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--- manual:chapter3:units [2018/10/18 09:29]
claudio
+++ manual:chapter3:units [2019/11/19 13:34] (current)
claudio [Custom Units]
@@ Line 1: / Line 1: @@
-| **Length** ||||
+===== Units =====
-| Unit | Unit name | Definition | Description |
-| 1_m | metre | 1_m  | Base SI unit of length |
+----
-| 1_cm | centimetre | 1_cm = 0.01_m | SI length |
-| 1_mm | millimetre | 1_mm = 0.001_m | SI length |
+==== Entering Units ====
-| 1_yd | yard | 1_yd = 3_ft | US customary length |
-| 1_ft | foot | 1_ft = 12_in | US customary length |
+Units are objects that have 2 separate parts: the value and the unit per se. The value is usually a real number, but units may be attached to other numerical objects as well, for example complex numbers and vectors or matrices can have associated units. The unit part is entered using the underscore ''_'', which acts as both a separator and an operator.
-| 1_in | inch | 1_in = 0.0254_m | US customary length |
-| 1_Mpc | megaparsec | 1_Mpc = 1000000_pc | Astronomical length |
+The unit is an algebraic expression combining one or more unit symbols, and may be optionally surrounded by square brackets. The brackets are optional for stack input, but mandatory when units are used within a symbolic expression (to separate the unit from other algebraic operations). The ''_'' key will always add the brackets regardless of whether the user is typing an algebraic expression or not, the use of the brackets is encouraged.
-| 1_pc | parsec | 1_pc = (648000/π)_au | Astronomical length |
-| 1_lyr | light year | 1_lyr = 365.25_d*c | Astronomical length |
+Examples of valid unit objects are: \\
-| 1_au | astronomical unit | 1_au = 149597870700_m | SI astronomical length |
+''1_m'' \\
-| 1_km | kilometre | 1_km = 1000_m | SI length |
+''3_m/s^2'' \\
-| 1_mi | mile | 1_mi = 5280_ft | US customary length |
+''3_[m/s^2]'' \\
-| 1_nmi | nautical mile | 1_nmi = 1852_m | non-SI length |
+''[1 2 3]_kg'' \\
-| 1_miUS | surveying mile | 1_miUS = 5280_ftUS | US surveying length |
+''%%'3_[m]+2_[ft]'%%'' \\
-| 1_chain | surveying chain | 1_chain = 66_ftUS | US surveying length |
+in all cases above, the underscore is being used as a separator, but can be used as an operator as well: \\
-| 1_rd | rod | 1_rd =16.5_ft | US customary length |
+''_[m]'' \\
-| 1_rdUS | surveying rod | 1_rdUS =16.5_ftUS | US surveying length |
+''[1 2 3] _[kg]'' \\
-| 1_fath | surveying fathom | 1_fath =6_ftUS | US surveying length |
+in the first case, the operator ''_[]'' will apply the unit to some value previously left on the stack. In the second case, the value (a vector) is separated with a space, therefore it will be compiled as a separate object, put on the stack, and then the underscore operator will apply the unit.
-| 1_ftUS | surveying foot | 1_ftUS = 1200/3937_m | US surveying length |
-| 1_mil | thousandth of inch | 1_mil = 0.001_in | US customary length |
+Exponents within a unit expression can be integer numbers, real numbers or fractional exponents. The use of fractional exponents is always preferred over real numbers because they can be simplified in an exact way. \\
-| 1_µ | micron or micrometre | 1_µ = 1_µm | SI length |
+''3_[m^(2/3)]'' is always preferred to ''3_[m^0.666667]'' \\
-| 1_Å | Ångström | 1_Å = 0.1_nm | non-SI length |
-| 1_fermi | fermi or femtometre | 1_fermi = 1_fm | SI length |
+==== Multiples and Submultiples ====
+Multiples and submultiples of many units are obtained by prefixing them with the standard SI prefixes. Therefore, ''1_dal'' is equivalent to ''10_l'' and ''3.5_MHz'' is equivalent to ''3500000_Hz''.
+SI prefixes can be used with all SI units and many non-SI units where SI prefixes are common usage. Other non-SI units do not accept prefixes.
