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# 74. unit

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## 74.1 Introduction to Units

The unit package enables the user to convert between arbitrary units and work with dimensions in equations. The functioning of this package is radically different from the original Maxima units package - whereas the original was a basic list of definitions, this package uses rulesets to allow the user to chose, on a per dimension basis, what unit final answers should be rendered in. It will separate units instead of intermixing them in the display, allowing the user to readily identify the units associated with a particular answer. It will allow a user to simplify an expression to its fundamental Base Units, as well as providing fine control over simplifying to derived units. Dimensional analysis is possible, and a variety of tools are available to manage conversion and simplification options. In addition to customizable automatic conversion, units also provides a traditional manual conversion option.

Note - when unit conversions are inexact Maxima will make approximations resulting in fractions. This is a consequence of the techniques used to simplify units. The messages warning of this type of substitution are disabled by default in the case of units (normally they are on) since this situation occurs frequently and the warnings clutter the output. (The existing state of ratprint is restored after unit conversions, so user changes to that setting will be preserved otherwise.) If the user needs this information for units, they can set unitverbose:on to reactivate the printing of warnings from the unit conversion process.

unit is included in Maxima in the share/contrib/unit directory. It obeys normal Maxima package loading conventions:

    (%i1) load("unit")$
    *******************************************************************
    *                       Units version 0.50                        *
    *          Definitions based on the NIST Reference on             *
    *              Constants, Units, and Uncertainty                  *
    *       Conversion factors from various sources including         *
    *                   NIST and the GNU units package                *
    *******************************************************************

    Redefining necessary functions...
    WARNING: DEFUN/DEFMACRO: redefining function TOPLEVEL-MACSYMA-EVAL ...
    WARNING: DEFUN/DEFMACRO: redefining function MSETCHK ...
    WARNING: DEFUN/DEFMACRO: redefining function KILL1 ...
    WARNING: DEFUN/DEFMACRO: redefining function NFORMAT ...
    Initializing unit arrays...
    Done.

The WARNING messages are expected and not a cause for concern - they indicate the unit package is redefining functions already defined in Maxima proper. This is necessary in order to properly handle units. The user should be aware that if other changes have been made to these functions by other packages those changes will be overwritten by this loading process.

The unit.mac file also loads a lisp file unit-functions.lisp which contains the lisp functions needed for the package.

Clifford Yapp is the primary author. He has received valuable assistance from Barton Willis of the University of Nebraska at Kearney (UNK), Robert Dodier, and other intrepid folk of the Maxima mailing list.

There are probably lots of bugs. Let me know. `float` and `numer` don't do what is expected.

TODO : dimension functionality, handling of temperature, showabbr and friends. Show examples with addition of quantities containing units.

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## 74.2 Functions and Variables for Units

Function: setunits (list)

  By default, the unit package does not use any derived dimensions, but will convert all units to the seven fundamental dimensions using MKS units.

      (%i2) N;
                                           kg m
      (%o2)                                ----
                                             2
                                            s
      (%i3) dyn;
                                         1      kg m
      (%o3)                           (------) (----)
                                       100000     2
                                                 s
      (%i4) g;
                                          1
      (%o4)                             (----) (kg)
                                         1000
      (%i5) centigram*inch/minutes^2;
                                        127        kg m
      (%o5)                       (-------------) (----)
                                   1800000000000     2
                                                    s

  In some cases this is the desired behavior. If the user wishes to use other units, this is achieved with the `setunits` command:

      (%i6) setunits([centigram,inch,minute]);
      (%o6)                                done
      (%i7) N;
                                  1800000000000   %in cg
      (%o7)                      (-------------) (------)
                                       127            2
                                                  %min
      (%i8) dyn;
                                     18000000   %in cg
      (%o8)                         (--------) (------)
                                       127          2
                                                %min
      (%i9) g;
      (%o9)                             (100) (cg)
      (%i10) centigram*inch/minutes^2;
                                          %in cg
      (%o10)                              ------
                                              2
                                          %min

