English Customary Units¶

English customary units differ in various ways in different regions. In Britain a complex system of volume measurements featured different gallons for different materials such as a wine gallon and ale gallon that different by twenty percent. This complexity was swept away in 1824 by a reform that created an entirely new gallon, the British Imperial gallon defined as the volume occupied by ten pounds of water. Meanwhile in the USA the gallon is derived from the 1707 Winchester wine gallon, which is 231 cubic inches. These gallons differ by about twenty percent. By default if units runs in the ‘en_GB’ locale you will get the British volume measures. If it runs in the ‘en_US’ locale you will get the US volume measures. In other locales the default values are the US definitions. If you wish to force different definitions, then set the environment variable UNITS_ENGLISH to either ‘US’ or ‘GB’ to set the desired definitions independent of the locale.

Before 1959, the value of a yard (and other units of measure defined in terms of it) differed slightly among English-speaking countries. In 1959, Australia, Canada, New Zealand, the United Kingdom, the United States, and South Africa adopted the Canadian value of 1 yard = 0.9144 m (exactly), which was approximately halfway between the values used by the UK and the US; it had the additional advantage of making 1 inch = 2.54 cm (exactly). This new standard was termed the International Yard. Australia, Canada, and the UK then defined all customary lengths in terms of the International Yard (Australia did not define the furlong or rod); because many US land surveys were in terms of the pre-1959 units, the US continued to define customary surveyors' units (furlong, chain, rod, and link) in terms of the previous value for the foot, which was termed the US survey foot. The US defined a US survey mile as 5280 US survey feet, and defined a statute mile as a US survey mile. The US values for these units differ from the international values by about 2 ppm.

The units program uses the international values for these units; the US values can be obtained by using either the ‘US’ or the ‘survey’ prefix. In either case, the simple familiar relationships among the units are maintained, e.g., 1 ‘furlong’ = 660 ‘ft’, and 1 ‘USfurlong’ = 660 ‘USft’, though the metric equivalents differ slightly between the two cases. The ‘US’ prefix or the ‘survey’ prefix can also be used to obtain the US survey mile and the value of the US yard prior to 1959, e.g., ‘USmile’ or ‘surveymile’ (but not ‘USsurveymile’). To get the US value of the statute mile, use either ‘USstatutemile’ or ‘USmile’.

Except for distances that extend over hundreds of miles (such as in the US State Plane Coordinate System), the differences in the miles are usually insignificant:

You have: 100 surveymile - 100 mile
You want: inch


        * 12.672025


        / 0.078913984

The pre-1959 UK values for these units can be obtained with the prefix ‘UK’.

In the US, the acre is officially defined in terms of the US survey foot, but units uses a definition based on the international foot. If you want the official US acre use ‘USacre’ and similarly use ‘USacrefoot’ for the official US version of that unit. The difference between these units is about 4 parts per million.

Operators¶

You can enter more complicated units by combining units with operations such as multiplication, division, powers, addition, subtraction, and parentheses for grouping. You can use the customary symbols for these operators when units is invoked with its default options. Additionally, units supports some extensions, including high priority multiplication using a space, and a high priority numerical division operator (‘|’) that can simplify some expressions.

You multiply units using a space or an asterisk (‘*’). The next example shows both forms:

You have: arabicfoot * arabictradepound * force
You want: ft lbf


        * 0.7296


        / 1.370614

You can divide units using the slash (‘/’) or with ‘per’:

You have: furlongs per fortnight
You want: m/s


        * 0.00016630986


        / 6012.8727

You can use parentheses for grouping:

You have: (1/2) kg / (kg/meter)
You want: league


        * 0.00010356166


        / 9656.0833

White space surrounding operators is optional, so the previous example could have used ‘(1/2)kg/(kg/meter)’. As a consequence, however, hyphenated spelled-out numbers (e.g., ‘forty-two’) cannot be used; ‘forty-two’ is interpreted as ‘40 - 2’.

Multiplication using a space has a higher precedence than division using a slash and is evaluated left to right; in effect, the first ‘/’ character marks the beginning of the denominator of a unit expression. This makes it simple to enter a quotient with several terms in the denominator: ‘J / mol K’. The ‘*’ and ‘/’ operators have the same precedence, and are evaluated left to right; if you multiply with ‘*’, you must group the terms in the denominator with parentheses: ‘J / (mol * K)’.

The higher precedence of the space operator may not always be advantageous. For example, ‘m/s s/day’ is equivalent to ‘m / s s day’ and has dimensions of length per time cubed. Similarly, ‘1/2 meter’ refers to a unit of reciprocal length equivalent to 0.5/meter, perhaps not what you would intend if you entered that expression. The get a half meter you would need to use parentheses: ‘(1/2) meter’. The ‘*’ operator is convenient for multiplying a sequence of quotients. For example, ‘m/s * s/day’ is equivalent to ‘m/day’. Similarly, you could write ‘1/2 * meter’ to get half a meter.

The units program supports another option for numerical fractions: you can indicate division of numbers with the vertical bar (‘|’), so if you wanted half a meter you could write ‘1|2 meter’. You cannot use the vertical bar to indicate division of non-numerical units (e.g., ‘m|s’ results in an error message).

Powers of units can be specified using the ‘^’ character, as shown in the following example, or by simple concatenation of a unit and its exponent: ‘cm3’ is equivalent to ‘cm^3’; if the exponent is more than one digit, the ‘^’ is required. You can also use ‘**’ as an exponent operator.

You have: cm^3
You want: gallons


        * 0.00026417205


        / 3785.4118

Concatenation only works with a single unit name: if you write ‘(m/s)2’, units will treat it as multiplication by 2. When a unit includes a prefix, exponent operators apply to the combination, so ‘centimeter3’ gives cubic centimeters. If you separate the prefix from the unit with any multiplication operator (e.g., ‘centi meter^3’), the prefix is treated as a separate unit, so the exponent applies only to the unit without the prefix. The second example is equivalent to ‘centi * (meter^3)’, and gives a hundredth of a cubic meter, not a cubic centimeter. The units program is limited internally to products of 99 units; accordingly, expressions like ‘meter^100’ or ‘joule^34’ (represented internally as ‘kg^34 m^68 / s^68’) will fail.

The ‘|’ operator has the highest precedence, so you can write the square root of two thirds as ‘2|3^1|2’. The ‘^’ operator has the second highest precedence, and is evaluated right to left, as usual:

You have: 5 * 2^3^2
You want:


        Definition: 2560

With a dimensionless base unit, any dimensionless exponent is meaningful (e.g., ‘pi^exp(2.371)’). Even though angle is sometimes treated as dimensionless, exponents cannot have dimensions of angle:

You have: 2^radian


                 ^
Exponent not dimensionless

If the base unit is not dimensionless, the exponent must be a rational number p/q, and the dimension of the unit must be a power of q, so ‘gallon^2|3’ works but ‘acre^2|3’ fails. An exponent using the slash (‘/’) operator (e.g., ‘gallon^(2/3)’) is also acceptable; the parentheses are needed because the precedence of ‘^’ is higher than that of ‘/’. Since units cannot represent dimensions with exponents greater than 99, a fully reduced exponent must have q < 100. When raising a non-dimensionless unit to a power, units attempts to convert a decimal exponent to a rational number with q < 100. If this is not possible units displays an error message:

You have: ft^1.234
Base unit not dimensionless; rational exponent required

A decimal exponent must match its rational representation to machine precision, so ‘acre^1.5’ works but ‘gallon^0.666’ does not.

Sums and Differences of Units¶

You may sometimes want to add values of different units that are outside the SI. You may also wish to use units as a calculator that keeps track of units. Sums of conformable units are written with the ‘+’ character, and differences with the ‘-’ character.

