## Units

### Entering Units

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.

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.

Examples of valid unit objects are:

`1_m`

`3_m/s^2`

`3_[m/s^2]`

`[1 2 3]_kg`

`'3_[m]+2_[ft]'`

in all cases above, the underscore is being used as a separator, but can be used as an operator as well:

`_[m]`

`[1 2 3] _[kg]`

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.

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.

`3_[m^(2/3)]`

is always preferred to `3_[m^0.666667]`

### 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 | |

1_cm | centimetre | 1_cm = 0.01_m | SI, metric | |

1_mm | millimetre | 1_mm = 0.001_m | SI, metric | |

1_yd | yard | 1_yd = 3_ft | US Customary | |

1_ft | foot | 1_ft = 12_in | US Customary | |

1_in | inch | 1_in = 0.0254_m | US Customary | |

1_Mpc | megaparsec | 1_Mpc = 1000000_pc | Astronomical length | |

1_pc | parsec | 1_pc = (648000/π)_au | Astronomical length | |

1_lyr | light year | 1_lyr = 9460730472580800_m | Astronomical length | |

1_au | astronomical unit | 1_au = 149597870700_m | SI astronomical length | |

1_km | kilometre | 1_km = 1000_m | SI, metric | |

1_mi | mile | 1_mi = 5280_ft | US Customary | |

1_nmi | nautical mile | 1_nmi = 1852_m | non-SI length | |

1_miUS | surveying mile | 1_miUS = 5280_ftUS | US surveying length | |

1_chain | surveying chain | 1_chain = 66_ftUS | US surveying length | |

1_rd | rod | 1_rd =16.5_ft | US customary length | |

1_rdUS | surveying rod | 1_rdUS =16.5_ftUS | US surveying length | |

1_fath | surveying fathom | 1_fath =6_ftUS | US surveying length | |

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 | |

1_µ | micron | 1_µ = 1_µm | SI length | |

1_Å | ångström | 1_Å = 0.1_nm | non-SI length | |

1_fermi | fermi | 1_fermi = 1_fm | SI length |

Area |
|||

Unit | Unit name | Definition | Description |

1_m^2 | square metre | 1_m^2 | Base SI unit of area |

1_cm^2 | square centimetre | 1_cm^2 = 0.0001_m^2 | SI area |

1_b | barn | 1_b = 1E-28_m^2 | non-SI area |

1_yd^2 | square yard | 1_yd^2 = 9_ft^2 | US customary area |

1_ft^2 | square foot | 1_ft^2 = 144_in^2 | US customary area |

1_in^2 | square inch | 1_in^2 = 6.4516_cm^2 | US customary area |

1_km^2 | square kilometre | 1_km^2 = 1E6_m^2 | SI area |

1_ha | hectare | 1_ha = 1_hm^2 | SI area (deprecated) |

1_a | are | 1_a = 100_m^2 | non-SI area |

1_mi^2 | square mile | 1_mi^2 = 9_ft^2 | US customary area |

1_miUS^2 | survey square mile | 1_miUS^2 = 9_ftUS^2 | US surveying area |

1_acre | acre | 1_acre = 4840_yd^2 | US customary area |

Volume |
|||

Unit | Unit name | Definition | Description |

1_m^3 | cubic metre | 1_m^3 | Base SI unit of volume |

1_cm^3 | cubic centimetre | 1_cm^3 = 1E-6_m^3 | SI volume |

1_ml | millilitre | 1_ml = 1_cm^3 | SI volume |

1_l | litre | 1_l = 0.001_m^3 | SI volume |

1_st | stere | 1_st = 1_m^3 | non-SI volume |

1_yd^3 | cubic yard | 1_yd^3 = 27_ft^3 | US customary volume |

1_ft^3 | cubic foot | 1_ft^3 = 1728_in^3 | US customary volume |

1_in^3 | cubic inch | 1_in^3 = 16.387064_cm^3 | US customary volume |

1_gal | gallon | 1_gal = 231_in^3 | US customary volume |

1_qt | quart | 1_qt = 1/4_gal | US customary volume |

1_pt | pint | 1_pt = 1/8_gal | US customary volume |

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 | tablespoon | 1_tbsp = 1/2_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 |