  The setting of units is quite flexible. For example, if we want to get back to kilograms, meters, and seconds as defaults for those dimensions we can do:

      (%i11) setunits([kg,m,s]);
      (%o11)                               done
      (%i12) centigram*inch/minutes^2;
                                        127        kg m
      (%o12)                      (-------------) (----)
                                   1800000000000     2
                                                    s

  Derived units are also handled by this command:

      (%i17) setunits(N);
      (%o17)                               done
      (%i18) N;
      (%o18)                                 N
      (%i19) dyn;
                                          1
      (%o19)                           (------) (N)
                                        100000
      (%i20) kg*m/s^2;
      (%o20)                                 N
      (%i21) centigram*inch/minutes^2;
                                          127
      (%o21)                        (-------------) (N)
                                     1800000000000

  Notice that the unit package recognized the non MKS combination of mass, length, and inverse time squared as a force, and converted it to Newtons. This is how Maxima works in general. If, for example, we prefer dyne to Newtons, we simply do the following:

      (%i22) setunits(dyn);
      (%o22)                               done
      (%i23) kg*m/s^2;
      (%o23)                          (100000) (dyn)
      (%i24) centigram*inch/minutes^2;
                                        127
      (%o24)                         (--------) (dyn)
                                      18000000

  To discontinue simplifying to any force, we use the uforget command:

      (%i26) uforget(dyn);
      (%o26)                               false
      (%i27) kg*m/s^2;
                                           kg m
      (%o27)                               ----
                                             2
                                            s
      (%i28) centigram*inch/minutes^2;
                                        127        kg m
      (%o28)                      (-------------) (----)
                                   1800000000000     2
                                                    s

  This would have worked equally well with `uforget(N)` or `uforget(%force)`.

  See also `uforget`. To use this function write first `load("unit")`.

Function: uforget (list)

  By default, the unit package converts all units to the seven fundamental dimensions using MKS units. This behavior can be changed with the `setunits` command. After that, the user can restore the default behavior for a particular dimension by means of the `uforget` command:

      (%i13) setunits([centigram,inch,minute]);
      (%o13)                               done
      (%i14) centigram*inch/minutes^2;
                                          %in cg
      (%o14)                              ------
                                              2
                                          %min
      (%i15) uforget([cg,%in,%min]);
      (%o15)                      [false, false, false]
      (%i16) centigram*inch/minutes^2;
                                        127        kg m
      (%o16)                      (-------------) (----)
                                   1800000000000     2
                                                    s

  `uforget` operates on dimensions, not units, so any unit of a particular dimension will work. The dimension itself is also a legal argument.

  See also `setunits`. To use this function write first `load("unit")`.

Function: convert (expr, list)

  When resetting the global environment is overkill, there is the `convert` command, which allows one time conversions. It can accept either a single argument or a list of units to use in conversion. When a convert operation is done, the normal global evaluation system is bypassed, in order to avoid the desired result being converted again. As a consequence, for inexact calculations "rat" warnings will be visible if the global environment controlling this behavior (`ratprint`) is true. This is also useful for spot-checking the accuracy of a global conversion. Another feature is `convert` will allow a user to do Base Dimension conversions even if the global environment is set to simplify to a Derived Dimension.

      (%i2) kg*m/s^2;
                                           kg m
      (%o2)                                ----
                                             2
                                            s
      (%i3) convert(kg*m/s^2,[g,km,s]);
                                           g km
      (%o3)                                ----
                                             2
                                            s
      (%i4) convert(kg*m/s^2,[g,inch,minute]);

      `rat' replaced 39.37007874015748 by 5000//127 = 39.37007874015748
                                    18000000000   %in g
      (%o4)                        (-----------) (-----)
                                        127           2
                                                  %min
      (%i5) convert(kg*m/s^2,[N]);
      (%o5)                                  N
      (%i6) convert(kg*m^2/s^2,[N]);
      (%o6)                                 m N
      (%i7) setunits([N,J]);
      (%o7)                                done
      (%i8) convert(kg*m^2/s^2,[N]);
      (%o8)                                 m N
      (%i9) convert(kg*m^2/s^2,[N,inch]);

