# Units of measurement#

This is the units package. It contains information about many units and conversions between them.

TUTORIAL:

To return a unit:

```sage: units.length.meter
meter
```
```>>> from sage.all import *
>>> units.length.meter
meter
```

This unit acts exactly like a symbolic variable:

```sage: s = units.length.meter
sage: s^2
meter^2
sage: s + var('x')
meter + x
```
```>>> from sage.all import *
>>> s = units.length.meter
>>> s**Integer(2)
meter^2
>>> s + var('x')
meter + x
```

Units have additional information in their docstring:

```sage: # You would type: units.force.dyne?
sage: print(units.force.dyne.__doc__)
CGS unit for force defined to be gram*centimeter/second^2.
Equal to 10^-5 newtons.
```
```>>> from sage.all import *
>>> # You would type: units.force.dyne?
>>> print(units.force.dyne.__doc__)
CGS unit for force defined to be gram*centimeter/second^2.
Equal to 10^-5 newtons.
```

You may call the convert function with units:

```sage: t = units.mass.gram*units.length.centimeter/units.time.second^2
sage: t.convert(units.mass.pound*units.length.foot/units.time.hour^2)
5400000000000/5760623099*(foot*pound/hour^2)
sage: t.convert(units.force.newton)
1/100000*newton
```
```>>> from sage.all import *
>>> t = units.mass.gram*units.length.centimeter/units.time.second**Integer(2)
>>> t.convert(units.mass.pound*units.length.foot/units.time.hour**Integer(2))
5400000000000/5760623099*(foot*pound/hour^2)
>>> t.convert(units.force.newton)
1/100000*newton
```

Calling the convert function with no target returns base SI units:

```sage: t.convert()
1/100000*kilogram*meter/second^2
```
```>>> from sage.all import *
>>> t.convert()
1/100000*kilogram*meter/second^2
```

Giving improper units to convert to raises a ValueError:

```sage: t.convert(units.charge.coulomb)
Traceback (most recent call last):
...
ValueError: Incompatible units
```
```>>> from sage.all import *
>>> t.convert(units.charge.coulomb)
Traceback (most recent call last):
...
ValueError: Incompatible units
```

Converting temperatures works as well:

```sage: s = 68*units.temperature.fahrenheit
sage: s.convert(units.temperature.celsius)
20*celsius
sage: s.convert()
293.150000000000*kelvin
```
```>>> from sage.all import *
>>> s = Integer(68)*units.temperature.fahrenheit
>>> s.convert(units.temperature.celsius)
20*celsius
>>> s.convert()
293.150000000000*kelvin
```

Trying to multiply temperatures by another unit then converting raises a ValueError:

```sage: wrong = 50*units.temperature.celsius*units.length.foot
sage: wrong.convert()
Traceback (most recent call last):
...
ValueError: cannot convert
```
```>>> from sage.all import *
>>> wrong = Integer(50)*units.temperature.celsius*units.length.foot
>>> wrong.convert()
Traceback (most recent call last):
...
ValueError: cannot convert
```

AUTHORS:

• David Ackerman

• William Stein

class sage.symbolic.units.UnitExpression[source]#

Bases: `Expression`

A symbolic unit.

EXAMPLES:

```sage: acre = units.area.acre
sage: type(acre)
<class 'sage.symbolic.units.UnitExpression'>
```
```>>> from sage.all import *
>>> acre = units.area.acre
>>> type(acre)
<class 'sage.symbolic.units.UnitExpression'>
```
class sage.symbolic.units.Units(data, name='')[source]#

Bases: `ExtraTabCompletion`

A collection of units of some type.

EXAMPLES:

```sage: units.power
Collection of units of power: cheval_vapeur horsepower watt
```
```>>> from sage.all import *
>>> units.power
Collection of units of power: cheval_vapeur horsepower watt
```
sage.symbolic.units.base_units(unit)[source]#

Converts unit to base SI units.

