4.1 Physical Constants

Many constants of nature are defined, including these:

pi          ratio of circumference of a circle to its diameter
c           speed of light
e           charge on an electron
force       acceleration of gravity
mole        Avogadro’s number
water       pressure per unit height of water
Hg          pressure per unit height of mercury
au          astronomical unit
k           Boltzman’s constant
mu0         permeability of vacuum
epsilon0    permittivity of vacuum
G           Gravitational constant
mach        speed of sound

The standard data file includes numerous other constants. Also included are the densities of various ingredients used in baking so that ‘2 cups flour_sifted’ can be converted to ‘grams’. This is not an exhaustive list. Consult the units data file to see the complete list, or to see the definitions that are used.

4.2 Atomic Masses of the Elements

The data file elements.units includes atomic masses for most elements and most known isotopes. If the mole fractions of constituent isotopes are known, an elemental mass is calculated from the sum of the products of the mole fractions and the masses of the constituent isotopes. If the mole fractions are not known, the mass of the most stable isotope—if known—is given as the elemental mass. For radioactive elements with atomic numbers 95 or greater, the mass number of the most stable isotope is not specified, because the list of studied isotopes is still incomplete. If no stable isotope is known, no elemental mass is given, and you will need to choose the most appropriate isotope.

The data are obtained from the US National Institute for Standards and Technology (NIST): https://physics.nist.gov/cgi-bin/Compositions/stand_alone.pl?ele=&all=all&ascii=ascii2&isotype=all. The elements.units file can be generated from these data using the elemcvt command included with the distribution.

4.3 Currency Exchange Rates and Consumer Price Index

The data file currency.units includes currency conversion rates; the file cpi.units includes the US Consumer Price Index (CPI), published by the US Bureau of Labor Statistics. The data are updated monthly by the BLS; see Updating Currency Exchange Rates and CPI for information on updating currency.units and cpi.units.

4.4 English Customary Units

English customary units differ in various ways among 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, pole, perch, 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 differed from the international values by about 2 ppm.

The 1959 redefinition of the foot was legally binding in the US but allowed continued use of the previous definition of the foot for geodetic surveying. It was assumed that this use would be temporary, but use persisted, leading to confusion and errors, and it was at odds with the intent of uniform standards. Since January 1, 2023, the US survey foot has been officially deprecated (85 FR 62698), with its use limited to historical and legacy applications.

The units program has always used the international values for these units; the legacy 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’. The pre-1959 UK values for these units can be obtained with the prefix ‘UK’.

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

4.5 Miscellaneous Notes on Unit Definitions

The ‘pound’ is a unit of mass. To get force, multiply by the force conversion unit ‘force’ or use the shorthand ‘lbf’. (Note that ‘g’ is already taken as the standard abbreviation for the gram.) The unit ‘ounce’ is also a unit of mass. The fluid ounce is ‘fluidounce’ or ‘floz’. When British capacity units differ from their US counterparts, such as the British Imperial gallon, the unit is defined both ways with ‘br’ and ‘us’ prefixes. Your locale settings will determine the value of the unprefixed unit. Currency is prefixed with its country name: ‘belgiumfranc’, ‘britainpound’.

When searching for a unit, if the specified string does not appear exactly as a unit name, then the units program will try to remove a trailing ‘s’, ‘es’. Next units will replace a trailing ‘ies’ with ‘y’. If that fails, units will check for a prefix. The database includes all of the standard metric prefixes. Only one prefix is permitted per unit, so ‘micromicrofarad’ will fail. However, prefixes can appear alone with no unit following them, so ‘micro*microfarad’ will work, as will ‘micro microfarad’.

To find out which units and prefixes are available, read the default units data files; the main data file is extensively annotated.

5.1 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.

5.2 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
                                      ^
Invalid 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
                   ^
Invalid 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.

5.3 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’.

5.4 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

5.5 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

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.

5.6 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. You might want 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

5.7 Variables Assigned at Run Time

Unit definitions are fixed once units has finished reading the units data file(s), but at run time you can assign unit expressions to variables whose names begin with an underscore, using the syntax

_name = <unit expression>

This can help manage a long calculation by saving intermediate quantities as variables that you can use later. For example, to determine the shot-noise-limited signal-to-noise ratio (SNR) of an imaging system using a helium–neon laser, you could do

You have: _lambda = 632.8 nm            # laser wavelength
You have: _nu = c / _lambda             # optical frequency
You have: _photon_energy = h * _nu
You have: _power = 550 uW
You have: _photon_count = _power * 500 ns / _photon_energy
You have: _snr = sqrt(_photon_count)
You have: _snr
You want:
        Definition: sqrt(_photon_count) = 29597.922

Except for beginning with an underscore, runtime variables follow the same naming rules as units. Because names beginning with ‘_’ are reserved for these variables and unit names cannot begin with ‘_’, runtime variables can never hide unit definitions. Runtime variables are undefined until you make an assignment to them, so if you give a name beginning with an underscore and no assignment has been made, you get an error message.