You have: 2 hours + 23 minutes + 32 seconds
You want: seconds


        * 8612


        / 0.00011611705

You have: 12 ft + 3 in
You want: cm


        * 373.38


        / 0.0026782366

You have: 2 btu + 450 ft lbf
You want: btu


        * 2.5782804


        / 0.38785542

The expressions that are added or subtracted must reduce to identical expressions in primitive units, or an error message will be displayed:

You have: 12 printerspoint - 4 heredium


                                      ^
Illegal sum of non-conformable units

If you add two values of vastly different scale you may exceed the available precision of floating point (about 15 digits). The effect is that the addition of the smaller value makes no change to the larger value; in other words, the smaller value is treated as if it were zero.

You have: lightyear + cm

No warning is given, however. As usual, the precedence for ‘+’ and ‘-’ is lower than that of the other operators. A fractional quantity such as 2 1/2 cups can be given as ‘(2+1|2) cups’; the parentheses are necessary because multiplication has higher precedence than addition. If you omit the parentheses, units attempts to add ‘2’ and ‘1|2 cups’, and you get an error message:

You have: 2+1|2 cups


                   ^
Illegal sum or difference of non-conformable units

The expression could also be correctly written as ‘(2+1/2) cups’. If you write ‘2 1|2 cups’ the space is interpreted as multiplication so the result is the same as ‘1 cup’.

The ‘+’ and ‘-’ characters sometimes appears in exponents like ‘3.43e+8’. This leads to an ambiguity in an expression like ‘3e+2 yC’. The unit ‘e’ is a small unit of charge, so this can be regarded as equivalent to ‘(3e+2) yC’ or ‘(3 e)+(2 yC)’. This ambiguity is resolved by always interpreting ‘+’ and ‘-’ as part of an exponent if possible.

Numbers as Units¶

For units, numbers are just another kind of unit. They can appear as many times as you like and in any order in a unit expression. For example, to find the volume of a box that is 2 ft by 3 ft by 12 ft in steres, you could do the following:

You have: 2 ft 3 ft 12 ft
You want: stere


        * 2.038813


        / 0.49048148
You have: $ 5 / yard
You want: cents / inch


        * 13.888889


        / 0.072

And the second example shows how the dollar sign in the units conversion can precede the five. Be careful: units will interpret ‘$5’ with no space as equivalent to ‘dollar^5’.

Built-in Functions¶

Several built-in functions are provided: ‘sin’, ‘cos’, ‘tan’, ‘asin’, ‘acos’, ‘atan’, ‘sinh’, ‘cosh’, ‘tanh’, ‘asinh’, ‘acosh’, ‘atanh’, ‘exp’, ‘ln’, ‘log’, ‘abs’, ‘round’, ‘floor’, ‘ceil’, ‘factorial’, ‘Gamma’, ‘lnGamma’, ‘erf’, and ‘erfc’; the function ‘lnGamma’ is the natural logarithm of the ‘Gamma’ function.

The ‘sin’, ‘cos’, and ‘tan’ functions require either a dimensionless argument or an argument with dimensions of angle.

You have: sin(30 degrees)
You want:


        Definition: 0.5
You have: sin(pi/2)
You want:


        Definition: 1
You have: sin(3 kg)


                  ^
Unit not dimensionless

The other functions on the list require dimensionless arguments. The inverse trigonometric functions return arguments with dimensions of angle.

The ‘ln’ and ‘log’ functions give natural log and log base 10 respectively. To obtain logs for any integer base, enter the desired base immediately after ‘log’. For example, to get log base 2 you would write ‘log2’ and to get log base 47 you could write ‘log47’.

You have: log2(32)
You want: 


        Definition: 5
You have: log3(32)
You want: 


        Definition: 3.1546488
You have: log4(32)
You want: 


        Definition: 2.5
You have: log32(32)
You want: 


        Definition: 1
You have: log(32)
You want: 


        Definition: 1.50515
You have: log10(32)
You want: 


        Definition: 1.50515

If you wish to take roots of units, you may use the ‘sqrt’ or ‘cuberoot’ functions. These functions require that the argument have the appropriate root. You can obtain higher roots by using fractional exponents:

You have: sqrt(acre)
You want: feet


        * 208.71074


        / 0.0047913202
You have: (400 W/m^2 / stefanboltzmann)^(1/4)
You have:


        Definition: 289.80882 K
You have: cuberoot(hectare)


                          ^
Unit not a root

Previous Result¶

You can insert the result of the previous conversion using the underscore (‘_’). It is useful when you want to convert the same input to several different units, for example

You have: 2.3 tonrefrigeration
You want: btu/hr


        * 27600


        / 3.6231884e-005
You have: _
You want: kW


        * 8.0887615


        / 0.12362832

Suppose you want to do some deep frying that requires an oil depth of 2 inches. You have 1/2 gallon of oil, and want to know the largest-diameter pan that will maintain the required depth. The nonlinear unit ‘circlearea’ gives the radius of the circle (see Other Nonlinear Units, for a more detailed description) in SI units; you want the diameter in inches:

You have: 1|2 gallon / 2 in
You want: circlearea


        0.10890173 m
You have: 2 _
You want: in


        * 8.5749393


        / 0.1166189

In most cases, surrounding white space is optional, so the previous example could have used ‘2_’. If ‘_’ follows a non-numerical unit symbol, however, the space is required:

You have: m_


           ^
Parse error

When ‘_’ is followed by a digit, the operation is multiplication rather than exponentiation, so that ‘_2’, is equivalent to ‘_ * 2’ rather than ‘_^2’.

You can use the ‘_’ symbol any number of times; for example,

You have: m
You want:


        Definition: 1 m
You have: _ _
You want:


        Definition: 1 m^2

Using ‘_’ before a conversion has been performed (e.g., immediately after invocation) generates an error:

You have: _


          ^
No previous result; '_' not set

Accordingly, ‘_’ serves no purpose when units is invoked non-interactively.

If units is invoked with the ‘--verbose’ option (see Invoking Units), the value of ‘_’ is not expanded:

You have: mile
You want: ft


        mile = 5280 ft


        mile = (1 / 0.00018939394) ft
You have: _
You want: m


        _ = 1609.344 m


        _ = (1 / 0.00062137119) m

You can give ‘_’ at the ‘You want:’ prompt, but it usually is not very useful.

Complicated Unit Expressions¶

The units program is especially helpful in ensuring accuracy and dimensional consistency when converting lengthy unit expressions. For example, one form of the Darcy-Weisbach fluid-flow equation is

where Delta P is the pressure drop, rho is the mass density, f is the (dimensionless) friction factor, L is the length of the pipe, Q is the volumetric flow rate, and d is the pipe diameter. It might be desired to have the equation in the form

that accepted the user’s normal units; for typical units used in the US, the required conversion could be something like

You have: (8/pi^2)(lbm/ft^3)ft(ft^3/s)^2(1/in^5)
You want: psi


        * 43.533969


        / 0.022970568

The parentheses allow individual terms in the expression to be entered naturally, as they might be read from the formula. Alternatively, the multiplication could be done with the ‘*’ rather than a space; then parentheses are needed only around ‘ft^3/s’ because of its exponent:

You have: 8/pi^2 * lbm/ft^3 * ft * (ft^3/s)^2 /in^5
You want: psi


        * 43.533969


        / 0.022970568

Without parentheses, and using spaces for multiplication, the previous conversion would need to be entered as

You have: 8 lb ft ft^3 ft^3 / pi^2 ft^3 s^2 in^5
You want: psi


        * 43.533969


        / 0.022970568

Backwards Compatibility: ‘*’ and ‘-’¶

The original units assigned multiplication a higher precedence than division using the slash. This differs from the usual precedence rules, which give multiplication and division equal precedence, and can be confusing for people who think of units as a calculator.

