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4 Unit expressions

In order to enter more complicated units or fractions, you will need to use operations such as powers, products and division. Powers of units can be specified using the ^ character as shown in the following example, or by simple concatenation: 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. The ^ operator has the second highest precedence.

         You have: cm^3
         You want: gallons
                 * 0.00026417205
                 / 3785.4118
     
         You have: arabicfoot * arabictradepound * force
         You want: ft lbf
                 * 0.7296
                 / 1.370614

Multiplication of units can be specified by using spaces, or an asterisk (*). If units is invoked with the --product option then the hyphen (-) also acts as a multiplication operator. Division of units is indicated by the slash (/) or by per.

         You have: furlongs per fortnight
         You want: m/s
                 * 0.00016630986
                 / 6012.8727

Multiplication has a higher precedence than division and is evaluated left to right, so 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 .5/meter, which is probably not what you would intend if you entered that expression. You can indicate division of numbers with the vertical dash (|). This operator has the highest precedence so the square root of two thirds could be written 2|3^1|2.

         You have: 1|2 inch
         You want: cm
                 * 1.27
                 / 0.78740157

Parentheses can be used for grouping as desired.

         You have: (1/2) kg / (kg/meter)
         You want: league
                 * 0.00010356166
                 / 9656.0833

Prefixes are defined separately from base units. In order to get centimeters, the units database defines centi- and c- as prefixes. Prefixes can appear alone with no unit following them. An exponent applies only to the immediately preceding unit and its prefix so that cm^3 or centimeter^3 refer to cubic centimeters but centi*meter^3 refers to hundredths of cubic meters. Only one prefix is permitted per unit, so micromicrofarad will fail, but micro*microfarad will work, as will micro microfarad..

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 which 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 dollars^5.

Outside of the SI system, it is often desirable to add values of different units together. You may also wish to use units as a calculator that keeps track of units. Sums of conformable units are written 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 which are added together 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

Historically - has been used for products of units, which complicates its iterpretation in units. 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. This behavior can be altered using the --product option which causes units to treat the binary - operator as a product operator. Note that when - is a multiplication operator it has the same precedence as *, but when - is a subtraction operator it has the lower precedence as the addition operator.

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.

The + character 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 + as part of an exponent if possible.

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. Higher roots can be obtained 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

Nonlinear units are represented using functional notation. They make possible nonlinear unit conversions such temperature. This is different from the linear units that convert temperature differences. Note the difference below. The absolute temperature conversions are handled by units starting with temp, and you must use functional notation. The temperature differences 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 which indicates that x should have units of tempF attached to it. See Nonlinear units. The first conversion shows that if it's 45 degrees Fahrehneit outside it's 7.2 degrees Celsius. The second conversions indicates that a change of 45 degrees Fahrenheit corresponds to a change of 25 degrees Celsius.

Some other examples of nonlinears units are ring size and wire gauge. There are numerous different gauges and ring sizes. See the units database for more details. Note that 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 units database.

         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