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 section Predefined Units. You can also define your own unit names;
see section 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 section 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 section 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

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,

`si`

units
except that `base`

uses `si`

uses
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.

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