This manual describes the units command for units conversion and how you can use it as a powerful scientific calculator that keeps track of units. This is Edition 2.01 of The Units Conversion Manual for units Version 2.00.
The units program converts quantities expressed in various systems of measurement to their equivalents in other systems of measurement. Like many similar programs, it can handle multiplicative scale changes. It can also handle nonlinear conversions such as Fahrenheit to Celsius.1 See Temperature Conversions. The program can also perform conversions from and to sums of units, such as converting between meters and feet plus inches.
Beyond simple unit conversions, units can be used as a general-purpose scientific calculator that keeps track of units in its calculations. You can form arbitrary complex mathematical expressions of dimensions including sums, products, quotients, powers, and even roots of dimensions. Thus you can ensure accuracy and dimensional consistency when working with long expressions that involve many different units that may combine in complex ways.
The units are defined in an external data file. You can use the extensive data file that comes with this program, or you can provide your own data file to suit your needs. You can also use your own data file to supplement the standard data file.
Basic operation is simple: you enter the units that you want to convert from and the units that you want to convert to. You can use the program interactively with prompts, or you can use it from the command line.
To invoke units for interactive use, type units at your shell prompt. The program will print something like this:
Currency exchange rates from 04/23/12 2516 units, 85 prefixes, 65 nonlinear units You have:
At the ‘You have:’ prompt, type the quantity and units that you are converting from. For example, if you want to convert ten meters to feet, type 10 meters. Next, units will print ‘You want:’. You should type the units you want to convert to. To convert to feet, you would type feet. If the readline library was compiled in then the tab key can be used to complete unit names. See Readline Support, for more information about readline. To quit the program press Ctrl-C or Ctrl-D under Unix. Under Windows press Ctrl-Z.
The answer will be displayed in two ways. The first line of output, which is marked with a ‘*’ to indicate multiplication, gives the result of the conversion you have asked for. The second line of output, which is marked with a ‘/’ to indicate division, gives the inverse of the conversion factor. If you convert 10 meters to feet, units will print
* 32.808399 / 0.03048
which tells you that 10 meters equals about 32.8 feet. The second number gives the conversion in the opposite direction. In this case, it tells you that 1 foot is equal to about 0.03 dekameters since the dekameter is 10 meters. It also tells you that 1/32.8 is about 0.03.
The units program prints the inverse because sometimes it is a more convenient number. In the example above, for example, the inverse value is an exact conversion: a foot is exactly 0.03048 dekameters. But the number given the other direction is inexact.
If you convert grains to pounds, you will see the following:
You have: grains You want: pounds * 0.00014285714 / 7000
From the second line of the output you can immediately see that a grain is equal to a seven thousandth of a pound. This is not so obvious from the first line of the output. If you find the output format confusing, try using the --verbose option:
You have: grain You want: aeginamina grain = 0.00010416667 aeginamina grain = (1 / 9600) aeginamina
If you request a conversion between units that measure reciprocal dimensions, then units will display the conversion results with an extra note indicating that reciprocal conversion has been done:
You have: 6 ohms You want: siemens reciprocal conversion * 0.16666667 / 6
Reciprocal conversion can be suppressed by using the --strict option. As usual, use the --verbose option to get more comprehensible output:
You have: tex You want: typp reciprocal conversion 1 / tex = 496.05465 typp 1 / tex = (1 / 0.0020159069) typp You have: 20 mph You want: sec/mile reciprocal conversion 1 / 20 mph = 180 sec/mile 1 / 20 mph = (1 / 0.0055555556) sec/mile
If you enter incompatible unit types, the units program will print a message indicating that the units are not conformable and it will display the reduced form for each unit:
You have: ergs/hour You want: fathoms kg^2 / day conformability error 2.7777778e-11 kg m^2 / sec^3 2.1166667e-05 kg^2 m / sec
If you only want to find the reduced form or definition of a unit, simply press <Enter> at the ‘You want:’ prompt. Here is an example:
You have: jansky You want: Definition: fluxunit = 1e-26 W/m^2 Hz = 1e-26 kg / s^2
The output from units indicates that the jansky is defined to be equal to a fluxunit which in turn is defined to be a certain combination of watts, meters, and hertz. The fully reduced (and in this case somewhat more cryptic) form appears on the far right.
Some named units are treated as dimensionless in some situations. These units include the radian and steradian. These units will be treated as equal to 1 in units conversions. Power is equal to torque times angular velocity. This conversion can only be performed if the radian is dimensionless.