When you assign a unit expression to a runtime variable, units checks the expression to determine whether it is valid, but the resulting definition is stored as a text string that is not reduced to primitive units. The text will be processed anew each time you use the variable in a conversion or calculation; this means that if your definition depends on other runtime variables (or the special variable ‘_’), the result of calculating with your variable will change if any of those variables change. A dependence need not be direct.

Continuing the example of the laser above, suppose you have done the calculation as shown. You now wonder what happens if you switch to an argon laser:

You have: _lambda = 454.6 nm
You have: _snr
You want:
        Definition: sqrt(_photon_count) = 25086.651

If you then change the power:

You have: _power = 1 mW
You have: _snr
You want:
        Definition: sqrt(_photon_count) = 33826.834

Instead of having to reenter or edit a lengthy expression when you perform another calculation, you need only enter values that change; in this respect, runtime variables are similar to a spreadsheet.

The more times a variable appears in an expression that depends on it, the greater the benefit of having a calculation using that expression reflect changes to that variable. For example, the length of daylight—the time the Sun is above the horizon—at a given latitude and declination of the Sun is given by

where \(L\) is the day length, \(h\) is the Sun’s altitude (elevation angle), \(\phi\) is the location’s latitude, and \(\delta\) is the Sun’s declination (angle with the equatorial plane). The result above is in sidereal time; the length in solar time is obtained by multiplying by

siderealday / day

By convention, the Sun’s altitude at rise or set is \(-50^\prime\) to allow for atmospheric refraction and the semidiameter of its disk. At the summer solstice in the northern hemisphere, the Sun’s declination is approximately \(23.44^\circ\); to find the length of the longest day of the year for a latitude of \(55^\circ\), you could do

You have: _alt = -50 arcmin
You have: _lat = 55 deg
You have: _decl = 23.44 deg
You have: _num = sin(_alt) - sin(_lat) sin(_decl)
You have: _denom = cos(_lat) cos(_decl)
You have: _sday = 2 (acos(_num / _denom) / circle) 24 hr
You have: _day = _sday siderealday / day
You have: _day
You want: hms
        17 hr + 19 min + 34.895151 sec

At the winter solstice, the Sun’s declination is approximately \(-23.44^\circ\), so you could calculate the length of the shortest day of the year using:

You have: _decl = -23.44 deg
You have: _day
You want: hms
        7 hr + 8 min + 40.981084 sec

Latitude and declination each appear twice in the expression for _day; the result in the examples above is updated by changing only the value of the declination.

It may seem easier—and less subject to error—to simply specify the new value of _decl as the negative of the current value (e.g., ‘_decl = -_decl’). This doesn’t work; when you make an assignment with the ‘=’ operator, the definition is stored as entered, including possible dependencies on variables. But if you attempt an assignment that is ultimately self-referential, the current definition is retained, and you get an error message. For example,

You have: _decl = 23.44 deg
You have: _decl = -_decl
Circular unit definition

You can overcome this by using the ‘:=’ operator, which reduces the right hand side to primitive units before making the assignment, eliminating any dependencies on variables. Returning to the example above,

You have: _decl = 23.44 deg
You have: _decl = -_decl
Circular unit definition
You have: _decl := -_decl
You have: _decl
You want: deg
        * -23.44
        / -0.042662116

This works to much the same effect as if the assignment had been entered literally, e.g.,

You have: _decl = -23.44 deg

but the actual definition is in primitive units—in this case, radians:

You have: _decl = 23.44 deg
You have: _decl := -_decl
You have: _decl
You want:
        Definition: -0.40910517666747087 radian = -0.40910518 radian

Definitions are text strings, and a redefinition using ‘:=’ is given with enough digits maintain the full precision of the current definition when converted back to a number; because it is a string, all digits are displayed when showing the definition, regardless of the numerical display precision, so you may see more digits than expected.

A runtime variable must be assigned before it can be used in an assignment; in the first of the three examples above, giving the general equation before the values for _alt, _lat, and _decl had been assigned would result in an error message.

5.8 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.