The star operator (‘*’) included in this units program has, by default, the same precedence as division, and hence follows the usual precedence rules. For backwards compatibility you can invoke units with the ‘--oldstar’ option. Then ‘*’ has a higher precedence than division, and the same precedence as multiplication using the space.

Historically, the hyphen (‘-’) has been used in technical publications to indicate products of units, and the original units program treated it as a multiplication operator. Because units provides several other ways to obtain unit products, and because ‘-’ is a subtraction operator in general algebraic expressions, units treats the binary ‘-’ as a subtraction operator by default. For backwards compatibility use the ‘--product’ option, which causes units to treat the binary ‘-’ operator as a product operator. When ‘-’ is a multiplication operator it has the same precedence as multiplication with a space, giving it a higher precedence than division.

When ‘-’ is used as a unary operator it negates its operand. Regardless of the units options, if ‘-’ appears after ‘(’ or after ‘+’, then it will act as a negation operator. So you can always compute 20 degrees minus 12 minutes by entering ‘20 degrees + -12 arcmin’. You must use this construction when you define new units because you cannot know what options will be in force when your definition is processed.

Temperature Conversions¶

Conversions between temperatures are different from linear conversions between temperature increments—see the example below. The absolute temperature conversions are handled by units starting with ‘temp’, and you must use functional notation. The temperature-increment conversions are done using units starting with ‘deg’ and they do not require functional notation.

You have: tempF(45)
You want: tempC


        7.2222222
You have: 45 degF
You want: degC


        * 25


        / 0.04

Think of ‘tempF(x)’ not as a function but as a notation that indicates that x should have units of ‘tempF’ attached to it. See Defining Nonlinear Units. The first conversion shows that if it’s 45 degrees Fahrenheit outside, it’s 7.2 degrees Celsius. The second conversion indicates that a change of 45 degrees Fahrenheit corresponds to a change of 25 degrees Celsius. The conversion from ‘tempF(x)’ is to absolute temperature, so that

You have: tempF(45)
You want: degR


        * 504.67


        / 0.0019814929

gives the same result as

You have: tempF(45)
You want: tempR


        * 504.67


        / 0.0019814929

But if you convert ‘tempF(x)’ to ‘degC’, the output is probably not what you expect:

You have: tempF(45)
You want: degC


        * 280.37222


        / 0.0035666871

The result is the temperature in K, because ‘degC’ is defined as ‘K’, the Kelvin. For consistent results, use the ‘tempX’ units when converting to a temperature rather than converting a temperature increment.

The ‘tempC()’ and ‘tempF()’ definitions are limited to positive absolute temperatures, and giving a value that would result in a negative absolute temperature generates an error message:

You have: tempC(-275)


                    ^
Argument of function outside domain

Other Nonlinear Units¶

Some other examples of nonlinear units are numerous different ring sizes and wire gauges, the grit sizes used for abrasives, the decibel scale, shoe size, scales for the density of sugar (e.g., baume). The standard data file also supplies units for computing the area of a circle and the volume of a sphere. See the standard units data file for more details. Wire gauges with multiple zeroes are signified using negative numbers where two zeroes is ‘-1’. Alternatively, you can use the synonyms ‘g00’, ‘g000’, and so on that are defined in the standard units data file.

You have: wiregauge(11)
You want: inches


        * 0.090742002


        / 11.020255
You have: brwiregauge(g00)
You want: inches


        * 0.348


        / 2.8735632
You have: 1 mm
You want: wiregauge


        18.201919
You have: grit_P(600)
You want: grit_ansicoated


        342.76923

The last example shows the conversion from P graded sand paper, which is the European standard and may be marked “P600” on the back, to the USA standard.

You can compute the area of a circle using the nonlinear unit, ‘circlearea’. You can also do this using the circularinch or circleinch. The next example shows two ways to compute the area of a circle with a five inch radius and one way to compute the volume of a sphere with a radius of one meter.

You have: circlearea(5 in)
You want: in2


        * 78.539816


        / 0.012732395
You have: 10^2 circleinch
You want: in2


        * 78.539816


        / 0.012732395
You have: spherevol(meter)
You want: ft3


        * 147.92573


        / 0.0067601492

The inverse of a nonlinear conversion is indicated by prefixing a tilde (‘~’) to the nonlinear unit name:

You have: ~wiregauge(0.090742002 inches)
You want:


        Definition: 11

You can give a nonlinear unit definition without an argument or parentheses, and press Enter at the ‘You want:’ prompt to get the definition of a nonlinear unit; if the definition is not valid for all real numbers, the range of validity is also given. If the definition requires specific units this information is also displayed:

You have: tempC


        Definition: tempC(x) = x K + stdtemp


                    defined for x >= -273.15
You have: ~tempC


        Definition: ~tempC(tempC) = (tempC +(-stdtemp))/K


                    defined for tempC >= 0 K
You have: circlearea


        Definition: circlearea(r) = pi r^2


                    r has units m

To see the definition of the inverse use the ‘~’ notation. In this case the parameter in the functional definition will usually be the name of the unit. Note that the inverse for ‘tempC’ shows that it requires units of ‘K’ in the specification of the allowed range of values. Nonlinear unit conversions are described in more detail in Defining Nonlinear Units.

Cooking Measure¶

In North America, recipes for cooking typically measure ingredients by volume, and use units that are not always convenient multiples of each other. Suppose that you have a recipe for 6 and you wish to make a portion for 1. If the recipe calls for 2 1/2 cups of an ingredient, you might wish to know the measurements in terms of measuring devices you have available, you could use units and enter

You have: (2+1|2) cup / 6
You want: cup;1|2 cup;1|3 cup;1|4 cup;tbsp;tsp;1|2 tsp;1|4 tsp


        1|3 cup + 1 tbsp + 1 tsp

By default, if a unit in a list begins with fraction of the form 1|x and its multiplier is an integer, the fraction is given as the product of the multiplier and the numerator; for example,

You have: 12.28125 ft
You want: ft;in;1|8 in;


        12 ft + 3 in + 3|8 in

In many cases, such as the example above, this is what is wanted, but sometimes it is not. For example, a cooking recipe for 6 might call for 5 1/4 cup of an ingredient, but you want a portion for 2, and your 1-cup measure is not available; you might try

You have: (5+1|4) cup / 3
You want: 1|2 cup;1|3 cup;1|4 cup


        3|2 cup + 1|4 cup

This result might be fine for a baker who has a 1 1/2-cup measure (and recognizes the equivalence), but it may not be as useful to someone with more limited set of measures, who does want to do additional calculations, and only wants to know “How many 1/2-cup measures to I need to add?” After all, that’s what was actually asked. With the ‘--show-factor’ option, the factor will not be combined with a unity numerator, so that you get

You have: (5+1|4) cup / 3
You want: 1|2 cup;1|3 cup;1|4 cup


        3 * 1|2 cup + 1|4 cup

A user-specified fractional unit with a numerator other than 1 is never overridden, however—if a unit list specifies ‘3|4 cup;1|2 cup’, a result equivalent to 1 1/2 cups will always be shown as ‘2 * 3|4 cup’ whether or not the ‘--show-factor’ option is given.

Unit List Aliases¶

A unit list such as

cup;1|2 cup;1|3 cup;1|4 cup;tbsp;tsp;1|2 tsp;1|4 tsp

can be tedious to enter. The units program provides shorthand names for some common combinations:

hms         hours, minutes, seconds
dms         angle: degrees, minutes, seconds
time        years, days, hours, minutes and seconds
usvol       US cooking volume: cups and smaller
ftin        feet, inches and 1/8 inches
inchfine    inches subdivided to 1/64 inch

Using these shorthands, or unit list aliases, you can do the following conversions:

You have: anomalisticyear
You want: time


        1 year + 25 min + 3.4653216 sec
You have: 1|6 cup
You want: usvol


        2 tbsp + 2 tsp

You can define your own unit list aliases; see Defining Unit List Aliases.