You have: (14 ft lbf) (12 radians/sec) You want: watts * 227.77742 / 0.0043902509
Named dimensionless units are not treated as dimensionless in other contexts. They cannot be used as exponents so for example, ‘meter^radian’ is not allowed. If you want a list of options you can type ? at the ‘You want:’ prompt. The program will display a list of named units that are conformable with the unit that you entered at the ‘You have:’ prompt above. Conformable unit combinations will not appear on this list.
Typing help at either prompt displays a short help message. You can also type help followed by a unit name. This will invoke a pager on the units data base at the point where that unit is defined. You can read the definition and comments that may give more details or historical information about the unit. (You can generally quit out of the page by pressing ‘q’.)
Typing search text will display a list of all of the units whose names contain text as a substring along with their definitions. This may help in the case where you aren't sure of the right unit name.
The units program can perform units conversions non-interactively from the command line. To do this, type the command, type the original unit expression, and type the new units you want. If a units expression contains non-alphanumeric characters, you may need to protect it from interpretation by the shell using single or double quote characters.
If you type
units "2 liters" quarts
then units will print
* 2.1133764 / 0.47317647
and then exit. The output tells you that 2 liters is about 2.1 quarts, or alternatively that a quart is about 0.47 times 2 liters.
If the conversion is successful, then units will return success (zero) to the calling environment. If you enter non-conformable units then units will print a message giving the reduced form of each unit and it will return failure (nonzero) to the calling environment.
When you invoke units with only one argument, it will print out the definition of the specified unit. It will return failure if the unit is not defined and success if the unit is defined.
The conversion information is read from a units data file that is called definitions.units and is usually located in the /usr/share/units directory. If you invoke units with the -V option, it will print the location of this file. The default file includes definitions for all familiar units, abbreviations and metric prefixes. It also includes many obscure or archaic units.
Many constants of nature are defined, including these:
pi ratio of circumference to 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 atomic masses for all of the elements and 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.
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’. British capacity units that differ from their US counterparts, such as the British Imperial gallon, are prefixed with ‘br’. 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 standard units data file, which is extensively annotated.
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.
You can enter more complicated units by combining units with operations such as powers, multiplication, division, 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.
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. An exponent like ‘2^3^2’ is evaluated right to left as usual. The ‘^’ operator has the second highest precedence. You can also use ‘**’ as an exponent operator.
You have: cm^3 You want: gallons * 0.00026417205 / 3785.4118 You have: arabicfoot * arabictradepound * force You want: ft lbf * 0.7296 / 1.370614
You multiply units using a space or an asterisk (‘*’). The example above shows both forms. You can divide units using the slash (‘/’) or with ‘per’.
You have: furlongs per fortnight You want: m/s * 0.00016630986 / 6012.8727
When a unit includes a prefix, exponent operators apply to the combination, so ‘centimeter^3’ gives cubic centimeters. If you separate the prefix from the unit with any multiplication operator, such as ‘centi meter^3’, then the prefix is treated as a separate unit, so the exponent does not apply. The second example would be a hundredth of a cubic meter, not a centimeter.
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: ‘W / m^2 Hz’. If you multiply with ‘*’ then you must group the terms in the denominator with parentheses: ‘W / (m^2 * Hz)’.
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 ‘*’ operator is convenient for multiplying a sequence of quotients. With the ‘*’ operator, the example above becomes ‘m/s * s/day’, which is equivalent to ‘m/day’. Similarly, you could write ‘1/2 * meter’ to get half a meter. Alternatively, parentheses can be used for grouping: you could write ‘(1/2) meter’ to get half a meter. See Complicated Unit Expressions, for an illustration of the various options.
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’. This operator has the highest precedence, so you can write the square root of two thirds ‘2|3^1|2’. You cannot use the vertical bar to indicate division of non-numerical units (e.g., ‘m|s’ results in an error message).
You have: 1|2 inch You want: cm * 1.27 / 0.78740157
You can use parentheses for grouping:
You have: (1/2) kg / (kg/meter) You want: league * 0.00010356166 / 9656.0833
Outside of the SI, it is sometimes desirable to add values of different units. 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
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.
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’.
Several built-in functions are provided: ‘sin’, ‘cos’, ‘tan’, ‘ln’, ‘log’, ‘log2’, ‘exp’, ‘acos’, ‘atan’ and ‘asin’. 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.
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
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
Delta P = (8/pi^2) rho f L (Q^2 / d^5)
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
Delta P = A1 rho f L (Q^2 / d^5)
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
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.
