One special interpretation of algebraic formulas is as numbers with units.
For example, the formula ``5 m / s^2'` can be read "five meters
per second squared." The commands in this chapter help you
manipulate units expressions in this form. Units-related commands
begin with the `u` prefix key.

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.

The `u v` (`calc-enter-units-table`

) command displays the units table
in another buffer called `*Units Table*`

. Each entry in this table
gives the unit name as it would appear in an expression, the definition
of the unit in terms of simpler units, and a full name or description of
the unit. Fundamental units are defined as themselves; these are the
units produced by the `u b` command. The fundamental units are
meters, seconds, grams, kelvins, amperes, candelas, moles, radians,
and steradians.

The Units Table buffer also displays the Unit Prefix Table. Note that
two prefixes, "kilo" and "hecto," accept either upper- or lower-case
prefix letters. ``Meg'` is also accepted as a synonym for the ``M'`
prefix. Whenever a unit name can be interpreted as either a built-in name
or a prefix followed by another built-in name, the former interpretation
wins. For example, ``2 pt'` means two pints, not two pico-tons.

The Units Table buffer, once created, is not rebuilt unless you define
new units. To force the buffer to be rebuilt, give any numeric prefix
argument to `u v`.

The `u V` (`calc-view-units-table`

) command is like `u v` except
that the cursor is not moved into the Units Table buffer. You can
type `u V` again to remove the Units Table from the display. To
return from the Units Table buffer after a `u v`, type `M-# c`
again or use the regular Emacs `C-x o` (`other-window`

)
command. You can also kill the buffer with `C-x k` if you wish;
the actual units table is safely stored inside the Calculator.

The `u g` (`calc-get-unit-definition`

) command retrieves a unit's
defining expression and pushes it onto the Calculator stack. For example,
`u g in` will produce the expression ``2.54 cm'`. This is the
same definition for the unit that would appear in the Units Table buffer.
Note that this command works only for actual unit names; `u g km`
will report that no such unit exists, for example, because `km`

is
really the unit `m`

with a `k`

("kilo") prefix. To see a
definition of a unit in terms of base units, it is easier to push the
unit name on the stack and then reduce it to base units with `u b`.

The `u e` (`calc-explain-units`

) command displays an English
description of the units of the expression on the stack. For example,
for the expression ``62 km^2 g / s^2 mol K'`, the description is
"Square-Kilometer Gram per (Second-squared Mole Degree-Kelvin)." This
command uses the English descriptions that appear in the righthand
column of the Units Table.

Since the exact definitions of many kinds of units have evolved over the
years, and since certain countries sometimes have local differences in
their definitions, it is a good idea to examine Calc's definition of a
unit before depending on its exact value. For example, there are three
different units for gallons, corresponding to the US (`gal`

),
Canadian (`galC`

), and British (`galUK`

) definitions. Also,
note that `oz`

is a standard ounce of mass, `ozt`

is a Troy
ounce, and `ozfl`

is a fluid ounce.

The temperature units corresponding to degrees Kelvin and Centigrade
(Celsius) are the same in this table, since most units commands treat
temperatures as being relative. The `calc-convert-temperature`

command has special rules for handling the different absolute magnitudes
of the various temperature scales.

The unit of volume "liters" can be referred to by either the lower-case
`l`

or the upper-case `L`

.

The unit `A`

stands for Amperes; the name `Ang`

is used

The unit `pt`

stands for pints; the name `point`

stands for
a typographical point, defined by ``72 point = 1 in'`. There is
also `tpt`

, which stands for a printer's point as defined by the
TeX typesetting system: ``72.27 tpt = 1 in'`.

The unit `e`

stands for the elementary (electron) unit of charge;
because algebra command could mistake this for the special constant
e, Calc provides the alternate unit name `ech`

which is
preferable to `e`

.

The name `g`

stands for one gram of mass; there is also `gf`

,
one gram of force. (Likewise for `lb`, pounds, and `lbf`.)
Meanwhile, one "g" of acceleration is denoted `ga`

.

The unit `ton`

is a U.S. ton of ``2000 lb'`, and `t`

is
a metric ton of ``1000 kg'`.

The names `s`

(or `sec`

) and `min`

refer to units of
time; `arcsec`

and `arcmin`

are units of angle.

Some "units" are really physical constants; for example, `c`

represents the speed of light, and `h`

represents Planck's
constant. You can use these just like other units: converting
``.5 c'` to ``m/s'` expresses one-half the speed of light in
meters per second. You can also use this merely as a handy reference;
the `u g` command gets the definition of one of these constants
in its normal terms, and `u b` expresses the definition in base
units.

Two units, `pi`

and `fsc`

(the fine structure constant,
approximately *1/137*) are dimensionless. The units simplification
commands simply treat these names as equivalent to their corresponding
values. However you can, for example, use `u c` to convert a pure
number into multiples of the fine structure constant, or `u b` to
convert this back into a pure number. (When `u c` prompts for the
"old units," just enter a blank line to signify that the value
really is unitless.)

Calc provides ways to get quick access to your selected "favorite" units, as well as ways to define your own new units.

To select your favorite units, store a vector of unit names or
expressions in the Calc variable `Units`

. The `u 1`
through `u 9` commands (`calc-quick-units`

) provide access
to these units. If the value on the top of the stack is a plain
number (with no units attached), then `u 1` gives it the
specified units. (Basically, it multiplies the number by the
first item in the `Units`

vector.) If the number on the
stack *does* have units, then `u 1` converts that number
to the new units. For example, suppose the vector ``[in, ft]'`
is stored in `Units`

. Then `30 u 1` will create the
expression ``30 in'`, and `u 2` will convert that expression
to ``2.5 ft'`.

The `u 0` command accesses the tenth element of `Units`

.
Only ten quick units may be defined at a time. If the `Units`

variable has no stored value (the default), or if its value is not
a vector, then the quick-units commands will not function. The
`s U` command is a convenient way to edit the `Units`

variable; see section Other Operations on Variables.

The `u d` (`calc-define-unit`

) command records the units
expression on the top of the stack as the definition for a new,
user-defined unit. For example, putting ``16.5 ft'` on the stack and
typing `u d rod` defines the new unit ``rod'` to be equivalent to
16.5 feet. The unit conversion and simplification commands will now
treat `rod`

just like any other unit of length. You will also be
prompted for an optional English description of the unit, which will
appear in the Units Table.

The `u u` (`calc-undefine-unit`

) command removes a user-defined
unit. It is not possible to remove one of the predefined units,
however.

If you define a unit with an existing unit name, your new definition
will replace the original definition of that unit. If the unit was a
predefined unit, the old definition will not be replaced, only
"shadowed." The built-in definition will reappear if you later use
`u u` to remove the shadowing definition.

To create a new fundamental unit, use either 1 or the unit name itself
as the defining expression. Otherwise the expression can involve any
other units that you like (except for composite units like ``mfi'`).
You can create a new composite unit with a sum of other units as the
defining expression. The next unit operation like `u c` or `u v`
will rebuild the internal unit table incorporating your modifications.
Note that erroneous definitions (such as two units defined in terms of
each other) will not be detected until the unit table is next rebuilt;
`u v` is a convenient way to force this to happen.

Temperature units are treated specially inside the Calculator; it is not possible to create user-defined temperature units.

The `u p` (`calc-permanent-units`

) command stores the user-defined
units in your ``.emacs'` file, so that the units will still be
available in subsequent Emacs sessions. If there was already a set of
user-defined units in your ``.emacs'` file, it is replaced by the
new set. (See section General Mode Commands, for a way to tell Calc to use
a different file instead of ``.emacs'`.)

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