INPUT:

• `unit` – a unit

OUTPUT:

• a symbolic expression

EXAMPLES:

```sage: sage.symbolic.units.base_units(units.length.foot)
381/1250*meter
```
```>>> from sage.all import *
>>> sage.symbolic.units.base_units(units.length.foot)
381/1250*meter
```

If unit is already a base unit, it just returns that unit:

```sage: sage.symbolic.units.base_units(units.length.meter)
meter
```
```>>> from sage.all import *
>>> sage.symbolic.units.base_units(units.length.meter)
meter
```

Derived units get broken down into their base parts:

```sage: sage.symbolic.units.base_units(units.force.newton)
kilogram*meter/second^2
sage: sage.symbolic.units.base_units(units.volume.liter)
1/1000*meter^3
```
```>>> from sage.all import *
>>> sage.symbolic.units.base_units(units.force.newton)
kilogram*meter/second^2
>>> sage.symbolic.units.base_units(units.volume.liter)
1/1000*meter^3
```

Returns variable if ‘unit’ is not a unit:

```sage: sage.symbolic.units.base_units(var('x'))
x
```
```>>> from sage.all import *
>>> sage.symbolic.units.base_units(var('x'))
x
```
sage.symbolic.units.convert(expr, target)[source]#

Converts units between expr and target. If target is None then converts to SI base units.

INPUT:

• `expr` – the symbolic expression converting from

• `target` – (default None) the symbolic expression converting to

OUTPUT:

• a symbolic expression

EXAMPLES:

```sage: sage.symbolic.units.convert(units.length.foot, None)
381/1250*meter
sage: sage.symbolic.units.convert(units.mass.kilogram, units.mass.pound)
100000000/45359237*pound
```
```>>> from sage.all import *
>>> sage.symbolic.units.convert(units.length.foot, None)
381/1250*meter
>>> sage.symbolic.units.convert(units.mass.kilogram, units.mass.pound)
100000000/45359237*pound
```

Raises ValueError if expr and target are not convertible:

```sage: sage.symbolic.units.convert(units.mass.kilogram, units.length.foot)
Traceback (most recent call last):
...
ValueError: Incompatible units
sage: sage.symbolic.units.convert(units.length.meter^2, units.length.foot)
Traceback (most recent call last):
...
ValueError: Incompatible units
```
```>>> from sage.all import *
>>> sage.symbolic.units.convert(units.mass.kilogram, units.length.foot)
Traceback (most recent call last):
...
ValueError: Incompatible units
>>> sage.symbolic.units.convert(units.length.meter**Integer(2), units.length.foot)
Traceback (most recent call last):
...
ValueError: Incompatible units
```

Recognizes derived unit relationships to base units and other derived units:

```sage: sage.symbolic.units.convert(units.length.foot/units.time.second^2, units.acceleration.galileo)
762/25*galileo
sage: sage.symbolic.units.convert(units.mass.kilogram*units.length.meter/units.time.second^2, units.force.newton)
newton
sage: sage.symbolic.units.convert(units.length.foot^3, units.area.acre*units.length.inch)
1/3630*(acre*inch)
sage: sage.symbolic.units.convert(units.charge.coulomb, units.current.ampere*units.time.second)
(ampere*second)
sage: sage.symbolic.units.convert(units.pressure.pascal*units.si_prefixes.kilo, units.pressure.pounds_per_square_inch)
1290320000000/8896443230521*pounds_per_square_inch
```
```>>> from sage.all import *
>>> sage.symbolic.units.convert(units.length.foot/units.time.second**Integer(2), units.acceleration.galileo)
762/25*galileo
>>> sage.symbolic.units.convert(units.mass.kilogram*units.length.meter/units.time.second**Integer(2), units.force.newton)
newton
>>> sage.symbolic.units.convert(units.length.foot**Integer(3), units.area.acre*units.length.inch)
1/3630*(acre*inch)
>>> sage.symbolic.units.convert(units.charge.coulomb, units.current.ampere*units.time.second)
(ampere*second)
>>> sage.symbolic.units.convert(units.pressure.pascal*units.si_prefixes.kilo, units.pressure.pounds_per_square_inch)
1290320000000/8896443230521*pounds_per_square_inch
```

```sage: sage.symbolic.units.convert(units.pressure.pascal*units.si_prefixes.kilo, units.pressure.pounds_per_square_inch)*1.0
0.145037737730209*pounds_per_square_inch
```
```>>> from sage.all import *
>>> sage.symbolic.units.convert(units.pressure.pascal*units.si_prefixes.kilo, units.pressure.pounds_per_square_inch)*RealNumber('1.0')
0.145037737730209*pounds_per_square_inch
```