Nonlinear units are represented using functional notation. They make possible nonlinear unit conversions such as temperature.
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.
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
Outside of the SI, it is sometimes desirable to convert a single unit to a sum of units—for example, feet to feet plus inches. The conversion from sums of units was described in Sums and Differences of Units, and is a simple matter of adding the units with the ‘+’ sign:
You have: 12 ft + 3 in + 3|8 in You want: ft * 12.28125 / 0.081424936
Although you can similarly write a sum of units to convert to, the result will not be the conversion to the units in the sum, but rather the conversion to the particular sum that you have entered:
You have: 12.28125 ft You want: ft + in + 1|8 in * 11.228571 / 0.089058524
The unit expression given at the ‘You want:’ prompt is equivalent to asking for conversion to multiples of ‘1 ft + 1 in + 1|8 in’, which is 1.09375 ft, so the conversion in the previous example is equivalent to
You have: 12.28125 ft You want: 1.09375 ft * 11.228571 / 0.089058524
In converting to a sum of units like miles, feet and inches, you typically want the largest integral value for the first unit, followed by the largest integral value for the next, and the remainder converted to the last unit. You can do this conversion easily with units using a special syntax for lists of units. You must list the desired units in order from largest to smallest, separated by the semicolon (‘;’) character:
You have: 12.28125 ft You want: ft;in;1|8 in 12 ft + 3 in + 3|8 in
The conversion always gives integer coefficients on the units in the list, except possibly the last unit when the conversion is not exact:
You have: 12.28126 ft You want: ft;in;1|8 in 12 ft + 3 in + 3.00096 * 1|8 in
The order in which you list the units is important:
You have: 3 kg You want: oz;lb 105 oz + 0.051367866 lb You have: 3 kg You want: lb;oz 6 lb + 9.8218858 oz
Listing ounces before pounds produces a technically correct result, but not a very useful one. You must list the units in descending order of size in order to get the most useful result.
Ending a unit list with the separator ‘;’ has the same effect as repeating the last unit on the list, so ‘ft;in;1|8 in;’ is equivalent to ‘ft;in;1|8 in;1|8 in’. With the example above, this gives
You have: 12.28126 ft You want: ft;in;1|8 in; 12 ft + 3 in + 3|8 in + 0.00096 * 1|8 in
in effect separating the integer and fractional parts of the coefficient for the last unit. If you instead prefer to round the last coefficient to an integer you can do this with the --round (-r) option. With the previous example, the result is
You have: 12.28126 ft You want: ft;in;1|8 in 12 ft + 3 in + 3|8 in (rounded down to nearest 1|8 in)
When you use the -r option, repeating the last unit on the list has no effect (e.g., ‘ft;in;1|8 in;1|8 in’ is equivalent to ‘ft;in;1|8 in’), and hence neither does ending a list with a ‘;’. With a single unit and the -r option, a terminal ‘;’ does have an effect: it causes units to treat the single unit as a list and produce a rounded value for the single unit. Without the extra ‘;’, the -r option has no effect on single unit conversions. This example shows the ouput using the -r option:
You have: 12.28126 ft You want: in * 147.37512 / 0.0067854058 You have: 12.28126 ft You want: in; 147 in (rounded down to nearest in)
Each unit that appears in the list must be conformable with the first unit on the list, and of course the listed units must also be comformable with the You have unit that you enter.
You have: meter You want: ft;kg ^ conformability error ft = 0.3048 m kg = 1 kg You have: meter You want: lb;oz conformability error 1 m 0.45359237 kg
In the first case, units reports the disagreement between units appearing on the list. In the second case, units reports disagreement between the unit you entered and the desired conversion. This conformability error is based on the first unit on the unit list.
Other common candidates for conversion to sums of units are angles and time:
You have: 23.437754 deg You want; deg;arcmin;arcsec 23 deg + 26 arcmin + 15.9144 arcsec You have: 7.2319 hr You want: hr;min;sec 7 hr + 13 min + 54.84 sec
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.
Some applications for unit lists may be less obvious. Suppose that you have a postal scale and wish to ensure that it's accurate at 1 oz, but have only metric calibration weights. You might try
You have: 1 oz You want: 100 g;50 g; 20 g;10 g;5 g;2 g;1 g; 20 g + 5 g + 2 g + 1 g + 0.34952312 * 1 g
You might then place one each of the 20 g, 5 g, 2 g, and 1 g weights on the scale and hope that it indicates close to
You have: 20 g + 5 g + 2 g + 1 g You want: oz; 0.98767093 oz
Appending ‘;’ to ‘oz’ forces a one-line display that includes the unit; here the integer part of the result is zero, so it is not displayed.