+^  SI Prefix  ^  Factor  ^  Name  ^
+|  Y   |  $10^{24}$   |  yotta-  |
+|  Z   |  $10^{21}$   |  zetta-  |
+|  E   |  $10^{18}$   |  exa-    |
+|  P   |  $10^{15}$   |  peta-   |
+|  T   |  $10^{12}$   |  tera-   |
+|  G   |  $10^{9}$    |  giga-   |
+|  M   |  $10^{6}$    |  mega-   |
+|  k   |  $10^{3}$    |  kilo-   |
+|  h   |  $10^{2}$    |  hecto-  |
+|  da  |  $10^{1}$    |  deka-   |
+|  d   |  $10^{-1}$   |  deci-   |
+|  c   |  $10^{-2}$   |  centi-  |
+|  m   |  $10^{-3}$   |  milli-  |
+|  µ   |  $10^{-6}$   |  micro-  |
+|  n   |  $10^{-9}$   |  nano-   |
+|  p   |  $10^{-12}$  |  pico-   |
+|  f   |  $10^{-15}$  |  femto-  |
+|  a   |  $10^{-18}$  |  atto-   |
+|  z   |  $10^{-21}$  |  zepto-  |
+|  y   |  $10^{-24}$  |  yocto-  |
+==== Operating with Units ====
+Unit objects can be operated upon normally, and the system will perform all the necessary unit conversions in the background. Unit consistency is strictly enforced for addition and subtraction, therefore if one variable uses units in an algebraic expression, all variables should have proper units attached.
+Results of an addition or subtraction between consistent units will produce a result with the units of the first argument (argument in level 2 of the stack at the time of the operation, or the left argument in an algebraic expression).
+Unit conversions can be performed directly using the ''CONVERT'' command, which is accessible directly from the keyboard (**LS-6**). For example to convert 100 ml into m^3 simply place the original unit and the new unit on the stack: ''100_ml'' ''1_m^3'' ''CONVERT''.
+To reduce a unit to its base components, use the command ''UBASE''. For example ''1_N'' ''UBASE'' will return ''1_kg*m/s^2''.
+To extract the value part of a unit object, use the ''UVAL'' command, which will return the object without its unit.
+==== Custom Units ====
+Units in newRPL can be arbitrary symbols, not just the ones predefined in the system. For example, adding ''3_apples'' and ''2_apples'' will produce ''5_apples'' as expected. In this case, ''apples'' is a custom unit without a definition. Units without a definition are considered base units. The system already includes base units for mass, length, time, etc. but any number of new base units can be added simply by using new symbols.
+Custom units can also be defined as derived units, in terms of other base units. This is done with the ''UDEFINE'' command. For example typing ''750_ml'' ''%%'mug'%%'' ''UDEFINE'' would define a unit named ''mug'' as 750 ml. From that moment ''mug'' can be used in the exact same way as any system-defined unit, and it will be considered consistent with all other volume units.
+Custom units may also accept SI prefixes, but this needs to be explicitly enabled during the unit definition. To create a custom unit that accepts SI prefixes, the name of the unit needs to be preceded by a question mark ''?'' when the unit is defined. For example, following the previous example, for the unit ''mug'' to accept SI prefixes, the definition should be done typing ''750_ml'' ''%%'?mug'%%'' ''UDEFINE'' creates a unit ''mug'' exactly as before, except now ''1_[Mmug]'' ''UBASE'' will produce ''750_m^3'', as expected.
+New units can also be defined in terms of new (unknown) base units. For example typing ''3_apples'' ''%%'oranges'%%'' ''UDEFINE'' will define ''oranges'' as equivalent to 3 ''apples'', where ''apples'' is a base unit (amount of fruit?) that may or may not have been explicitly defined before.
+From that moment the addition ''2_[oranges]+3_[apples]'' will result in ''3_oranges'' since the units of the result will match the units of the left operand, and the 3 apples were automatically converted to 1 orange.
+Custom units work systemwide, the command ''ULIST'' returns a list of all the custom units that are defined in the system. To remove a unit definition use the ''UPURGE'' command. Following the previous example, typing ''%%'oranges'%%'' ''UPURGE'' will undefine the unit. Notice that the unit ''oranges'' definition was removed, which means this symbol now represents an unknown base unit, then ''2_[oranges]+3_[apples]'' cannot proceed since oranges and apples are now considered as unrelated base units.
+==== Available Units ====
+This is a list of the units available in the ''Units'' menu on newRPL. All the conversion factors are meant to be exact, not approximated.