You cannot combine a unit list alias with other units: it must appear alone at the ‘You want:’ prompt.

You can display the definition of a unit list alias by entering it at the ‘You have:’ prompt:

You have: dms


        Definition: unit list, deg;arcmin;arcsec

When you specify compact output with ‘--compact’, ‘--terse’ or ‘-t’ and perform conversion to a unit list, units lists the conversion factors for each unit in the list, separated by semicolons.

You have: year
You want: day;min;sec
365;348;45.974678

Unlike the case of regular output, zeros are included in this output list:

You have: liter
You want: cup;1|2 cup;1|4 cup;tbsp
4;0;0;3.6280454

CGS Units¶

The SI—an extension of the MKS (meter–kilogram–second) system—has largely supplanted the older CGS (centimeter–gram–second) system, but CGS units are still used in a few specialized fields, especially in physics where they lead to a more elegant formulation of Maxwell’s equations. Conversions between SI and CGS involving mechanical units are straightforward, involving powers of 10 (e.g., 1 m = 100 cm). Conversions involving electromagnetic units are more complicated, and units supports four different systems of CGS units: electrostatic units (ESU), electromagnetic units (EMU), the Gaussian system and the Heaviside–Lorentz system. The differences between these systems arise from different choices made for proportionality constants in electromagnetic equations. Coulomb’s law gives electrostatic force between two charges separated by a distance r: $r$ :

Ampere’s law gives the electromagnetic force per unit length between two current-carrying conductors separated by a distance r: $r$ :

The two constants, k_C and k_A, $k_C$ and $k_A$ , are related by the square of the speed of light: k_A = k_C / c^2. $k_A=k_C/c^2$ .

In the SI, the constants have dimensions, and an additional base unit, the ampere, measures electric current. The CGS systems do not define new base units, but express charge and current as derived units in terms of mass, length, and time. In the ESU system, the constant for Coulomb’s law is chosen to be unity and dimensionless, which defines the unit of charge. In the EMU system, the constant for Ampere’s law is chosen to be unity and dimensionless, which defines a unit of current. The Gaussian system usually uses the ESU units for charge and current; it chooses another constant so that the units for the electric and magnetic fields are the same. The Heaviside–Lorentz system is “rationalized” so that factors of 4{pi} do not appear in Maxwell’s equations. The SI system is similarly rationalized, but the other CGS systems are not. In the Heaviside–Lorentz (HLU) system the factor of 4{pi} appears in Coulomb’s law instead; this system differs from the Gaussian system by factors of the square root of 4{pi}

The dimensions of electrical quantities in the various CGS systems are different from the SI dimensions for the same units; strictly, conversions between these systems and SI are not possible. But units in different systems relate to the same physical quantities, so there is a correspondence between these units. The units program defines the units so that you can convert between corresponding units in the various systems.

The CGS definitions involve cm^(1/2) and g^(1/2), ${}^{1/2}$ and g ${}^{1/2}$ , which is problematic because units does not normally support fractional roots of base units. The ‘--units’ (‘-u’) option allows selection of a CGS unit system and works around this restriction by introducing base units for the square roots of length and mass: ‘sqrt_cm’ and ‘sqrt_g’. The centimeter then becomes ‘sqrt_cm^2’ and the gram, ‘sqrt_g^2’. This allows working from equations using the units in the CGS system, and enforcing dimensional conformity within that system. Recognized CGS arguments to the ‘--units’ option are ‘gauss[ian]’, ‘esu’, ‘emu’, ‘lhu’; the argument is case insensitive. You can also give ‘si’ which just enforces the default SI mode and displays ‘(SI)’ at the ‘You have:’ prompt to emphasize the units mode. Some other types of units are also supported as described below. Giving an unrecognized system generates a warning, and units uses SI units.

The changes resulting from the ‘--units’ option are actually controlled by the UNITS_SYSTEM environment variable. If you frequently work with one of the supported CGS units systems, you may set this environment variable rather than giving the ‘--units’ option at each invocation. As usual, an option given on the command line overrides the setting of the environment variable. For example, if you would normally work with Gaussian units but might occasionally work with SI, you could set UNITS_SYSTEM to ‘gaussian’ and specify SI with the ‘--units’ option. Unlike the argument to the ‘--units’ option, the value of UNITS_SYSTEM is case sensitive, so setting a value of ‘EMU’ will have no effect other than to give an error message and set SI units.

The CGS definitions appear as conditional settings in the standard units data file, which you can consult for more information on how these units are defined, or on how to define an alternate units system.

The ESU system derives the electromagnetic units from its unit of charge, the statcoulomb, which is defined from Coulomb’s law. The statcoulomb equals dyne^(1/2) cm, or cm^(3/2) g^(1/2) s^(−1). ${\mathrm{dyne}}^{1/2}\mathrm{cm}$ , ${\mathrm{cm}}^{3/2}{g}^{1/2}{s}^{-1}$ . The unit of current, the statampere, is statcoulomb sec, analogous to the relationship in SI. Other electrical units are then derived in a manner similar to that for SI units; the units use the SI names prefixed by ‘stat-’, e.g., ‘statvolt’ or ‘statV’. The prefix ‘st-’ is also recognized (e.g., ‘stV’).

The EMU system derives the electromagnetic units from its unit of current, the abampere, which is defined in terms of Ampere’s law. The abampere is equal to dyne^(1/2), or cm^(1/2) g^(1/2) s^(−1). ${\mathrm{dyne}}^{1/2}$ , ${\mathrm{cm}}^{1/2}{g}^{1/2}{s}^{-1}$ . The unit of charge, the abcoulomb, is abampere sec, again analogous to the SI relationship. Other electrical units are then derived in a manner similar to that for SI units; the units use the SI names prefixed by ‘ab-’, e.g., ‘abvolt’ or ‘abV’. The magnetic field units include the gauss, the oersted and the maxwell.

The Gaussian units system, which was also known as the Symmetric System, uses the same charge and current units as the ESU system (e.g., ‘statC’, ‘statA’); it differs by defining the magnetic field so that it has the same units as the electric field. The resulting magnetic field units are the same ones used in the EMU system: the gauss, the oersted and the maxwell.

The Heaviside–Lorentz system appears to lack named units. We define five basic units, ‘hlu_charge’, ‘hlu_current’, ‘hlu_volt’, ‘hlu_efield’ and ‘hlu_bfield’ for conversions with this system. It is important to remember that with all of the CGS systems, the units may look the same but mean something different. The HLU system and Gaussian systems both measure magnetic field using the same CGS dimensions, but the amount of magnetic field with the same units is different in the two systems.

The CGS systems define units that measure the same thing but may have conflicting dimensions. Furthermore, the dimensions of the electromagnetic CGS units are never compatible with SI. But if you measure charge in two different systems you have measured the same physical thing, so there is a correspondence between the units in the different systems, and units supports conversions between corresponding units. When running with SI, units defines all of the CGS units in terms of SI. When you select a CGS system, units defines the SI units and the other CGS system units in terms of the system you have selected.

(Gaussian) You have: statA


           You want: abA


        * 3.335641e-11


        / 2.9979246e+10
(Gaussian) You have: abA


           You want: sqrt(dyne)
conformability error


        2.9979246e+10 sqrt_cm^3 sqrt_g / s^2


        1 sqrt_cm sqrt_g / s

In the above example, units converts between the current units statA and abA even though the abA, from the EMU system, has incompatible dimensions. This works because in Gaussian mode, the abA is defined in terms of the statA, so it does not have the correct definition for EMU; consequently, you cannot convert the abA to its EMU definition.