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
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 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
You invoke units like this:
units [options] [from-unit [to-unit]]
If the from-unit and to-unit are omitted, then the program will use interactive prompts to determine which conversions to perform. See Interactive Use. If both from-unit and to-unit are given, units will print the result of that single conversion and then exit. If only from-unit appears on the command line, units will display the definition of that unit and exit. Units specified on the command line may need to be quoted to protect them from shell interpretation and to group them into two arguments. See Command Line Use.
The following options allow you to read in an alternative units file, check your units file, or change the output format:
filename. You can specify up to 25 units files on the command line. When you use this option, units will load only the files you list on the command line; it will not load the standard file or your personal units file unless you explicitly list them. If filename is the empty string (‘-f ""’), the default units file (or that specified by UNITSFILE) will be loaded in addition to any others specified with ‘-f’.
The units and prefixes that units can convert are defined in the units data file, typically /usr/share/units/definitions.units. 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 when 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
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.)
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 and HOMEPATH. Running units -V will display the location and name of your personal units file.
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.
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 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 (‘.’). Names cannot begin with a digit, and if a name ends in a digit other than zero, 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’ would be valid names but ‘foo2’ or ‘foo_a2’ would be invalid. 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, 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’.
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 0.0254 m # Inch defined in terms of meters ft 12 inches # The foot defined in terms of inches mile 5280 ft # And the mile
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)’.
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 data file, so that an eventual conversion to primitive units is possible.
Here is an example nonlinear unit definition:
tempF(x) units=[1;K] (x+(-32)) degF + stdtemp ; \ (tempF+(-stdtemp))/degF + 32
A nonlinear unit definition comprises a unit name, a dummy parameter name, two functions, and two corresponding units. The functions tell units how to convert to and from the new unit. In order to produce valid results, the arguments of these functions need to have the correct dimensions. To facilitate error checking, you may optionally indicate units for these arguments.
The definition begins with the unit name followed immediately (with no spaces) by a ‘(’ character. In parentheses is the name of the parameter. Next is an optional specification of the units required by the functions in this definition. In the example above, the ‘tempF’ function requires an input argument conformable with ‘1’. For normal nonlinear units definitions the forward function will always take a dimensionless argument. The inverse function requires an input argument conformable with ‘K’. In general the inverse function will need units that match the quantity measured by your nonlinear unit. The purpose of the expression in brackets to enable units to perform error checking on function arguments, and also to assign units to range and domain 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.
In order to make conversions to Fahrenheit possible, 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, but then conversions to the unit will be impossible. If the inverse is omitted then 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.
If you wish to make synonyms for nonlinear units, you still need to define 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)
You may define a function whose range and domain do not cover all of the real numbers. In this case units can handle errors better if you specify an appropriate range and domain. You specify the range and domain 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 the ‘domain=’ with the endpoints given in brackets. One of the end points can be omitted to get an interval that goes to infinity. So the range could be specified as nonnegative by writing ‘range=[0,]’. Both the range and domain are optional and can appear independently and in any order along with the ‘units’ specification. The values in the range and domain are attached to the units given in the ‘units’ specification. If you don't specify the units then the parameter inputs are reduced to primitive units for the numeric comparison to the values you give in the range or domain. In this case you should only use ‘range’ or ‘domain’ if the endpoints are zero and infinity.
Specifying the range and domain allows units to perform better error checking and give more helpful error messages when you invoke nonlinear units conversions outside of their bounds. It also enables the -c option to find a point in the domain to use for its point check of your inverse definition.
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)
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 illegal; 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. The --check option will print a warning if a non-monotonic piecewise linear unit is encountered.
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.
By default, results of conversions are shown to eight significant figures; this can be changed with the --exponential and --output-format options. The former sets an exponential format (i.e., scientific notation) like that used in the original Unix units program; the latter allows the format to be given as that of the printf function in the ANSI C standard.
The format recognized with the --output-format option is a
subset of that for printf(). Only a floating-point format of the form
is allowed: it must begin with ‘%’, and must end with a
floating-point type specifier (‘E’ or ‘e’ for scientific
notation, ‘f’ for fixed-point decimal, or ‘G’ or ‘g’ to
specify the number of significant figures). The format specification
may include one optional flag (‘+’, ‘-’, ‘#’, or a
space), followed by an optional value for the minimum field width, and
an optional precision specification that begins with a period (e.g.,
‘.6’). In addition to the digits, the field width includes the
decimal point, the exponent, and the sign if any of these are shown. A
width specification is typically used with fixed-point decimal to have
columns of numbers align at the decimal point; it normally is not
useful with units. Non-floating-point type specifiers make
no sense for units, and are forbidden.