      `rat' replaced 39.37007874015748 by 5000//127 = 39.37007874015748
                                       5000
      (%o9)                           (----) (%in N)
                                       127
      (%i10) convert(kg*m^2/s^2,[J]);
      (%o10)                                 J
      (%i11) kg*m^2/s^2;
      (%o11)                                 J
      (%i12) setunits([g,inch,s]);
      (%o12)                               done
      (%i13) kg*m/s^2;
      (%o13)                                 N
      (%i14) uforget(N);
      (%o14)                               false
      (%i15) kg*m/s^2;
                                      5000000   %in g
      (%o15)                         (-------) (-----)
                                        127       2
                                                 s
      (%i16) convert(kg*m/s^2,[g,inch,s]);

      `rat' replaced 39.37007874015748 by 5000//127 = 39.37007874015748
                                      5000000   %in g
      (%o16)                         (-------) (-----)
                                        127       2
                                                 s

  See also `setunits` and `uforget`. To use this function write first `load("unit")`.

Optional variable: usersetunits

  Default value: none

  If a user wishes to have a default unit behavior other than that described, they can make use of maxima-init.mac and the usersetunits variable. The unit package will check on startup to see if this variable has been assigned a list. If it has, it will use setunits on that list and take the units from that list to be defaults. `uforget` will revert to the behavior defined by usersetunits over its own defaults. For example, if we have a maxima-init.mac file containing:

      usersetunits : [N,J];

  we would see the following behavior:

      (%i1) load("unit")$
      *******************************************************************
      *                       Units version 0.50                        *
      *          Definitions based on the NIST Reference on             *
      *              Constants, Units, and Uncertainty                  *
      *       Conversion factors from various sources including         *
      *                   NIST and the GNU units package                *
      *******************************************************************

      Redefining necessary functions...
      WARNING: DEFUN/DEFMACRO: redefining function
       TOPLEVEL-MACSYMA-EVAL ...
      WARNING: DEFUN/DEFMACRO: redefining function MSETCHK ...
      WARNING: DEFUN/DEFMACRO: redefining function KILL1 ...
      WARNING: DEFUN/DEFMACRO: redefining function NFORMAT ...
      Initializing unit arrays...
      Done.
      User defaults found...
      User defaults initialized.
      (%i2) kg*m/s^2;
      (%o2)                                  N
      (%i3) kg*m^2/s^2;
      (%o3)                                  J
      (%i4) kg*m^3/s^2;
      (%o4)                                 J m
      (%i5) kg*m*km/s^2;
      (%o5)                             (1000) (J)
      (%i6) setunits([dyn,eV]);
      (%o6)                                done
      (%i7) kg*m/s^2;
      (%o7)                           (100000) (dyn)
      (%i8) kg*m^2/s^2;
      (%o8)                     (6241509596477042688) (eV)
      (%i9) kg*m^3/s^2;
      (%o9)                    (6241509596477042688) (eV m)
      (%i10) kg*m*km/s^2;
      (%o10)                   (6241509596477042688000) (eV)
      (%i11) uforget([dyn,eV]);
      (%o11)                           [false, false]
      (%i12) kg*m/s^2;
      (%o12)                                 N
      (%i13) kg*m^2/s^2;
      (%o13)                                 J
      (%i14) kg*m^3/s^2;
      (%o14)                                J m
      (%i15) kg*m*km/s^2;
      (%o15)                            (1000) (J)

  Without `usersetunits`, the initial inputs would have been converted to MKS, and uforget would have resulted in a return to MKS rules. Instead, the user preferences are respected in both cases. Notice these can still be overridden if desired. To completely eliminate this simplification - i.e. to have the user defaults reset to factory defaults - the `dontusedimension` command can be used. `uforget` can restore user settings again, but only if `usedimension` frees it for use. Alternately, `kill(usersetunits)` will completely remove all knowledge of the user defaults from the session. Here are some examples of how these various options work.

