Knowledgebase Summary

AutoMathic's Converse mode allows you to specify problems and ask questions at a high level of abstraction.  Using an extensive knowledgebase, it automatically pulls in whatever background knowledge is needed at the time.

With AutoMathic's built-in knowledge, some problems require less specification than others, so the user can definitely benefit by being familiar with what AutoMathic already knows.  When posing questions, it is best to treat AutoMathic as a smart (but not very intelligent) assistant that has a lot of mathematical knowledge, but doesn't have much common-sense about the "real world".  For example, it understands "average speed", "distance", and "time", but does not know about planes, trains, and automobiles!

This knowledgebase (as well as the Lexicon and Sample Input) cover the extent of AutoMathic's mathematical knowledge.




#---------------------------------------------------
#
# Source for AutoMathic's external equation library
#
#                 version 20111201
#
#---------------------------------------------------
#       Copyright 1988, 2011  Kevin B. Belton
#---------------------------------------------------


################
#              #
# Length units #
#              #
################

There are a hundred centimeters in a meter
There are 2.54 centimeters per inch
There are twelve inches in a foot
Foot means feet
There are three feet in a yard
There are 220 yards in a furlong
There are 5280 feet in a mile
There are three miles in a league
There are 6076.1155 feet in a nautical mile
There are six feet in a fathom
There are a thousand meters in a kilometer
There are ten millimeters in a centimeter
There are a thousand microns in a millimeter
Micrometer means micron
There are a million nanometers in a millimeter
There are ten angstroms in a nanometer
There are 149578700 kilometers in an au
There are 63239.7 au in a light year
There are 3.26164 light years in a parsec


##############
#            #
# Time units #
#            #
##############

There are a billion nanoseconds per second
There are a million microseconds per second
There are a thousand milliseconds per second

There are sixty seconds in a minute
There are sixty minutes in an hour
There are 24 hours a day
There are seven days per week
There are two weeks in a fortnight
There are twelve months a year
There are 365.25 days a year

There are ten years in a decade
There are a hundred years in a century
There are a thousand years in a millennium


#############
#           #
# Frequency #
#           #
#############

#---------------------------------------------------
# Frequency
#
# Units: 1 / time (e.g. 1/s, Hz, GHz)
#---------------------------------------------------
# The frequencies of the musical notes on a piano's
# middle-C octave are built into the vocabulary:
#    C natural
#    C sharp, C#, D flat, Db
#    D natural
#    D sharp, D#, E flat, Eb
#    E natural
#    F natural
#    F sharp, F#, G flat
#    G natural
#    G sharp, G#, A flat, Ab
#    A natural
#    A sharp, A#, B flat, Bb
#    B natural
#---------------------------------------------------

Hertz means 1 / second

There are a thousand hertz in a kilohertz
There are a million hertz in a megahertz
There are a billion hertz in a gigahertz

   #-----------------------------------------------------------------
   # Custom time units supporting standard calendar frequencies. The
   # custom time units (dualweek, semimonth, etc.) are not meant to
   # be used directly by end-users.
   #
   # ("hourly", "daily", "weekly", etc. are also in the vocabulary)
   #-----------------------------------------------------------------

      #---------------------------------
      # Biweekly, bimonthly, biannually
      #---------------------------------

There are two weeks in a dualweek
There are two months in a dualmonth
There are two years in a dualyear

      #---------------------------------------
      # Semiweekly, semimonthly, semiannually
      #---------------------------------------

There are two semiweeks in a week
There are two semimonths in a month
There are two semiyears in a year

      #-----------
      # Quarterly
      #-----------

There are four quarteryears in a year


#####################
#                   #
# Temperature units #
#                   #
#####################

   #--------------------------------------------------------------
   # Temperatures must be referred to as "temperature" or "temp",
   # and specified with their units:
   #
   #    e.g.  Find temperature in unit Kelvins when temp measured
   #          in degrees F is 95.
   #
   # (See the "Units" documentation on "Temperature Conversion")
   #--------------------------------------------------------------