You can also convert quantities of units:

```sage: sage.symbolic.units.convert(cos(50) * units.angles.radian, units.angles.degree)
degree*(180*cos(50)/pi)
8.83795706233228*degree
sage: sage.symbolic.units.convert(50 * units.length.light_year / units.time.year, units.length.foot / units.time.second)
6249954068750/127*(foot/second)
```
```>>> from sage.all import *
degree*(180*cos(50)/pi)
8.83795706233228*degree
>>> sage.symbolic.units.convert(Integer(50) * units.length.light_year / units.time.year, units.length.foot / units.time.second)
6249954068750/127*(foot/second)
```

Quantities may contain variables (not for temperature conversion, though):

```sage: sage.symbolic.units.convert(50 * x * units.area.square_meter, units.area.acre)
acre*(1953125/158080329*x)
```
```>>> from sage.all import *
>>> sage.symbolic.units.convert(Integer(50) * x * units.area.square_meter, units.area.acre)
acre*(1953125/158080329*x)
```
sage.symbolic.units.convert_temperature(expr, target)[source]#

Function for converting between temperatures.

INPUT:

• `expr` – a unit of temperature

• `target` – a units of temperature

OUTPUT:

• a symbolic expression

EXAMPLES:

```sage: t = 32*units.temperature.fahrenheit
sage: t.convert(units.temperature.celsius)
0
sage: t.convert(units.temperature.kelvin)
273.150000000000*kelvin
```
```>>> from sage.all import *
>>> t = Integer(32)*units.temperature.fahrenheit
>>> t.convert(units.temperature.celsius)
0
>>> t.convert(units.temperature.kelvin)
273.150000000000*kelvin
```

If target is None then it defaults to kelvin:

```sage: t.convert()
273.150000000000*kelvin
```
```>>> from sage.all import *
>>> t.convert()
273.150000000000*kelvin
```

Raises ValueError when either input is not a unit of temperature:

```sage: t.convert(units.length.foot)
Traceback (most recent call last):
...
ValueError: cannot convert
sage: wrong = units.length.meter*units.temperature.fahrenheit
sage: wrong.convert()
Traceback (most recent call last):
...
ValueError: cannot convert
```
```>>> from sage.all import *
>>> t.convert(units.length.foot)
Traceback (most recent call last):
...
ValueError: cannot convert
>>> wrong = units.length.meter*units.temperature.fahrenheit
>>> wrong.convert()
Traceback (most recent call last):
...
ValueError: cannot convert
```

We directly call the convert_temperature function:

```sage: sage.symbolic.units.convert_temperature(37*units.temperature.celsius, units.temperature.fahrenheit)
493/5*fahrenheit
sage: 493/5.0
98.6000000000000
```
```>>> from sage.all import *
>>> sage.symbolic.units.convert_temperature(Integer(37)*units.temperature.celsius, units.temperature.fahrenheit)
493/5*fahrenheit
>>> Integer(493)/RealNumber('5.0')
98.6000000000000
```
sage.symbolic.units.evalunitdict()[source]#

Replace all the string values of the unitdict variable by their evaluated forms, and builds some other tables for ease of use. This function is mainly used internally, for efficiency (and flexibility) purposes, making it easier to describe the units.

EXAMPLES:

```sage: sage.symbolic.units.evalunitdict()
```
```>>> from sage.all import *
>>> sage.symbolic.units.evalunitdict()
```
sage.symbolic.units.is_unit(s)[source]#

Return a boolean when asked whether the input is in the list of units.