      (%i2) kg*m/s^2;
      (%o2)                                  N
      (%i3) kg*m^2/s^2;
      (%o3)                                  J
      (%i4) setunits([dyn,eV]);
      (%o4)                                done
      (%i5) kg*m/s^2;
      (%o5)                           (100000) (dyn)
      (%i6) kg*m^2/s^2;
      (%o6)                     (6241509596477042688) (eV)
      (%i7) uforget([dyn,eV]);
      (%o7)                          [false, false]
      (%i8) kg*m/s^2;
      (%o8)                                  N
      (%i9) kg*m^2/s^2;
      (%o9)                                  J
      (%i10) dontusedimension(N);
      (%o10)                             [%force]
      (%i11) dontusedimension(J);
      (%o11)                         [%energy, %force]
      (%i12) kg*m/s^2;
                                           kg m
      (%o12)                               ----
                                             2
                                            s
      (%i13) kg*m^2/s^2;
                                               2
                                           kg m
      (%o13)                               -----
                                             2
                                            s
      (%i14) setunits([dyn,eV]);
      (%o14)                               done
      (%i15) kg*m/s^2;
                                           kg m
      (%o15)                               ----
                                             2
                                            s
      (%i16) kg*m^2/s^2;
                                               2
                                           kg m
      (%o16)                               -----
                                             2
                                            s
      (%i17) uforget([dyn,eV]);
      (%o17)                         [false, false]
      (%i18) kg*m/s^2;
                                           kg m
      (%o18)                               ----
                                             2
                                            s
      (%i19) kg*m^2/s^2;
                                               2
                                           kg m
      (%o19)                               -----
                                             2
                                            s
      (%i20) usedimension(N);
      Done.  To have Maxima simplify to this dimension, use
      setunits([unit]) to select a unit.
      (%o20)                               true
      (%i21) usedimension(J);
      Done.  To have Maxima simplify to this dimension, use
      setunits([unit]) to select a unit.
      (%o21)                               true
      (%i22) kg*m/s^2;
                                           kg m
      (%o22)                               ----
                                             2
                                            s
      (%i23) kg*m^2/s^2;
                                               2
                                           kg m
      (%o23)                               -----
                                             2
                                            s
      (%i24) setunits([dyn,eV]);
      (%o24)                               done
      (%i25) kg*m/s^2;
      (%o25)                          (100000) (dyn)
      (%i26) kg*m^2/s^2;
      (%o26)                    (6241509596477042688) (eV)
      (%i27) uforget([dyn,eV]);
      (%o27)                           [false, false]
      (%i28) kg*m/s^2;
      (%o28)                                 N
      (%i29) kg*m^2/s^2;
      (%o29)                                 J
      (%i30) kill(usersetunits);
      (%o30)                               done
      (%i31) uforget([dyn,eV]);
      (%o31)                          [false, false]
      (%i32) kg*m/s^2;
                                           kg m
      (%o32)                               ----
                                             2
                                            s
      (%i33) kg*m^2/s^2;
                                               2
                                           kg m
      (%o33)                               -----
                                             2
                                            s

  Unfortunately this wide variety of options is a little confusing at first, but once the user grows used to them they should find they have very full control over their working environment.

Function: metricexpandall (x)

  Rebuilds global unit lists automatically creating all desired metric units. x is a numerical argument which is used to specify how many metric prefixes the user wishes defined. The arguments are as follows, with each higher number defining all lower numbers' units:

                 0 - none. Only base units
                 1 - kilo, centi, milli
      (default)  2 - giga, mega, kilo, hecto, deka, deci, centi, milli,
                     micro, nano
                 3 - peta, tera, giga, mega, kilo, hecto, deka, deci,
                     centi, milli, micro, nano, pico, femto
                 4 - all

  Normally, Maxima will not define the full expansion since this results in a very large number of units, but `metricexpandall` can be used to rebuild the list in a more or less complete fashion. The relevant variable in the unit.mac file is %unitexpand.

Variable: %unitexpand

  Default value: `2`

  This is the value supplied to `metricexpandall` during the initial loading of unit.

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