6.1 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

6.2 US Consumer Price Index

units includes the US Consumer Price Index published by the US Bureau of Labor Statistics. Several functions that use this value are provided: ‘cpi’, ‘cpi_now’, ‘inflation_since’, and ‘dollars_in’.

The ‘cpi’ function gives the CPI for a specified decimal year. A decimal year is given as the year plus the fractional part of the year; because of leap years and the different lengths of months, calculating an exact value for the fractional part can be tedious, but for the purposes of CPI, an approximate value is usually adequate. For example, 1 January 2000 is 2000.0, 1 April 2000 is 2000.25, 1 July 2000 is 2000.4986, and 1 October 2000 is 2000.75. Note also that the CPI data update monthly; values in between months are linearly interpolated.

In the middle of 1975, the CPI was

You have: cpi(1975.5)
You want:
        Definition: 53.6

The value of the CPI for a month is usually published sometime around the 20th day of the following month; the latest value of the CPI is available with ‘cpi_now’. On 7 January 2024, the value was

You have: cpi_now
You want:
        Definition: UScpi_now = 307.051

This means that the CPI was 307.015 on 1 December 2023. The ‘cpi_now’ variable can only present the most recent data available, so it can lag the current CPI by several weeks. The decimal year of the last update is available with ‘cpi_lastdate’.

The ‘inflation_since’ function provides a convenient way to determine the inflation factor from a specified decimal year to the latest value in the CPI table. For example, on 7 January 2024:

You have: inflation_since(1970)
You want:
        Definition: 8.1445889

In other words, goods that cost 1 US$ in 1970 would cost 8.14 US$ on 1 December 2023.

The ‘inflation_since’ function can be used to determine an annual rate of inflation. The earliest US CPI data are from about 1913.1; the approximate time between then and 7 January 2024 is 110.9 years. The approximate annual inflation rate for that period is then

You have: inflation_since(1913.1)^1|110.9 - 1
You want: %
        * 3.1548115
        / 0.31697614

The inflation rate for any time period can be found from the ratio of the CPI at the end of the period to that of the beginning:

You have: (cpi(1982)/cpi(1972))^1|10 - 1
You want: %
        * 8.6247033
        / 0.11594602

The period 1972–1982 was indeed one of high inflation.

The ‘dollars_in’ function is similar to ‘inflation_since’ but its output is in US$ rather than dimensionless:

You have: dollars_in(1970)
You want:
        Definition: 8.1445889 US$

A typical use might be

You have: 250 dollars_in(1970)
You want: $
        * 2036.1472
        / 0.00049112362

Because ‘dollars_in’ includes the units, you should not include them at the ‘You have:’ prompt. You can also use ‘dollars_in’ to convert between two specified years:

You have: 250 dollars_in(1970)
You want: dollars_in(1950)
        * 156.49867
        / 0.0063898305

which shows that 250 US$ in 1970 would have equivalent purchasing power to 156 US$ in 1950.

6.3 Other Nonlinear Units

Some other examples of nonlinear units are numerous different ring sizes and wire gauges, screw gauges, pipe and tubing sizes, 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.

Diameters of American wire sizes can be found using the ‘wiregauge()’ function or its alias ‘awg()’:

You have: wiregauge(11)
You want: inches
        * 0.090742002
        / 11.020255

You have: 1 mm
You want: wiregauge
        18.201919

Wire and screw gauges with multiple zeroes are signified using negative numbers, where two zeroes (“00”; “2/0”) is ‘-1’, three zeros (“000”; “3/0”) is ‘-2’, and so on. Alternatively, you can use the synonyms ‘g00’, ‘g000’, or ‘g2_0’, ‘g3_0’, and so on that are defined in the standard units data file.

You have: brwiregauge(g00)
You want: inches
        * 0.348
        / 2.8735632

In North America, wire sizes larger than 0000 (“4/0”) are usually given in terms of area, either in kcmil or the older initialism MCM (thousand circular mils). Outside of North America, all wire sizes are usually given in terms of area in mm². Wire area can be obtained using ‘wiregaugeA()’ or its alias ‘awgA()’:

You have: awgA(g6_0)
You want: kcmil
        * 336.45718
        / 0.0029721464
You have: awgA(12)
You want: mm^2
        * 3.3087729
        / 0.30222685

The closest standard metric sizes are 2.5 mm² and 4 mm²; in general, there isn’t an exact correlation between American and metric wire sizes.