+=== Length ===
+^ Unit ^ Unit name ^ Definition ^ Description ^ Prefixable ^
+| 1_m | metre | 1_m  | Base SI unit of length | {{fa>check?lg&color=green}} |
+| 1_cm | centimetre | 1_cm = 0.01_m | SI, metric | {{fa>times?lg&color=red}} |
+| 1_mm | millimetre | 1_mm = 0.001_m | SI, metric | {{fa>times?lg&color=red}} |
+| 1_yd | yard | 1_yd = 3_ft | US Customary | {{fa>times?lg&color=red}} |
+| 1_ft | foot | 1_ft = 12_in | US Customary | {{fa>times?lg&color=red}} |
+| 1_in | inch | 1_in = 0.0254_m | US Customary | {{fa>times?lg&color=red}} |
+| 1_Mpc | megaparsec | 1_Mpc = 1000000_pc | Astronomical length | {{fa>times?lg&color=red}} |
+| 1_pc | parsec | 1_pc = (648000/π)_au | Astronomical length | {{fa>times?lg&color=red}} |
+| 1_lyr | light year | 1_lyr = 9460730472580800_m | Astronomical length | {{fa>times?lg&color=red}} |
+| 1_au | astronomical unit | 1_au = 149597870700_m | SI astronomical length | {{fa>times?lg&color=red}} |
+| 1_km | kilometre | 1_km = 1000_m | SI, metric | {{fa>times?lg&color=red}} |
+| 1_mi | mile | 1_mi = 5280_ft | US Customary | {{fa>times?lg&color=red}} |
+| 1_nmi | nautical mile | 1_nmi = 1852_m | non-SI length | {{fa>times?lg&color=red}} |
+| 1_miUS | surveying mile | 1_miUS = 5280_ftUS | US surveying length | {{fa>times?lg&color=red}} |
+| 1_chain | surveying chain | 1_chain = 66_ftUS | US surveying length | {{fa>times?lg&color=red}} |
+| 1_rd | rod | 1_rd =16.5_ft | US customary length | {{fa>times?lg&color=red}} |
+| 1_rdUS | surveying rod | 1_rdUS =16.5_ftUS | US surveying length | {{fa>times?lg&color=red}} |
+| 1_fath | surveying fathom | 1_fath =6_ftUS | US surveying length | {{fa>times?lg&color=red}} |
+| 1_ftUS | surveying foot | 1_ftUS = 1200/3937_m | US surveying length | {{fa>times?lg&color=red}} |
+| 1_mil | thousandth of inch | 1_mil = 0.001_in | US customary length | {{fa>times?lg&color=red}} |
+| 1_µ | micron | 1_µ = 1_µm | SI length | {{fa>times?lg&color=red}} |
+| 1_Å | ångström | 1_Å = 0.1_nm | non-SI length | {{fa>times?lg&color=red}} |
+| 1_fermi | fermi  | 1_fermi = 1_fm | SI length | {{fa>times?lg&color=red}} |
 | **Area** ||||
@@ Line 55: / Line 138: @@
 | 1_cu | cup | 1_cu = 1/16_gal | US customary volume |
 | 1_ozfl | fluid ounce | 1_ozfl = 1/128_gal | US customary volume |
-| 1_tbsp | table spoon | 1_tbsp = 1/2_ozfl | US customary volume |
+| 1_tbsp | tablespoon | 1_tbsp = 1/2_ozfl | US customary volume |
-| 1_tsp | tea spoon | 1_tsp = 1/6_ozfl | US customary volume |
+| 1_tsp | teaspoon | 1_tsp = 1/6_ozfl | US customary volume |
 | 1_fbm | board foot | 1_fbm = 144_in%%^%%3 | US customary volume |
 | 1_galUK | gallon (UK) | 1_galUK = 0.00454609_m%%^%%3 | Imperial volume |
@@ Line 76: / Line 159: @@
 | **Time** ||||
 | Unit | Unit name | Definition | Description |
-| 1_yr | tropical year | 1_yr = 31556925.9747_sper  | 1952 Ephemeris Time |
+| 1_yr | tropical year | 1_yr = 31556925.9747_s  |  per 1952 Ephemeris Time |
