calc: Basic Operations on Units
12.1 Basic Operations on Units
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A “units expression” is a formula which is basically a number multiplied
and/or divided by one or more “unit names”, which may optionally be
raised to integer powers. Actually, the value part need not be a
number; any product or quotient involving unit names is a units
expression. Many of the units commands will also accept any formula,
where the command applies to all units expressions which appear in the
formula.
A unit name is a variable whose name appears in the “unit table”, or
a variable whose name is a prefix character like ‘k’ (for “kilo”) or ‘u’
(for “micro”) followed by a name in the unit table. A substantial table
of built-in units is provided with Calc; 
Predefined Units. You
can also define your own unit names; User-Defined Units.
Note that if the value part of a units expression is exactly ‘1’, it
will be removed by the Calculator’s automatic algebra routines: The
formula ‘1 mm’ is “simplified” to ‘mm’. This is only a display anomaly,
however; ‘mm’ will work just fine as a representation of one millimeter.
You may find that Algebraic mode (Algebraic Entry) makes
working with units expressions easier. Otherwise, you will have to
remember to hit the apostrophe key every time you wish to enter units.
The ‘u s’ (‘calc-simplify-units’) [‘usimplify’] command simplifies a
units expression. It uses Calc’s algebraic simplifications to simplify
the expression first as a regular algebraic formula; it then looks for
features that can be further simplified by converting one object’s units
to be compatible with another’s. For example, ‘5 m + 23 mm’ will
simplify to ‘5.023 m’. When different but compatible units are added,
the righthand term’s units are converted to match those of the lefthand
term. Simplification Modes, for a way to have this done
automatically at all times.
Units simplification also handles quotients of two units with the
same dimensionality, as in ‘2 in s/L cm’ to ‘5.08 s/L’; fractional
powers of unit expressions, as in ‘sqrt(9 mm^2)’ to ‘3 mm’ and ‘sqrt(9
acre)’ to a quantity in meters; and ‘floor’, ‘ceil’, ‘round’, ‘rounde’,
‘roundu’, ‘trunc’, ‘float’, ‘frac’, ‘abs’, and ‘clean’ applied to units
expressions, in which case the operation in question is applied only to
the numeric part of the expression. Finally, trigonometric functions of
quantities with units of angle are evaluated, regardless of the current
angular mode.
The ‘u c’ (‘calc-convert-units’) command converts a units expression
to new, compatible units. For example, given the units expression ‘55
mph’, typing ‘u c m/s <RET>’ produces ‘24.5872 m/s’. If you have
previously converted a units expression with the same type of units (in
this case, distance over time), you will be offered the previous choice
of new units as a default. Continuing the above example, entering the
units expression ‘100 km/hr’ and typing ‘u c <RET>’ (without specifying
new units) produces ‘27.7777777778 m/s’.
The ‘u c’ command treats temperature units (like ‘degC’ and ‘K’) as
relative temperatures. For example, ‘u c’ converts ‘10 degC’ to ‘18
degF’: A change of 10 degrees Celsius corresponds to a change of 18
degrees Fahrenheit. To convert absolute temperatures, you can use the
‘u t’ (‘calc-convert-temperature’) command. The value on the stack must
be a simple units expression with units of temperature only. This
command would convert ‘10 degC’ to ‘50 degF’, the equivalent temperature
on the Fahrenheit scale.
While many of Calc’s conversion factors are exact, some are
necessarily approximate. If Calc is in fraction mode (Fraction
Mode), then unit conversions will try to give exact, rational
conversions, but it isn’t always possible. Given ‘55 mph’ in fraction
mode, typing ‘u c m/s <RET>’ produces ‘15367:625 m/s’, for example,
while typing ‘u c au/yr <RET>’ produces ‘5.18665819999e-3 au/yr’.
If the units you request are inconsistent with the original units,
the number will be converted into your units times whatever “remainder”
units are left over. For example, converting ‘55 mph’ into acres
produces ‘6.08e-3 acre / (m s)’. Remainder units are expressed in terms
of “fundamental” units like ‘m’ and ‘s’, regardless of the input units.
If you intend that your new units be consistent with the original
units, the ‘u n’ (‘calc-convert-exact-units’) command will check the
units before the conversion. For example, to change ‘mi/hr’ to ‘km/hr’,
you could type ‘u c km <RET>’, but ‘u n km <RET>’ would signal an error.
You would need to type ‘u n km/hr <RET>’.