Temperature measured in Fahrenheit is 32 plus nine fifths temperature measured in Celsius
Temperature measured in Kelvin is temperature measured in Celsius plus 273.16


#######################
#                     #
# Weight & Mass units #
#                     #
#######################

   #--------------------------------------------------------------------------
   # USAGE NOTE:
   #
   # For convenience, AutoMathic treats English weight units (pounds, ounces,
   # etc.) as masses, not forces!  This makes calculations and conversions
   # involving English weights simpler by assuming an acceleration of 1 G.
   #
   # This is also reflected in higher-level concepts such as Force, Work,
   # Energy, etc. where mass is a component.
   #--------------------------------------------------------------------------

There are sixteen ounces in a pound
There are two thousand pounds in a ton
There are 2.2 pounds in a kilogram
There are a thousand kilograms in a metric ton
There are a thousand metric tons in a kiloton
There are a thousand grams in a kilogram
There are a thousand milligrams in a gram


#########################
#                       #
# Liquid / Volume units #
#                       #
#########################

There are three teaspoons in a tablespoon
There are two tablespoons in a fluid ounce
There are eight fluid ounces in a cup
There are two cups in a pint
There are two pints in a quart
There are four quarts per gallon
There are 3.785 liters in a gallon
There are a thousand milliliters in a liter

   #----------------------------------------------------------------------
   # This definition of a "drop" is based on water, but is "close-enough"
   # for calculations of other liquids in real-world conditions.
   #----------------------------------------------------------------------

There are twenty drops in a milliliter


#########################
#                       #
# Chemistry measurement #
#                       #
#########################

   #----------------------------------------------------------
   # "Particles" is used as the general term for items being
   # counted.  In practice it means whole atoms or molecules.
   #----------------------------------------------------------

Avogadro's number is the number of particles per mole

   #--------------------------------------------------
   # Molarity
   #
   # Units: number / volume (e.g. moles/L, moles/gal)
   #--------------------------------------------------

Molarity is moles per liter

   #--------------------------------------------------------------------------
   # Formula/molecular/atomic Weight/Mass
   #
   # Units: (these weights/masses are dimensionless)
   #--------------------------------------------------------------------------
   # "Formula/molecular/atomic weight/mass" is really measured in AMU (Atomic
   # Mass Units) but is used here and in practice as a dimensionless number.
   #--------------------------------------------------------------------------

Moles is grams / formula weight

Avogadro's number is the number of AMU per gram


#######################
#                     #
# Angular Measurement #
#                     #
#######################

   #---------------
   # Angular units
   #---------------

There are two Pi radians in 360 degrees
There are nine degrees in ten grads

      #---------------------------------------------------------
      # "Arcminutes" and "arcseconds" are used to differentiate
      # angular measure from "minutes" and "seconds" of time!
      #---------------------------------------------------------

There are sixty arcminutes in a degree
There are sixty arcseconds in an arcminute

   #-----------------------
   # Common rotation terms
   #-----------------------

There are ninety degrees in a right angle
There are 360 degrees per revolution
Rotation means revolution
A circle is a revolution


###################
#                 #
# Circle formulas #
#                 #
###################

A diameter is twice the radius
Circumference is two Pi times the radius

   #-------------------------------------------------------------------------
   # Distance, circumference, and revolutions/rotations/circles are related.
   #-------------------------------------------------------------------------

Circumference times revolutions is distance


###########################
#                         #
# Uniform Motion formulas #
#                         #
###########################

   #----------------------------------------------------------------------
   # Average speed
   #
   # Units: length / time (e.g. mph, kph, feet per sec)
   #----------------------------------------------------------------------
   # "Average speed" refers to non-accelerated (possibly constant) speed.
   #----------------------------------------------------------------------

Distance is average speed times time

   #----------------------------------------------------------
   # Acceleration
   #
   # Units: speed / time (e.g. m/s/s, ft/s/s, G's)
   #----------------------------------------------------------
   # "Speed" refers to variable or instantaneous speed due to
   # acceleration/deceleration.
   #----------------------------------------------------------