One challenge of conversion is that because the CGS system has fewer base units, quantities that have different dimensions in SI may have the same dimension in a CGS system. And yet, they may not have the same conversion factor. For example, the unit for the E field and B fields are the same in the Gaussian system, but the conversion factors to SI are quite different. This means that correct conversion is only possible if you keep track of what quantity is being measured. You cannot convert statV/cm to SI without indicating which type of field the unit measures. To aid in dimensional analysis, units defines various dimension units such as LENGTH, TIME, and CHARGE to be the appropriate dimension in SI. The electromagnetic dimensions such as B_FIELD or E_FIELD may be useful aids both for conversion and dimensional analysis in CGS. You can convert them to or from CGS in order to perform SI conversions that in some cases will not work directly due to dimensional incompatibilities. This example shows how the Gaussian system uses the same units for all of the fields, but they all have different conversion factors with SI.

(Gaussian) You have: statV/cm


           You want: E_FIELD


        * 29979.246


        / 3.335641e-05
(Gaussian) You have: statV/cm


           You want: B_FIELD


        * 0.0001


        / 10000
(Gaussian) You have: statV/cm


           You want: H_FIELD


        * 79.577472


        / 0.012566371
(Gaussian) You have: statV/cm


           You want: D_FIELD


        * 2.6544187e-07


        / 3767303.1

The next example shows that the oersted cannot be converted directly to the SI unit of magnetic field, A/m, because the dimensions conflict. We cannot redefine the ampere to make this work because then it would not convert with the statampere. But you can still do this conversion as shown below.

(Gaussian) You have: oersted


           You want: A/m
conformability error


        1 sqrt_g / s sqrt_cm


        29979246 sqrt_cm sqrt_g / s^2
(Gaussian) You have: oersted


           You want: H_FIELD


        * 79.577472


        / 0.012566371

Natural Units¶

Like the CGS units, “natural” units are an alternative to the SI system used primarily physicists in different fields, with different systems tailored to different fields of study. These systems are “natural” because the base measurements are defined using physical constants instead of arbitrary values such as the meter or second. In different branches of physics, different physical constants are more fundamental, which has given rise to a variety of incompatible natural unit systems.

The supported systems are the “natural” units (which seem to have no better name) used in high energy physics and cosmology, the Planck units, often used my scientists working with gravity, and the Hartree atomic units are favored by those working in physical chemistry and condensed matter physics.

You can select the various natural units using the ‘--units’ option in the same way that you select the CGS units. The “natural” units come in two types, a rationalized system derived from the Heaviside–Lorentz units and an unrationalized system derived from the Gaussian system. You can select these using ‘natural’ and ‘natural-gauss’ respectively. For conversions in SI mode, several unit names starting with ‘natural’ are available. This “natural” system is defined by setting {hbar}, c and the Boltzman constant to 1. Only a single base unit remains: the electron volt.

The Planck units exist in a variety of forms, and units supports two. Both supported forms are rationalized, in that factors of 4{pi} do not appear in Maxwell’s equations. However, Planck units can also differ based on how the gravitational constant is treated. This system is similar to the natural units in that c, {hbar}, and Boltzman’s constant are set to 1, but in this system, Newton’s gravitational constant, G is also fixed. In the “reduced” Planck system, 8{pi}G = 1 $8\pi G=1$ whereas in the unreduced system G = 1. $G=1$ . The reduced system eliminates factors of 8{pi} from the Einstein field equations for gravitation, so this is similar to the process of forming rationalized units to simplify Maxwell’s equations. To obtain the unreduced system use the name ‘planck’ and for the reduced Planck units, ‘planck-red’. Units such as ‘planckenergy’ and ‘planckenergy_red’ enable you to convert the unreduced and reduced Planck energy unit in SI mode between the various systems. In Planck units, all measurements are dimensionless.

The final natural unit system is the Hartree atomic units. Like the Planck units, all measurements in the Hartree units are dimensionless, but this system is defined by defined from completely different physical constants: the electron mass, Planck’s constant, the electron charge, and the Coulomb constant are the defining physical quantities, which are all set to unity. To invoke this system with the ‘--units’ option use the name ‘hartree’.

Prompt Prefix¶

If a unit system is specified with the ‘--units’ option, the selected system’s name is prepended to the ‘You have:’ prompt as a reminder, e.g.,

(Gaussian) You have: stC


           You want:


        Definition: statcoulomb = sqrt(dyne) cm = 1 sqrt_cm^3 sqrt_g / s

You can suppressed the prefix by including a line

!prompt

with no argument in a site or personal units data file. The prompt can be conditionally suppressed by including such a line within ‘!var’ ... ‘!endvar’ constructs, e.g.,

!var UNITS_SYSTEM gaussian gauss
!prompt
!endvar

This might be appropriate if you normally use Gaussian units and find the prefix distracting but want to be reminded when you have selected a different CGS system.

Units Data Files¶

The units and prefixes that units can convert are defined in the units data file, typically ‘/usr/share/units/definitions.units’. If you can’t find this file, run units --version to get information on the file locations for your installation. Although you can extend or modify this data file if you have appropriate user privileges, it’s usually better to put extensions in separate files so that the definitions will be preserved if you update units.

You can include additional data files in the units database using the ‘!include’ command in the standard units data file. For example

!include    /usr/local/share/units/local.units

might be appropriate for a site-wide supplemental data file. The location of the ‘!include’ statement in the standard units data file is important; later definitions replace earlier ones, so any definitions in an included file will override definitions before the ‘!include’ statement in the standard units data file. With normal invocation, no warning is given about redefinitions; to ensure that you don’t have an unintended redefinition, run units -c after making changes to any units data file.

If you want to add your own units in addition to or in place of standard or site-wide supplemental units data files, you can include them in the ‘.units’ file in your home directory. If this file exists it is read after the standard units data file, so that any definitions in this file will replace definitions of the same units in the standard data file or in files included from the standard data file. This file will not be read if any units files are specified on the command line. (Under Windows the personal units file is named ‘unitdef.units’.) Running units -V will display the location and name of your personal units file.

The units program first tries to determine your home directory from the HOME environment variable. On systems running Microsoft Windows, if HOME does not exist, units attempts to find your home directory from HOMEDRIVE, HOMEPATH and USERPROFILE. You can specify an arbitrary file as your personal units data file with the MYUNITSFILE environment variable; if this variable exists, its value is used without searching your home directory. The default units data files are described in more detail in Data Files.

Defining New Units and Prefixes¶

A unit is specified on a single line by giving its name and an equivalence. Comments start with a ‘#’ character, which can appear anywhere in a line. The backslash character (‘\’) acts as a continuation character if it appears as the last character on a line, making it possible to spread definitions out over several lines if desired. A file can be included by giving the command ‘!include’ followed by the file’s name. The ‘!’ must be the first character on the line. The file will be sought in the same directory as the parent file unless you give a full path. The name of the file to be included cannot contain spaces or the comment character ‘#’.

Unit names must not contain any of the operator characters ‘+’, ‘-’, ‘*’, ‘/’, ‘|’, ‘^’, ‘;’, ‘~’, the comment character ‘#’, or parentheses. They cannot begin or end with an underscore (‘_’), a comma (‘,’) or a decimal point (‘.’). The figure dash (U+2012), typographical minus ('-’; U+2212), and en dash ('--’; U+2013) are converted to the operator ‘-’, so none of these characters can appear in unit names. Names cannot begin with a digit, and if a name ends in a digit other than zero or one, the digit must be preceded by a string beginning with an underscore, and afterwards consisting only of digits, decimal points, or commas. For example, ‘foo_2’, ‘foo_2,1’, or ‘foo_3.14’ are valid names but ‘foo2’ or ‘foo_a2’ are invalid. The underscore is necessary because without it, units cannot determine whether ‘foo2’ is a unit name or represents ‘foo^2’. Zero and one are exceptions because units never interprets them as exponents.