The default format is ‘%.8g’; for greater precision, you could specify ‘-o %.15g’. The ‘G’ and ‘g’ formats use exponential format whenever the exponent would be less than −5, so the value 0.000013 displays as ‘1.3e-005’. If you prefer fixed-point display, you might specify ‘-o %.8f’; however, very small numbers may display very few significant figures, and for very small numbers, may show nothing but zeros.
See the documentation for printf() for more detailed descriptions of the format specification.
Some units have different values in different locations. The localization feature accommodates this by allowing a units data file to specify definitions that depend on the user's 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
_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
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 in the same directory, typically /usr/local/share/units, as the default units data files. The file includes entries for many combinations of language and country, and can be extended to include other combinations. The locale.map 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
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 accomodated 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.
The units program uses the following environment variables:
The standard units data file is written in Unicode using the UTF-8 encoding. Portions of the file that are not plain ASCII begin with ‘!utf8’ and end with ‘!endutf8’. As usual, the ‘!’ must appear as the first character on the line. If a line of a data file contains byte sequences that are invalid UTF-8 or non-printing UTF-8 then units ignores the entire line.
When units runs it checks the locale to determine the character set. If UTF-8 is listed, then units reads the utf8 definitions. If any other character set is in use, then units works in plain ASCII without support for extended characters.
If the readline package has been compiled in, then when units is used interactively, numerous command line editing features are available. To check if your version of units includes readline, invoke the program with the --version option.
For complete information about readline, consult the documentation for the readline package. Without any configuration, units will allow editing in the style of emacs. Of particular use with units are the completion commands.
If you type a few characters and then hit <ESC> followed by ? then units will display a list of all the units that start with the characters typed. For example, if you type metr and then request completion, you will see something like this:
You have: metr metre metriccup metrichorsepower metrictenth metretes metricfifth metricounce metricton metriccarat metricgrain metricquart metricyarncount You have: metr
If there is a unique way to complete a unitname, you can hit the <TAB> key and units will provide the rest of the unit name. If units beeps, it means that there is no unique completion. Pressing the <TAB> key a second time will print the list of all completions.
The units program includes currency exchange rates and prices for some precious metals in the database. Of course, these values change over time, sometimes very rapidly, and units cannot provide real time values. To update the exchange rates run the units_cur, which rewrites the files containing the currency rates, typically /usr/local/share/units/currency.units. This program must be run with suitable permissions to write the file. To keep the rates updated automatically, it could be run by a cron job on a Unix-like system, or a similar scheduling program on a different system. Currency exchange rates are taken from Time Genie (http://www.timegenie.com) and precious metals pricing from Packetizer (www.packetizer.com). These sites update once per day, so there is no benefit in running the update script more often than daily. You can run units_cur with a filename specified on the command line and it will write the data to that file. If you give ‘-’ for the file it will write to standard output.
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--check(option for units): Invoking Units
--check-verbose(option for units): Invoking Units
--compact(option for units): Invoking Units
--file(option for units): Invoking Units
--help(option for units): Invoking Units
--locale(option for units): Invoking Units
--minus(option for units): Invoking Units
--newstar(option for units): Invoking Units
--oldstar(option for units): Invoking Units
--one-line(option for units): Invoking Units
--output-format(option for units): Invoking Units
--product(option for units): Invoking Units
--quiet(option for units): Invoking Units
--silent(option for units): Invoking Units
--strict(option for units): Invoking Units
--terse(option for units): Invoking Units
--verbose(option for units): Invoking Units
--verbose-check(option for units): Invoking Units
--version(option for units): Invoking Units
-1(option for units): Invoking Units
-c(option for units): Invoking Units
-f(option for units): Invoking Units
-h(option for units): Invoking Units
-l(option for units): Invoking Units
-m(option for units): Invoking Units
-o(option for units): Invoking Units
-p(option for units): Invoking Units
-q(option for units): Invoking Units
-s(option for units): Invoking Units
-t(option for units): Invoking Units
-V(option for units): Invoking Units
-v(option for units): Invoking Units
 But Fahrenheit to Celsius is linear, you insist. Not so. A transformation T is linear if T(x+y)=T(x)+T(y) and this fails for T(x)=ax+b. This transformation is affine, but not linear.