Though based on the long-established iron pipe size (IPS) given in inches, nominal pipe size (NPS) is a dimensionless quantity that corresponds to the inch size. Pipe size can be equivalently specified using metric diamètre nominal (DN), which roughly corresponds to the diameter in mm. For a given pipe size, outside diameter is constant while inside diameter varies with schedule. For example, for NPS 2½ pipe,

You have: npsOD(2+1|2)
You want: in
        * 2.875
        / 0.34782609
You have: nps40(2+1|2)
You want: in
        * 2.469
        / 0.40502228
You have: nps80(2+1|2)
You want: in
        * 2.323
        / 0.43047783

Pipe size can be given equivalently in terms of the metric DN by using the ‘DN()’ function, which converts nominal metric size to nominal inch size:

You have: npsOD(DN(65))
You want: mm
        * 73.025
        / 0.01369394
You have: _
You want: in
        * 2.875
        / 0.34782609

Unlike with wire sizes, actual NPS and metric DN pipe dimensions are the same.

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.

7.1 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.

7.2 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         time: hours, minutes, seconds
dms         angle: degrees, minutes, seconds
time        time: years, days, hours, minutes and seconds
usvol       US cooking volume: cups and smaller
uswt        US weight: pounds and ounces
ftin        length: feet, inches and 1/8 inches
ftin2       length: feet, inches and 1/2 inches
ftin4       length: feet, inches and 1/4 inches
ftin8       length: feet, inches and 1/8 inches
ftin16      length: feet, inches and 1/16 inches
ftin32      length: feet, inches and 1/32 inches
ftin64      length: feet, inches and 1/64 inches
inchfine    length: 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

Suppose you want to drill a clearance hole for a #10 screw and have about 1/64 inch clearance; you could try

You have: screwgauge(10) + 1|64 in
You want: ftin64
        13.16 * 1|64 in
You have: _
You want: ftin32
        6.58 * 1|32 in

If a slightly tight fit is acceptable, a 13/64-inch drill would do the job; if not, a 7/32-inch drill would work with a slightly looser fit.

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

8.1 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 \(r\):

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

The two constants, \(k_{\rm C}\) and \(k_{\rm A}\), are related by the square of the speed of light:

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 \(\sqrt{4\pi}\).

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.

• Specifying CGS Units
• CGS Units Systems
• Conversions Between Different Systems

(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

(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

(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

8.2 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 by 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, \(8 \pi G=1\) whereas in the unreduced system \(G = 1\). 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’.

8.3 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.

14.1 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.

14.2 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 cannot begin or end with an underscore (‘_’), a comma (‘,’) or a decimal point (‘.’). Names must not contain any of the operator characters ‘+’, ‘-’, ‘*’, ‘/’, ‘|’, ‘^’, ‘;’, ‘~’, the comment character ‘#’, or parentheses. To facilitate copying and pasting from documents, several typographical characters are converted to operators: the figure dash (U+2012), minus (‘-’; U+2212), and en dash (‘–’; U+2013) are converted to the operator ‘-’; the multiplication sign (‘×’; U+00D7), N-ary times operator (U+2A09), dot operator (‘⋅’; U+22C5), and middle dot (‘·’; U+00B7) are converted to the operator ‘*’; the division sign (‘÷’; U+00F7) is converted to the operator ‘/’; and the fraction slash (U+2044) is converted to the operator ‘|’; accordingly, 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 and that there are no circular definitions that lead to endless loops. Because some errors may hide other errors, you should run units with the -c option again after correcting any errors, and keep doing so until no errors are displayed.

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)’.

14.3 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)

14.4 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.

14.5 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.

15.1 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 \(5\times 10^7\) displays as ‘50000000’ and the value \(5\times 10^8\) 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 \(5\times 10^{-8}\) 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 ‘'’ (apostrophe) 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.

15.2 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.

15.3 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

15.4 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.

16.1 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

Windows-locale   POSIX-locale

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

16.2 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 controlled 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.

19.1 Unicode Support on Windows

Microsoft Windows supports UTF-8 in console applications running in Windows Terminal but not in applications running in the older Windows Console Host. In Windows Terminal, the code page must be set to 65001 for UTF-8 to be enabled. With the UTF-8 code page, running units -V might show

GNU Units version 2.24
Without readline, with UTF-8, locale English_United States (en_US)

Two values are shown for the locale: the first is the one returned by the system; the second is the POSIX value to which the system value is mapped.

With a different code page, the result might be

GNU Units version 2.24
Without readline, with UTF-8 (disabled), locale English_United States (en_US)
To enable UTF-8: set code page to 65001

If units is running in Windows Console Host, regardless of the code page, the result might be

GNU Units version 2.24
Without readline, with UTF-8 (disabled), locale English_United States (en_US)
To enable UTF-8: run in Windows Terminal and set code page to 65001

The UTF-8 code page can be set by running chcp 65001.