 | 1_d | day | 1_d = 86400_s | SI time |
 | 1_h | hour | 1_h = 3600_s | SI time |
 | 1_min | minute | 1_min = 60_s | SI time |
 | 1_s | second | 1_s | Base SI unit of time |
-| 1_Hz | Hertz | 1_Hz = 1_1/s | SI frequency |
+| 1_Hz | hertz | 1_Hz = 1_1/s | SI frequency |
 | **Speed** ||||
@@ Line 107: / Line 190: @@
 | Unit | Unit name | Definition | Description |
 | 1_kg | kilogram | 1_kg | Base SI unit of mass |
-| 1_g | kilogram | 1_g = 0.001_kg | SI mass |
+| 1_g | gram | 1_g = 0.001_kg | SI mass |
 | 1_lb | pound | 1_lb = 7000_grain | US customary mass |
 | 1_oz | ounce | 1_oz = 1/16_lb | US customary mass |
@@ Line 114: / Line 197: @@
 | 1_ton | tonne | 1_ton = 2000_lb | US customary mass |
 | 1_tonUK | tonne (UK) | 1_tonUK = 2240_lb | Imperial mass |
-| 1_t | Metric tonne | 1_t = 1000_kg | non-SI mass |
+| 1_t | metric tonne | 1_t = 1000_kg | non-SI mass |
 | 1_ozt | troy ounce | 1_ozt = 480_grain | Imperial mass |
-| 1_ct | Metric carat | 1_ct = 200_mg | Metric System mass |
+| 1_ct | metric carat | 1_ct = 200_mg | Metric System mass |
 | 1_grain | grain | 1_grain = 64.79891_mg | Imperial mass |
 | 1_u | unified atomic mass | 1_u = 1.660539040E-27_kg | non-SI mass |
 | 1_mol | mole | 1_mol | Base SI amount of substance |
 | 1_gmol | gram mole | 1_gmol = 1_mol | non-SI amount of substance |
-| 1_lbmol | pound mole | 1_lbmol = 453.59237_mol | US customary substance |
+| 1_lbmol | pound mole | 1_lbmol = 453.59237_mol | US customary amount of substance |
 | **Force** ||||
 | Unit | Unit name | Definition | Description |
-| 1_N | Newton | 1_N = 1_kg*m/s%%^%%2 | SI force |
+| 1_N | newton | 1_N = 1_kg*m/s%%^%%2 | SI force |
 | 1_dyn | dyne | 1_dyn = 1_g*cm/s%%^%%2 | non-SI force |
 | 1_gf | gram-force | 1_gf = 1_g*ga | non-SI force |
 | 1_kip | kip | 1_kip = 1000_lbf | US customary force |
 | 1_lbf | pound-force | 1_lbf = 1_lb*ga | US customary force |
-| 1_pdl | poundal | 1_pdl = = 1_lb*ft/s%%^%%2 | US customary force |
+| 1_pdl | poundal | 1_pdl = 1_lb*ft/s%%^%%2 | US customary force |
 | **Energy** ||||
 | Unit | Unit name | Definition | Description |
-| 1_J | Joule | 1_J = 1_N*m | SI energy |
+| 1_J | joule | 1_J = 1_N*m | SI energy |
 | 1_kW*h | kilowatt-hour | 1_kW*h = 3.6_MJ | non-SI energy |
 | 1_erg | erg | 1_erg = 1_dyn*cm | non-SI energy |
-| 1_kcal | kilo calorie | 1_kcal = 1000_cal | non-SI energy |
+| 1_kcal | kilocalorie (thermochemical) | 1_kcal = 1000_cal | non-SI energy |
-| 1_cal | calorie | 1_cal = 4.184_J | non-SI energy |
+| 1_cal | calorie (thermochemical) | 1_cal = 4.184_J | non-SI energy |
 | 1_Btu | british thermal unit | 1_Btu = =1055.056_J | Imperial energy |
 | 1_ft*lbf | foot-pound | 1_ft*lbf = 1_lb*ft*ga | US customary energy |
 | 1_therm | therm | 1_therm = 100000_Btu | Imperial energy |
-| 1_MeV | mega electronvolt | 1_MeV = 1E6_eV | non-SI energy |
+| 1_MeV | megaelectronvolt | 1_MeV = 1E6_eV | non-SI energy |
