Basic Operations on Units
=========================

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;
See Predefined Units.  You can also define your own unit names;
See 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 (See 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 `a s' (`calc-simplify') 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.  See 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 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'.  (Recall that multiplication binds more strongly than
division in Calc formulas, so the units here are acres per
meter-second.)  Remainder units are expressed in terms of
"fundamental" units like `m' and `s', regardless of the input units.

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.  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'.

The `u c' and `u b' commands treat 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.

The `u t' (`calc-convert-temperature') command converts absolute
temperatures.  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.

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.