You could define nitrous oxide as

N2O     nitrogen 2  + oxygen

but would need to define nitrogen dioxide as

NO_2    nitrogen + oxygen 2

Be careful to define new units in terms of old ones so that a reduction leads to the primitive units, which are marked with ‘!’ characters. Dimensionless units are indicated by using the string ‘!dimensionless’ for the unit definition.

When adding new units, be sure to use the ‘-c’ option to check that the new units reduce properly. If you create a loop in the units definitions, then units will hang when invoked with the ‘-c’ option. You will need to use the ‘--check-verbose’ option, which prints out each unit as it is checked. The program will still hang, but the last unit printed will be the unit that caused the infinite loop.

If you define any units that contain ‘+’ characters in their definitions, carefully check them because the ‘-c’ option will not catch non-conformable sums. Be careful with the ‘-’ operator as well. When used as a binary operator, the ‘-’ character can perform addition or multiplication depending on the options used to invoke units. To ensure consistent behavior use ‘-’ only as a unary negation operator when writing units definitions. To multiply two units leave a space or use the ‘*’ operator with care, recalling that it has two possible precedence values and may require parentheses to ensure consistent behavior. To compute the difference of ‘foo’ and ‘bar’ write ‘foo+(-bar)’ or even ‘foo+-bar’.

You may wish to intentionally redefine a unit. When you do this, and use the ‘-c’ option, units displays a warning message about the redefinition. You can suppress these warnings by redefining a unit using a ‘+’ at the beginning of the unit name. Do not include any white space between the ‘+’ and the redefined unit name.

Here is an example of a short data file that defines some basic units:

m       !               # The meter is a primitive unit
sec     !               # The second is a primitive unit
rad     !dimensionless  # A dimensionless primitive unit
micro-  1e-6            # Define a prefix
minute  60 sec          # A minute is 60 seconds
hour    60 min          # An hour is 60 minutes
inch    72 m            # Inch defined incorrectly terms of meters
ft      12 inches       # The foot defined in terms of inches
mile    5280 ft         # And the mile
+inch   0.0254 m        # Correct redefinition, warning suppressed

A unit that ends with a ‘-’ character is a prefix. If a prefix definition contains any ‘/’ characters, be sure they are protected by parentheses. If you define ‘half- 1/2’, then ‘halfmeter’ would be equivalent to ‘1 / (2 meter)’.

Defining Nonlinear Units¶

Some unit conversions of interest are nonlinear; for example, temperature conversions between the Fahrenheit and Celsius scales cannot be done by simply multiplying by conversion factors.

When you give a linear unit definition such as ‘inch 2.54 cm’ you are providing information that units uses to convert values in inches into primitive units of meters. For nonlinear units, you give a functional definition that provides the same information.

Nonlinear units are represented using a functional notation. It is best to regard this notation not as a function call but as a way of adding units to a number, much the same way that writing a linear unit name after a number adds units to that number. Internally, nonlinear units are defined by a pair of functions that convert to and from linear units in the database, so that an eventual conversion to primitive units is possible.

Here is an example nonlinear unit definition:

tempF(x) units=[1;K] domain=[-459.67,) range=[0,) \


            (x+(-32)) degF + stdtemp ; (tempF+(-stdtemp))/degF + 32

A nonlinear unit definition comprises a unit name, a formal parameter name, two functions, and optional specifications for units, the domain, and the range (the domain of the inverse function). The functions tell units how to convert to and from the new unit. To produce valid results, the arguments of these functions need to have the correct dimensions and be within the domains for which the functions are defined.

The definition begins with the unit name followed immediately (with no spaces) by a ‘(’ character. In the parentheses is the name of the formal parameter. Next is an optional specification of the units required by the functions in the definition. In the example above, the ‘units=[1;K]’ specification indicates that the ‘tempF’ function requires an input argument conformable with ‘1’ (i.e., the argument is dimensionless), and that the inverse function requires an input argument conformable with ‘K’. For normal nonlinear units definition, the forward function will always take a dimensionless argument; in general, the inverse function will need units that match the quantity measured by your nonlinear unit. Specifying the units enables units to perform error checking on function arguments, and also to assign units to domain and range specifications, which are described later.

Next the function definitions appear. In the example above, the ‘tempF’ function is defined by

tempF(x) = (x+(-32)) degF + stdtemp

This gives a rule for converting ‘x’ in the units ‘tempF’ to linear units of absolute temperature, which makes it possible to convert from tempF to other units.

To enable conversions to Fahrenheit, you must give a rule for the inverse conversions. The inverse will be ‘x(tempF)’ and its definition appears after a ‘;’ character. In our example, the inverse is

x(tempF) = (tempF+(-stdtemp))/degF + 32

This inverse definition takes an absolute temperature as its argument and converts it to the Fahrenheit temperature. The inverse can be omitted by leaving out the ‘;’ character and the inverse definition, but then conversions to the unit will not be possible. If the inverse definition is omitted, the ‘--check’ option will display a warning. It is up to you to calculate and enter the correct inverse function to obtain proper conversions; the ‘--check’ option tests the inverse at one point and prints an error if it is not valid there, but this is not a guarantee that your inverse is correct.

With some definitions, the units may vary. For example, the definition

square(x)       x^2

can have any arbitrary units, and can also take dimensionless arguments. In such a case, you should not specify units. If a definition takes a root of its arguments, the definition is valid only for units that yield such a root. For example,

squirt(x)       sqrt(x)

is valid for a dimensionless argument, and for arguments with even powers of units.

Some definitions may not be valid for all real numbers. In such cases, units can handle errors better if you specify an appropriate domain and range. You specify the domain and range as shown below:

baume(d) units=[1;g/cm^3] domain=[0,130.5] range=[1,10] \


         (145/(145-d)) g/cm^3 ; (baume+-g/cm^3) 145 / baume

In this example the domain is specified after ‘domain=’ with the endpoints given in brackets. In accord with mathematical convention, square brackets indicate a closed interval (one that includes its endpoints), and parentheses indicate an open interval (one that does not include its endpoints). An interval can be open or closed on one or both ends; an interval that is unbounded on either end is indicated by omitting the limit on that end. For example, a quantity to which decibel (dB) is applied may have any value greater than zero, so the range is indicated by ‘(0,)’:

decibel(x) units=[1;1] range=(0,) 10^(x/10); 10 log(decibel)

If the domain or range is given, the second endpoint must be greater than the first.

The domain and range specifications can appear independently and in any order along with the units specification. The values for the domain and range endpoints are attached to the units given in the units specification, and if necessary, the parameter value is adjusted for comparison with the endpoints. For example, if a definition includes ‘units=[1;ft]’ and ‘range=[3,)’, the range will be taken as 3 ft to infinity. If the function is passed a parameter of ‘900 mm’, that value will be adjusted to 2.9527559 ft, which is outside the specified range. If you omit the units specification from the previous example, units can not tell whether you intend the lower endpoint to be 3 ft or 3 microfurlongs, and can not adjust the parameter value of 900 mm for comparison. Without units, numerical values other than zero or plus or minus infinity for domain or range endpoints are meaningless, and accordingly they are not allowed. If you give other values without units, then the definition will be ignored and you will get an error message.

Although the units, domain, and range specifications are optional, it’s best to give them when they are applicable; doing so allows units to perform better error checking and give more helpful error messages. Giving the domain and range also enables the ‘--check’ option to find a point in the domain to use for its point check of your inverse definition.