 | 1_eV | electronvolt | 1_eV = 1.6021766208E-19_J | non-SI energy |
 | **Power** ||||
 | Unit | Unit name | Definition | Description |
-| 1_W | Watt | 1_W = 1_J/s | SI power |
+| 1_W | watt | 1_W = 1_J/s | SI power |
 | 1_kW | kilowatt | 1_kW = 1000_W | SI power |
 | 1_MW | megawatt | 1_MW = 1000000_W | SI power |
@@ Line 155: / Line 238: @@
 | **Pressure** ||||
 | Unit | Unit name | Definition | Description |
-| 1_Pa | Pascal | 1_Pa = 1_N/m%%^%%2 | SI pressure |
+| 1_Pa | pascal | 1_Pa = 1_N/m%%^%%2 | SI pressure |
 | 1_atm | atmospheric pressure | 1_atm = 101325_Pa | non-SI pressure |
 | 1_bar | bar | 1_bar = 100000_Pa | non-SI pressure |
 | 1_psi | pound per square inch | 1_psi = 1_lbf/in%%^%%2 | US customary pressure |
-| 1_Torr | Torricelli | 1_Torr = 1/760_atm | non-SI pressure |
+| 1_Torr | torr | 1_Torr = 1/760_atm | non-SI pressure |
 | 1_mmHg | millimetres of mercury (at 0°C) | 1_mmHg = 13595.1_kg/m%%^%%3*mm*ga | non-SI pressure |
 | 1_inHg | inches of mercury (at 0°C) | 1_inHg = 13595.1_kg/m%%^%%3*in*ga | non-SI pressure |
@@ Line 166: / Line 249: @@
 | **Temperature** ||||
 | Unit | Unit name | Definition | Description |
-| 1_°C | celsius (or degree celsius) | 1_°C Zero: 0_°C → 273.15_K | Base SI temperature (rel. scale) |
+| 1_°C | celsius (or degree Celsius) | 1_°C Zero: 0_°C → 273.15_K | Base SI temperature (rel. scale) |
-| 1_Δ°C | celsius temperature change | 1_Δ°C=| 1_ΔKchange of temperature |
+| 1_Δ°C | celsius temperature change | 1_Δ°C = 1_ΔK | Change of temperature |
-| 1_°F | farenheit (or degree farenheit) | 1_°F Zero: 0_°F → 459.67_°R | US cust. temp. (rel. scale) |
+| 1_°F | farenheit (or degree Farenheit) | 1_°F Zero: 0_°F → 459.67_°R | US cust. temp. (rel. scale) |
-| 1_Δ°F | farenheit temperature change | 1_Δ°F=| 1_Δ°Rchange of temperature |
+| 1_Δ°F | farenheit temperature change | 1_Δ°F = 1_Δ°R | Change of temperature |
 | 1_K | kelvin | 1_K | Base SI temperature (abs. scale) |
 | 1_ΔK | kelvin temperature change | 1_ΔK | Base change of temperature |
 | 1_°R | rankine | 1_°R = 5/9_K | Imperial temperature (abs. scale) |
-| 1_Δ°R | rankine temperature change | 1_Δ°R = 5/9_ΔKchange of temperature |
+| 1_Δ°R | rankine temperature change | 1_Δ°R = 5/9_ΔK | Change of temperature |
 | **Electricity and Magnetism** ||||
@@ Line 185: / Line 268: @@
 | 1_Fdy | faraday | 1_Fdy = 96485.33289_C | non-SI electrical charge |
 | 1_H | henry | 1_H = 1_Wb/A | SI inductance |
-| 1_mho | mho (Siemens) | 1_mho = 1_S | non-SI conductance |
+| 1_mho | mho | 1_mho = 1_S | non-SI conductance |
 | 1_S | siemens | 1_S = 1_1/Ω | SI conductance |
 | 1_T | tesla | 1_T = 1_Wb/m%%^%%2 | SI magnetic flux density |
@@ Line 218: / Line 301: @@
 | 1_Bq | becquerel | 1_Bq = 1_1/s | SI radiation |
 | 1_Ci | curie | 1_Ci = 3.7E10_Bq | non-SI radiation |
-| 1_R | rontgen | 1_R = 2.58E-4_C/kg | non-SI radiation |
+| 1_R | roentgen | 1_R = 2.58E-4_C/kg | non-SI radiation |
 | **Viscosity** ||||