325 Pa

Natural units #

Name Symbol Value Unit
planckLength lP 1.61619997e-35 m
planckMass mP 2.1765113e-8 kg
planckTime tP 5.3910632e-44 s
planckCharge qP 1.87554595641e-18 C
planckTemperature TP 1.41683385e+32 K
Fork me on GitHub