You can make synonyms for nonlinear units by providing both the forward and inverse functions; inverse functions can be obtained using the ‘~’ operator. So to create a synonym for ‘tempF’ you could write

fahrenheit(x) units=[1;K] tempF(x); ~tempF(fahrenheit)

This is useful for creating a nonlinear unit definition that differs slightly from an existing definition without having to repeat the original functions. For example,

dBW(x)     units=[1;W] range=[0,) dB(x) W ;  ~dB(dBW/W) 

If you wish a synonym to refer to an existing nonlinear unit without modification, you can do so more simply by adding the synonym with appended parentheses as a new unit, with the existing nonlinear unit—without parentheses—as the definition. So to create a synonym for ‘tempF’ you could write

fahrenheit()  tempF

The definition must be a nonlinear unit; for example, the synonym

fahrenheit()  meter

will result in an error message when units starts.

You may occasionally wish to define a function that operates on units. This can be done using a nonlinear unit definition. For example, the definition below provides conversion between radius and the area of a circle. This definition requires a length as input and produces an area as output, as indicated by the ‘units=’ specification. Specifying the range as the nonnegative numbers can prevent cryptic error messages.

circlearea(r) units=[m;m^2] range=[0,)   pi r^2 ; sqrt(circlearea/pi)

Defining Piecewise Linear Units¶

Sometimes you may be interested in a piecewise linear unit such as many wire gauges. Piecewise linear units can be defined by specifying conversions to linear units on a list of points. Conversion at other points will be done by linear interpolation. A partial definition of zinc gauge is

zincgauge[in] 1 0.002, 10 0.02, 15 0.04, 19 0.06, 23 0.1

In this example, ‘zincgauge’ is the name of the piecewise linear unit. The definition of such a unit is indicated by the embedded ‘[’ character. After the bracket, you should indicate the units to be attached to the numbers in the table. No spaces can appear before the ‘]’ character, so a definition like ‘foo[kg meters]’ is invalid; instead write ‘foo[kg*meters]’. The definition of the unit consists of a list of pairs optionally separated by commas. This list defines a function for converting from the piecewise linear unit to linear units. The first item in each pair is the function argument; the second item is the value of the function at that argument (in the units specified in brackets). In this example, we define ‘zincgauge’ at five points. For example, we set ‘zincgauge(1)’ equal to ‘0.002 in’. Definitions like this may be more readable if written using continuation characters as

zincgauge[in] \


     1 0.002  \


    10 0.02   \


    15 0.04   \


    19 0.06   \


    23 0.1

With the preceding definition, the following conversion can be performed:

You have: zincgauge(10)
You want: in


    * 0.02


    / 50
You have: .01 inch
You want: zincgauge


    5

If you define a piecewise linear unit that is not strictly monotonic, then the inverse will not be well defined. If the inverse is requested for such a unit, units will return the smallest inverse.

After adding nonlinear units definitions, you should normally run units --check to check for errors. If the ‘units’ keyword is not given, the ‘--check’ option checks a nonlinear unit definition using a dimensionless argument, and then checks using an arbitrary combination of units, as well as the square and cube of that combination; a warning is given if any of these tests fail. For example,

Warning: function 'squirt(x)' defined as 'sqrt(x)'


         failed for some test inputs:


         squirt(7(kg K)^1): Unit not a root


         squirt(7(kg K)^3): Unit not a root

Running units --check will print a warning if a non-monotonic piecewise linear unit is encountered. For example, the relationship between ANSI coated abrasive designation and mean particle size is non-monotonic in the vicinity of 800 grit:

ansicoated[micron] \


     . . .


    600 10.55 \


    800 11.5 \


    1000 9.5 \

Running units --check would give the error message

Table 'ansicoated' lacks unique inverse around entry 800

Although the inverse is not well defined in this region, it’s not really an error. Viewing such error messages can be tedious, and if there are enough of them, they can distract from true errors. Error checking for nonlinear unit definitions can be suppressed by giving the ‘noerror’ keyword; for the examples above, this could be done as

squirt(x) noerror domain=[0,) range=[0,) sqrt(x); squirt^2
ansicoated[micron] noerror \


     . . .

Use the ‘noerror’ keyword with caution. The safest approach after adding a nonlinear unit definition is to run units --check and confirm that there are no actual errors before adding the ‘noerror’ keyword.

Defining Unit List Aliases¶

Unit list aliases are treated differently from unit definitions, because they are a data entry shorthand rather than a true definition for a new unit. A unit list alias definition begins with ‘!unitlist’ and includes the alias and the definition; for example, the aliases included in the standard units data file are

!unitlist   hms     hr;min;sec
!unitlist   time    year;day;hr;min;sec
!unitlist   dms     deg;arcmin;arcsec
!unitlist   ftin    ft;in;1|8 in
!unitlist   usvol   cup;3|4 cup;2|3 cup;1|2 cup;1|3 cup;1|4 cup;\


                    tbsp;tsp;1|2 tsp;1|4 tsp;1|8 tsp

Unit list aliases are only for unit lists, so the definition must include a ‘;’. Unit list aliases can never be combined with units or other unit list aliases, so the definition of ‘time’ shown above could not have been shortened to ‘year;day;hms’.

As usual, be sure to run units --check to ensure that the units listed in unit list aliases are conformable.

Format Specification¶

The format specification recognized with the ‘--output-format’ option is a subset of that for printf(). The format specification has the form %[flags][width][.precision]type; it must begin with ‘%’, and must end with a floating-point type specifier: ‘g’ or ‘G’ to specify the number of significant digits, ‘e’ or ‘E’ for scientific notation, and ‘f’ for fixed-point decimal. The ISO C99 standard added the ‘F’ type for fixed-point decimal and the ‘a’ and ‘A’ types for hexadecimal floating point; these types are allowed with compilers that support them. Type length modifiers (e.g., ‘L’ to indicate a long double) are inapplicable and are not allowed.

The default format for units is ‘%.8g’; for greater precision, you could specify ‘-o %.15g’. The ‘g’ and ‘G’ format types use exponential format whenever the exponent would be less than -4, so the value 0.000013 displays as ‘1.3e-005’. These types also use exponential notation when the exponent is greater than or equal to the precision, so with the default format, the value 5e7 displays as ‘50000000’ and the value 5e8 displays as ‘5e+008’. If you prefer fixed-point display, you might specify ‘-o %.8f’; however, small numbers will display very few significant digits, and values less than 0.5e-8 will show nothing but zeros.

The format specification may include one or more optional flags: ‘+’, ‘ ’ (space), ‘#’, ‘-’, or ‘0’ (the digit zero). The digit-grouping flag ‘'’ is allowed with compilers that support it. Flags are followed by an optional value for the minimum field width, and an optional precision specification that begins with a period (e.g., ‘.6’). The field width includes the digits, decimal point, the exponent, thousands separators (with the digit-grouping flag), and the sign if any of these are shown.

Flags¶

The ‘+’ flag causes the output to have a sign (‘+’ or ‘-’). The space flag ‘ ’ is similar to the ‘+’ flag, except that when the value is positive, it is prefixed with a space rather than a plus sign; this flag is ignored if the ‘+’ flag is also given. The ‘+’ or ‘ ’ flag could be useful if conversions might include positive and negative results, and you wanted to align the decimal points in exponential notation. The ‘#’ flag causes the output value to contain a decimal point in all cases; by default, the output contains a decimal point only if there are digits (which can be trailing zeros) to the right of the point. With the ‘g’ or ‘G’ types, the ‘#’ flag also prevents the suppression of trailing zeros. The digit-grouping flag ‘'’ shows a thousands separator in digits to the left of the decimal point. This can be useful when displaying large numbers in fixed-point decimal; for example, with the format ‘%f’,

You have: mile
You want: microfurlong


        * 8000000.000000


        / 0.000000

the magnitude of the first result may not be immediately obvious without counting the digits to the left of the decimal point. If the thousands separator is the comma (‘,’), the output with the format ‘%'f’ might be

You have: mile
You want: microfurlong


        * 8,000,000.000000


        / 0.000000

making the magnitude readily apparent. Unfortunately, few compilers support the digit-grouping flag.