As of late 2024, the Windows build of units does not identify characters—typically East Asian—that occupy more than one column, and error messages involving those characters may not be properly aligned.

21.1 Currency Exchange Rates

The units program database includes currency exchange rates and prices for some precious metals. Of course, these values change over time, sometimes very rapidly, and units cannot provide real-time values. To update the exchange rates, run units_cur, which rewrites the file containing the currency rates, typically /var/lib/units/currency.units or /usr/local/com/units/currency.units on a Unix-like system or C:\Program Files (x86)\GNU\units\definitions.units on a Windows system.

This program requires Python 3 (https://www.python.org). The program must be run with suitable permissions to write the file. To keep the rates updated automatically, run it using a cron job on a Unix-like system, or a similar scheduling program on a different system.

Reliable free sources of currency exchange rates have been annoyingly ephemeral. The program currently supports several sources:

• ExchangeRate-API.com (https://www.exchangerate-api.com).
  The default currency server. Allows open access without an API key, with unlimited API requests. Rates update once a day, the US dollar (‘USD’) is the default base currency, and you can choose your base currency with the -b option described below. You can optionally sign up for an API key to access paid benefits such as faster data update rates.
• FloatRates (https://www/floatrates.com).
  The US dollar (‘USD’) is the default base currency. You can change the base currency with the -b option described below. Allowable base currencies are listed on the FloatRates website. Exchange rates update daily.
• The European Central Bank (https://www.ecb.europa.eu).
  The base currency is always the euro (‘EUR’). Exchange rates update daily. This source offers a more limited list of currencies than the others.
• Fixer (https://fixer.io).
  Registration for a free API key is required. With a free API key, base currency is the euro; exchange rates are updated hourly, the service has a limit of 1,000 API calls per month, and SSL encryption (https protocol) is not available. Most of these restrictions are eliminated or reduced with paid plans.
• open exchange rates (https://openexchangerates.org).
  Registration for a free API key is required. With a free API key, the base currency is the US dollar; exchange rates are updated hourly, and there is a limit of 1,000 API calls per month. Most of these restrictions are eliminated or reduced with paid plans.

The default source is FloatRates; you can select a different one using -s option described below.

Precious metals pricing is obtained from Packetizer (www.packetizer.com). This site updates once per day.

21.2 US Consumer Price Index

The units program includes the US Consumer Price Index (CPI) published by the US Bureau of Labor Statistics: specifically, the Consumer Price Index for All Urban Consumers (CPI-U), not seasonally adjusted—Series CUUR0000SA0. The units_cur command updates the CPI and saves the result in cpi.units in the same location as currency.units. The data are obtained via the BLS Public Data API (https://www.bls.gov/developers/). These data update once a month. When units_cur runs it will only attempt to update the CPI data if the current CPI data file is from a previous month, or if the current date is after the 18th of the month.

21.3 Invoking units_cur

You invoke units_cur like this:

units_cur [options] [currency_file] [cpi_file]

By default, the output is written to the default currency and CPI files described above; this is usually what you want, because this is where units looks for the files. If you wish, you can specify different filenames on the command line and units_cur will write the data to those files. If you give ‘-’ for a file it will write to standard output.

The following options are available:

-h

--help

Print a summary of the options for units_cur.

-V

--version

Print the units_cur version number.

-v

--verbose

Give slightly more verbose output when attempting to update currency exchange rates.

-s source

--source source

Specify the source for currency exchange rates; currently supported values are ‘floatrates’ (for FloatRates), ‘eubank’ (for the European Central Bank), ‘fixer’ (for Fixer), and ‘openexchangerates’ (for open exchange rates); the last two require an API key to be given with the -k option.

-b base

--base base

Set the base currency (when allowed by the site providing the data). base should be a 3-letter ISO currency code, e.g., ‘USD’. The specified currency will be the primitive currency unit used by units. You may find it convenient to specify your local currency. Conversions may be more accurate and you will be able to convert to your currency by simply hitting Enter at the ‘You want:’ prompt. This option is ignored if the source does not allow specifying the base currency. (Currently only floatrates supports this option.)

-k key

--key key

Set the API key to key for currency sources that require it.

--blskey BLSkey

Set the US Bureau of Labor Statistics (BLS) key for fetching CPI data. Without a BLS key you should be able to fetch the CPI data exactly one time per day. If you want to use a key you must request a personal key from BLS.