One special exception is that if you specify a single unit name, and
a compatible unit appears somewhere in the units expression, then that
compatible unit will be converted to the new unit and the remaining
units in the expression will be left alone. For example, given the
input ‘980 cm/s^2’, the command ‘u c ms’ will change the ‘s’ to ‘ms’ to
get ‘9.8e-4 cm/ms^2’. The “remainder unit” ‘cm’ is left alone rather
than being changed to the base unit ‘m’.
You can use explicit unit conversion instead of the ‘u s’ command to
gain more control over the units of the result of an expression. For
example, given ‘5 m + 23 mm’, you can type ‘u c m’ or ‘u c mm’ to
express the result in either meters or millimeters. (For that matter,
you could type ‘u c fath’ to express the result in fathoms, if you
preferred!)
In place of a specific set of units, you can also enter one of the
units system names ‘si’, ‘mks’ (equivalent), or ‘cgs’. For example, ‘u
c si <RET>’ converts the expression into International System of Units
(SI) base units. Also, ‘u c base’ converts to Calc’s base units, which
are the same as ‘si’ units except that ‘base’ uses ‘g’ as the
fundamental unit of mass whereas ‘si’ uses ‘kg’.
The ‘u c’ command also accepts “composite units”, which are expressed
as the sum of several compatible unit names. For example, converting
‘30.5 in’ to units ‘mi+ft+in’ (miles, feet, and inches) produces ‘2 ft +
6.5 in’. Calc first sorts the unit names into order of decreasing
relative size. It then accounts for as much of the input quantity as it
can using an integer number times the largest unit, then moves on to the
next smaller unit, and so on. Only the smallest unit may have a
non-integer amount attached in the result. A few standard unit names
exist for common combinations, such as ‘mfi’ for ‘mi+ft+in’, and ‘tpo’
for ‘ton+lb+oz’. Composite units are expanded as if by ‘a x’, so that
‘(ft+in)/hr’ is first converted to ‘ft/hr+in/hr’.
If the value on the stack does not contain any units, ‘u c’ will
prompt first for the old units which this value should be considered to
have, then for the new units. (If the value on the stack can be
simplified so that it doesn’t contain any units, like ‘ft/in’ can be
simplified to 12, then ‘u c’ will still prompt for both old units and
new units. Assuming the old and new units you give are consistent with
each other, the result also will not contain any units. For example,
‘u c cm <RET> in <RET>’ converts the number 2 on the stack to 5.08.
The ‘u b’ (‘calc-base-units’) command is shorthand for ‘u c base’; it
converts the units expression on the top of the stack into ‘base’ units.
If ‘u s’ does not simplify a units expression as far as you would like,
try ‘u b’.
Like the ‘u c’ command, the ‘u b’ command treats temperature units as
relative temperatures.
The ‘u r’ (‘calc-remove-units’) command removes units from the
formula at the top of the stack. The ‘u x’ (‘calc-extract-units’)
command extracts only the units portion of a formula. These commands
essentially replace every term of the formula that does or doesn’t
(respectively) look like a unit name by the constant 1, then resimplify
the formula.
The ‘u a’ (‘calc-autorange-units’) command turns on and off a mode in
which unit prefixes like ‘k’ (“kilo”) are automatically applied to keep
the numeric part of a units expression in a reasonable range. This mode
affects ‘u s’ and all units conversion commands except ‘u b’. For
example, with autoranging on, ‘12345 Hz’ will be simplified to ‘12.345
kHz’. Autoranging is useful for some kinds of units (like ‘Hz’ and
‘m’), but is probably undesirable for non-metric units like ‘ft’ and
‘tbsp’. (Composite units are more appropriate for those; see above.)
Autoranging always applies the prefix to the leftmost unit name.
Calc chooses the largest prefix that causes the number to be greater
than or equal to 1.0. Thus an increasing sequence of adjusted times
would be ‘1 ms, 10 ms, 100 ms, 1 s, 10 s, 100 s, 1 ks’. Generally the
rule of thumb is that the number will be adjusted to be in the interval
‘[1 .. 1000)’, although there are several exceptions to this rule.
First, if the unit has a power then this is not possible; ‘0.1 s^2’
simplifies to ‘100000 ms^2’. Second, the “centi-” prefix is allowed to
form ‘cm’ (centimeters), but will not apply to other units. The
“deci-,” “deka-,” and “hecto-” prefixes are never used. Thus the
allowable interval is ‘[1 .. 10)’ for millimeters and ‘[1 .. 100)’ for
centimeters. Finally, a prefix will not be added to a unit if the
resulting name is also the actual name of another unit; ‘1e-15 t’ would
normally be considered a “femto-ton,” but it is written as ‘1000 at’
(1000 atto-tons) instead because ‘ft’ would be confused with feet.
Info Catalog
calc: Units
calc: Units
calc: The Units Table