With the ‘-’ flag, the output value is left aligned within the specified field width. If a field width greater than needed to show the output value is specified, the ‘0’ (zero) flag causes the output value to be left padded with zeros until the specified field width is reached; for example, with the format ‘%011.6f’,

You have: troypound
You want: grain


        * 5760.000000


        / 0000.000174

The ‘0’ flag has no effect if the ‘-’ (left align) flag is given.

Field Width¶

By default, the output value is left aligned and shown with the minimum width necessary for the specified (or default) precision. If a field width greater than this is specified, the value shown is right aligned, and padded on the left with enough spaces to provide the specified field width. A width specification is typically used with fixed-point decimal to have columns of numbers align at the decimal point; this arguably is less useful with units than with long columnar output, but it may nonetheless assist in quickly assessing the relative magnitudes of results. For example, with the format ‘%12.6f’,

You have: km
You want: in


        * 39370.078740


        /     0.000025
You have: km
You want: rod


        *   198.838782


        /     0.005029
You have: km
You want: furlong


        *     4.970970


        /     0.201168

Precision¶

The meaning of “precision” depends on the format type. With ‘g’ or ‘G’, it specifies the number of significant digits (like the ‘--digits’ option); with ‘e’, ‘E’, ‘f’, or ‘F’, it specifies the maximum number of digits to be shown after the decimal point.

With the ‘g’ and ‘G’ format types, trailing zeros are suppressed, so the results may sometimes have fewer digits than the specified precision (as indicated above, the ‘#’ flag causes trailing zeros to be displayed).

The default precision is 6, so ‘%g’ is equivalent to ‘%.6g’, and would show the output to six significant digits. Similarly, ‘%e’ or ‘%f’ would show the output with six digits after the decimal point.

The C printf() function allows a precision of arbitrary size, whether or not all of the digits are meaningful. With most compilers, the maximum internal precision with units is 15 decimal digits (or 13 hexadecimal digits). With the ‘--digits’ option, you are limited to the maximum internal precision; with the ‘--output-format’ option, you may specify a precision greater than this, but it may not be meaningful. In some cases, specifying excess precision can result in rounding artifacts. For example, a pound is exactly 7000 grains, but with the format ‘%.18g’, the output might be

You have: pound
You want: grain


        * 6999.9999999999991


        / 0.00014285714285714287

With the format ‘%.25g’ you might get the following:

You have: 1/3
You want:


        Definition: 0.333333333333333314829616256247

In this case the displayed value includes a series of digits that represent the underlying binary floating-point approximation to 1/3 but are not meaningful for the desired computation. In general, the result with excess precision is system dependent. The precision affects only the display of numbers; if a result relies on physical constants that are not known to the specified precision, the number of physically meaningful digits may be less than the number of digits shown.

See the documentation for printf() for more detailed descriptions of the format specification.

The ‘--output-format’ option is incompatible with the ‘--exponential’ or ‘--digits’ options; if the former is given in combination with either of the latter, the format is controlled by the last option given.

Locale¶

A locale is a subset of a user’s environment that indicates the user’s language and country, and some attendant preferences, such as the formatting of dates. The units program attempts to determine the locale from the POSIX setlocale function; if this cannot be done, units examines the environment variables LC_CTYPE and LANG. On POSIX systems, a locale is of the form language_country, where language is the two-character code from ISO 639-1 and country is the two-character code from ISO 3166-1; language is lower case and country is upper case. For example, the POSIX locale for the United Kingdom is en_GB.

On systems running Microsoft Windows, the value returned by setlocale() is different from that on POSIX systems; units attempts to map the Windows value to a POSIX value by means of a table in the file ‘locale_map.txt’ in the same directory as the other data files. The file includes entries for many combinations of language and country, and can be extended to include other combinations. The ‘locale_map.txt’ file comprises two tab-separated columns; each entry is of the form

where POSIX-locale is as described above, and Windows-locale typically spells out both the language and country. For example, the entry for the United States is

English_United States   en_US

You can force units to run in a desired locale by using the ‘-l’ option.

In order to create unit definitions for a particular locale you begin a block of definitions in a unit datafile with ‘!locale’ followed by a locale name. The ‘!’ must be the first character on the line. The units program reads the following definitions only if the current locale matches. You end the block of localized units with ‘!endlocale’. Here is an example, which defines the British gallon.

!locale en_GB
gallon       4.54609 liter
!endlocale

Additional Localization¶

Sometimes the locale isn’t sufficient to determine unit preferences. There could be regional preferences, or a company could have specific preferences. Though probably uncommon, such differences could arise with the choice of English customary units outside of English-speaking countries. To address this, units allows specifying definitions that depend on environment variable settings. The environment variables can be controled based on the current locale, or the user can set them to force a particular group of definitions.

A conditional block of definitions in a units data file begins with either ‘!var’ or ‘!varnot’ following by an environment variable name and then a space separated list of values. The leading ‘!’ must appear in the first column of a units data file, and the conditional block is terminated by ‘!endvar’. Definitions in blocks beginning with ‘!var’ are executed only if the environment variable is exactly equal to one of the listed values. Definitions in blocks beginning with ‘!varnot’ are executed only if the environment variable does not equal any of the list values.

The inch has long been a customary measure of length in many places. The word comes from the Latin uncia meaning “one twelfth,” referring to its relationship with the foot. By the 20th century, the inch was officially defined in English-speaking countries relative to the yard, but until 1959, the yard differed slightly among those countries. In France the customary inch, which was displaced in 1799 by the meter, had a different length based on a french foot. These customary definitions could be accommodated as follows:

!var INCH_UNIT usa
yard          3600|3937 m
!endvar
!var INCH_UNIT canada
yard          0.9144 meter
!endvar
!var INCH_UNIT uk
yard          0.91439841 meter
!endvar
!var INCH_UNIT canada uk usa
foot          1|3 yard
inch          1|12 foot
!endvar
!var INCH_UNIT france
foot          144|443.296 m
inch          1|12 foot
line          1|12 inch
!endvar
!varnot INCH_UNIT usa uk france canada
!message Unknown value for INCH_UNIT
!endvar

When units reads the above definitions it will check the environment variable INCH_UNIT and load only the definitions for the appropriate section. If INCH_UNIT is unset or is not set to one of the four values listed, then units will run the last block. In this case that block uses the ‘!message’ command to display a warning message. Alternatively that block could set default values.

In order to create default values that are overridden by user settings the data file can use the ‘!set’ command, which sets an environment variable only if it is not already set; these settings are only for the current units invocation and do not persist. So if the example above were preceded by ‘!set INCH_UNIT france’, then this would make ‘france’ the default value for INCH_UNIT. If the user had set the variable in the environment before invoking units, then units would use the user’s value.

To link these settings to the user’s locale you combine the ‘!set’ command with the ‘!locale’ command. If you wanted to combine the above example with suitable locales you could do by preceding the above definition with the following:

!locale en_US
!set INCH_UNIT usa
!endlocale
!locale en_GB
!set INCH_UNIT uk
!endlocale
!locale en_CA
!set INCH_UNIT canada
!endlocale
!locale fr_FR
!set INCH_UNIT france
!endlocale
!set INCH_UNIT france

These definitions set the overall default for INCH_UNIT to ‘france’ and set default values for four locales appropriately. The overall default setting comes last so that it only applies when INCH_UNIT was not set by one of the other commands or by the user.

If the variable given after ‘!var’ or ‘!varnot’ is undefined, then units prints an error message and ignores the definitions that follow. Use ‘!set’ to create defaults to prevent this situation from arising. The ‘-c’ option only checks the definitions that are active for the current environment and locale, so when adding new definitions take care to check that all cases give rise to a well defined set of definitions.