INPUT:

• `s` – an object

OUTPUT:

• a boolean

EXAMPLES:

```sage: sage.symbolic.units.is_unit(1)
False
sage: sage.symbolic.units.is_unit(units.length.meter)
True
```
```>>> from sage.all import *
>>> sage.symbolic.units.is_unit(Integer(1))
False
>>> sage.symbolic.units.is_unit(units.length.meter)
True
```

The square of a unit is not a unit:

```sage: sage.symbolic.units.is_unit(units.length.meter^2)
False
```
```>>> from sage.all import *
>>> sage.symbolic.units.is_unit(units.length.meter**Integer(2))
False
```

You can also directly create units using var, though they won’t have a nice docstring describing the unit:

```sage: sage.symbolic.units.is_unit(var('meter'))
True
```
```>>> from sage.all import *
>>> sage.symbolic.units.is_unit(var('meter'))
True
```
sage.symbolic.units.str_to_unit(name)[source]#

Create the symbolic unit with given name. A symbolic unit is a class that derives from symbolic expression, and has a specialized docstring.

INPUT:

• `name` – a string

OUTPUT:

EXAMPLES:

```sage: sage.symbolic.units.str_to_unit('acre')
acre
sage: type(sage.symbolic.units.str_to_unit('acre'))
<class 'sage.symbolic.units.UnitExpression'>
```
```>>> from sage.all import *
>>> sage.symbolic.units.str_to_unit('acre')
acre
>>> type(sage.symbolic.units.str_to_unit('acre'))
<class 'sage.symbolic.units.UnitExpression'>
```
sage.symbolic.units.unit_derivations_expr(v)[source]#

Given derived units name, returns the corresponding units expression. For example, given ‘acceleration’ output the symbolic expression length/time^2.

INPUT:

• `v` – a string, name of a unit type such as ‘area’, ‘volume’, etc.

OUTPUT:

• a symbolic expression

EXAMPLES:

```sage: sage.symbolic.units.unit_derivations_expr('volume')
length^3
sage: sage.symbolic.units.unit_derivations_expr('electric_potential')
length^2*mass/(current*time^3)
```
```>>> from sage.all import *
>>> sage.symbolic.units.unit_derivations_expr('volume')
length^3
>>> sage.symbolic.units.unit_derivations_expr('electric_potential')
length^2*mass/(current*time^3)
```

If the unit name is unknown, a `KeyError` is raised:

```sage: sage.symbolic.units.unit_derivations_expr('invalid')
Traceback (most recent call last):
...
KeyError: 'invalid'
```
```>>> from sage.all import *
>>> sage.symbolic.units.unit_derivations_expr('invalid')
Traceback (most recent call last):
...
KeyError: 'invalid'
```
sage.symbolic.units.unitdocs(unit)[source]#

Returns docstring for the given unit.

INPUT:

• `unit` – a unit

OUTPUT:

• a string

EXAMPLES:

```sage: sage.symbolic.units.unitdocs('meter')
'SI base unit of length.\nDefined to be the distance light travels in vacuum in 1/299792458 of a second.'
sage: sage.symbolic.units.unitdocs('amu')
'Abbreviation for atomic mass unit.\nApproximately equal to 1.660538782*10^-27 kilograms.'
```
```>>> from sage.all import *
>>> sage.symbolic.units.unitdocs('meter')
'SI base unit of length.\nDefined to be the distance light travels in vacuum in 1/299792458 of a second.'
>>> sage.symbolic.units.unitdocs('amu')
'Abbreviation for atomic mass unit.\nApproximately equal to 1.660538782*10^-27 kilograms.'
```

Units not in the list unit_docs will raise a ValueError:

```sage: sage.symbolic.units.unitdocs('earth')
Traceback (most recent call last):
...
ValueError: no documentation exists for the unit earth
```
```>>> from sage.all import *
>>> sage.symbolic.units.unitdocs('earth')
Traceback (most recent call last):
...
ValueError: no documentation exists for the unit earth
```
sage.symbolic.units.vars_in_str(s)[source]#

Given a string like ‘mass/(length*time)’, return the list [‘mass’, ‘length’, ‘time’].

INPUT:

• `s` – a string

OUTPUT:

• a list of strings (unit names)

EXAMPLES:

```sage: sage.symbolic.units.vars_in_str('mass/(length*time)')
['mass', 'length', 'time']
```
```>>> from sage.all import *
>>> sage.symbolic.units.vars_in_str('mass/(length*time)')
['mass', 'length', 'time']
```