1_qtUK | quart (UK) | 1_qtUK = 1/4_galUK | Imperial volume |

1_ptUK | pint (UK) | 1_ptUK = 1/8_galUK | Imperial volume |

1_ozUK | ounce (UK) | 1_ozUK = 1/160_galUK | US customary volume |

1_galC | gallon (Canada) | 1_galC = 1_galUK | Canadian volume |

1_qtC | quart (Canada) | 1_qtC = 1_qtUK | Canadian volume |

1_ptC | pint (Canada) | 1_ptC = 1_ptUK | Canadian volume |

1_ozC | ounce (Canada) | 1_ozC = 1_ozUK | Canadian volume |

1_bbl | oil barrel | 1_bbl = 42_gal | oil industry volume |

1_bu | bushel (US) | 1_bu = 2150.42_in^3 | US customary volume |

1_pk | peck (US) | 1_pk = 1/4_bu | US customary volume |

1_buUK | bushel (UK) | 1_buUK = 8_galUK | Imperial volume |

1_pkUK | peck (UK) | 1_pkUK = 1/4_buUK | Imperial volume |

1_buC | bushel (Canada) | 1_buC = 8_galC | Canadian volume |

1_pkC | peck (Canada) | 1_pkC = 1/4_buC | Canadian volume |

Time |
|||

Unit | Unit name | Definition | Description |

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 |

Speed |
|||

Unit | Unit name | Definition | Description |

1_m/s | metres per second | 1_m/s | Base SI unit of speed |

1_cm/s | centimetres per second | 1_cm/s = 0.01_m/s | SI speed |

1_ft/s | feet per second | 1_ft/s = 12_in/s | US customary speed |

1_kph | kilometres per hour | 1_kph = 1_km/h | SI speed |

1_mph | miles per hour | 1_mph = 1_mi/h | SI speed |

1_knot | nautical knot | 1_knot = 1_nmi/h | non-SI speed |

1_c | speed of light | 1_c = 299792458_m/s | non-SI speed |

1_r/s | radians per second | 1_r/s | Base SI angular speed |

1_rpm | revolutions per minute | 1_rpm = 1_tr/min | non-SI speed |

1_tr/s | turns per second | 1_tr/s = 2π_r/s | non-SI speed |

Acceleration |
|||

Unit | Unit name | Definition | Description |

1_m/s^2 | metre per second squared | 1_m/s^2 | SI acceleration |

1_cm/s^2 | centimetre per second squared | 1_cm/s^2 = 0.01_m/s^2 | SI acceleration |

1_ft/s^2 | foot per second squared | 1_ft/s^2 | US customary acceleration |

1_ga | acceleration of gravity | 1_ga = 9.80665_m/s^2 | non-SI acceleration |

1_r/s^2 | radian per second squared | 1_r/s^2 | SI angular acceleration |

Mass |
|||

Unit | Unit name | Definition | Description |

1_kg | kilogram | 1_kg | Base SI unit of 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 |

1_slug | slug | 1_slug = 1_lb*s^2/ft*ga | US customary mass |

1_lbt | troy pound | 1_lbt = 5760_grain | US customary mass |

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_ozt | troy ounce | 1_ozt = 480_grain | Imperial 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 amount of substance |

Force |
|||

Unit | Unit name | Definition | Description |

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 |

Energy |
|||

Unit | Unit name | Definition | Description |

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 | kilocalorie (thermochemical) | 1_kcal = 1000_cal | 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 | 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_kW | kilowatt | 1_kW = 1000_W | SI power |

1_MW | megawatt | 1_MW = 1000000_W | SI power |

1_hp | mechanical horsepower | 1_hp = 550_ft*lbf/s | Imperial power |

1_lbf*ft/s | pound-foot per second | 1_ft*lbf/s | Imperial power |

Pressure |
|||

Unit | Unit name | Definition | Description |

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 | 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 |

1_inH2O | inches of water (at 4°C) | 1_inH2O = 999.972_kg/m^3*in*ga | non-SI pressure |

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_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_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_ΔK | Change of temperature |

Electricity and Magnetism |
|||

Unit | Unit name | Definition | Description |

1_V | volt | 1_V = 1_J/C | SI potential difference |

1_A | ampere | 1_A | Base SI electric current |

1_C | coulomb | 1_C = 1_A*s | SI electrical charge |

1_Ω | ohm | 1_Ω = 1_V/A | SI electrical resistance |

1_F | farad | 1_F = 1_C/V | SI electrical capacity |

1_A*h | ampere-hour | 1_A*h = 3600_C | non-SI electrical charge |

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 | 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 |

1_Wb | weber | 1_Wb = 1_kg*m^2/A/s^2 | SI magnetic flux |

Angles |
|||

Unit | Unit name | Definition | Description |

1_° | degree | 1_° = 2π/360_r | non-SI plane angle |

1_r | radian | 1_r | Base SI plane angle |

1_grad | grad (gon) | 1_grad = 2π/400_r | non-SI plane angle |

1_tr | full turn (one revolution) | 1_tr = 2π_r | non-SI plane angle |

1_arcmin | arc minute | 1_arcmin = 1/60_° | non-SI plane angle |

1_arcs | arc second | 1_arcmin = 1/3600_° | non-SI plane angle |

1_sr | steradian | 1_sr | Base SI solid angle |

Light |
|||

Unit | Unit name | Definition | Description |

1_fc | foot-candle | 1_fc = 1_lm/ft^2 | non-SI illuminance |

1_flam | footlambert | 1_flam = 1/π_cd/ft^2 | non-SI illuminance |

1_lx | lux | 1_lx = 1_lm/m^2 | SI illuminance |

1_ph | phot | 1_ph = 1_lm/cm^2 | non-SI illuminance |

1_sb | stilb | 1_sb = 1_cd/cm^2 | non-SI illuminance |

1_lm | lumen | 1_lm = 1_cd*sr | SI luminous flux |

1_cd | candela | 1_cd | Base SI luminous intensity |

1_lam | lambert | 1_lam = 10000/π_cd/m^2 | non-SI illuminance |

Radiation |
|||

Unit | Unit name | Definition | Description |

1_Gy | gray | 1_Gy = 1_J/kg | SI radiation |

1_rad | rad | 1_rad = 0.01_Gy | non-SI radiation |

1_rem | roentgen equivalent man | 1_rem = 1_erg/g | SI radiation |

1_Bq | becquerel | 1_Bq = 1_1/s | SI radiation |

1_Ci | curie | 1_Ci = 3.7E10_Bq | non-SI radiation |

1_R | roentgen | 1_R = 2.58E-4_C/kg | non-SI radiation |

Viscosity |
|||

Unit | Unit name | Definition | Description |

1_P | poise | 1_P = 0.01_Pa*s | non-SI dynamic viscosity |

1_St | stokes | 1_St = 0.0001_m^2/s | non-SI kinematic viscosity |

1_m^2/s | square metre per second | 1_m^2/s | SI kinematic viscosity |

1_kg/(m*s) | kilogram per metre per second | 1_kg/(m*s) = 1_Pa*s | SI dynamic viscosity |