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. The `**' operator is provided as an alternative 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 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 Historically, multiplication in units was assigned a higher precedence than division. This disagrees with the usual precedence rules which give multiplication and division equal precedence, and it has been a source of confusion for people who think of units as a calculator. By default, multiplication using the star (`*') now has the same precedence as division and hence follows the usual precedence rules. If units is invoked with the the `--oldstar' option then then the old behavior is activated and `*' will have the same precedence as the other multiplication operators described next. Multiplication using a space or using the hyphen 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 (`|'), so if you wanted half a meter you could write `1|2 meter'. 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 Temperature Conversion Example .............................. 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.Note: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