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Units: A Unit Conversion Program

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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, a hyphen (`-') or an asterisk (`*'). 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.

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. 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 ```

Because `-' is used for products, it cannot also be used to form differences of units. If a `-' appears after `(' or after `+' then it will act as a negation operator. So you can compute 20 degrees minus 12 minutes by entering `20 degrees + -12 arcmin'. The `+' character is sometimes used 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 ```

In this case, 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 section 8. Defining nonlinear units.

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 ```

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