Acceleration is speed over time

   #-------------------------------------------------
   # This distance formula uses variable speed, and
   # assumes acceleration/deceleration from/to rest.
   #-------------------------------------------------

Distance is one half speed times time

   #---------------------------------------------------------
   # Force
   #
   # Units: mass x acceleration (e.g. kg x m/s/s, N, lb x G)
   #---------------------------------------------------------

Force is mass times acceleration

   #----------------------------------------------------
   # Centripetal acceleration
   #
   # Units: speed / time (e.g. m/s/s, ft/s/s, G's)
   #----------------------------------------------------

Centripetal acceleration is average speed times itself over radius


###################
#                 #
# Work and Energy #
#                 #
###################

   #--------------------------------------------------------------------
   # Work
   #
   # Units: force x length (e.g. kg x m x m/s/s, J, N x m, lb x ft x G)
   #--------------------------------------------------------------------

Work is force times distance

   #--------------------------------------------------------------------
   # Energy (Kinetic and Potential)
   #
   # Units: force x length (e.g. kg x m x m/s/s, J, N x m, lb x ft x G)
   #--------------------------------------------------------------------
   # Note that "potential energy" assumes an acceleration of 1 G
   #--------------------------------------------------------------------

Kinetic energy is half mass times average speed times itself

Potential energy is mass times 1 g times height

   #---------------------------------------------------------------------------
   # Power
   #
   # Units: energy / time (e.g. J/s, W, HP)
   #---------------------------------------------------------------------------
   # This formula works with generic energy...  If a particular KIND of energy
   # is being used or transformed, make sure you also say something like:
   #
   #    e.g.  The energy is potential energy.
   #    e.g.  Half the energy is kinetic energy.
   #    e.g.  Energy is 71% of the electrical energy.
   #---------------------------------------------------------------------------

Power is energy over time

   #-----------------------------------------------
   # Momentum
   #
   # Units: mass x speed (e.g. kg x m/s, lb x mph)
   #-----------------------------------------------

Momentum is mass times speed

   #--------------------------------------------------------------------
   # Torque
   #
   # Units: force x length (e.g. kg x m x m/s/s, J, N x m, lb x ft x G)
   #--------------------------------------------------------------------

Torque is force times radius

   #------------------------------------------------------------------
   # Springs
   #
   # Units: force / length (e.g. kg/s/s, N/m, lb x G / ft)
   #------------------------------------------------------------------
   # The unit definition is for the spring constant.
   #
   # "Displacement" is the absolute length change in the given units.
   #------------------------------------------------------------------

Force = spring constant x displacement


########################
#                      #
# Properties of Matter #
#                      #
########################

   #-------------------------------------------------------------------------
   # Density
   #
   # Units: mass / volume (e.g. gram/ml, lb/gal)
   #-------------------------------------------------------------------------
   # The density of several common substances are built into the vocabulary:
   #    water (H2O), mercury (Hg), alcohol, gasoline, air, helium (He),
   #    carbon dioxide (CO2), steam, aluminum (Al), iron (Fe), steel,
   #    copper (Cu), lead, gold (Au)
   #-------------------------------------------------------------------------

Density is the ratio of mass to volume

   #-----------------------------------------------------
   # Specific Gravity
   #
   # Units: (specific gravity is a dimensionless number)
   #-----------------------------------------------------

Specific Gravity = density / (1 [gram/ml])


########################################
#                                      #
# Atomic/Molecular/Chemical properties #
#                                      #
########################################

Atomic number is the number of protons in an atom
Mass number is the sum of the atomic number and neutrons in the atom
Mass number is the number of nucleons

   #--------------------------------------------------------
   # Ionic Charge/Number
   #
   # Units: (ionic charge/number is a dimensionless number)
   #--------------------------------------------------------
   # Ionic charge/number can be specified as "negative",
   # "neutral", "positive", -3, +2, etc.
   #--------------------------------------------------------

Ionic number is the difference between the number of protons per atom and electrons per atom

   #----------------------------------------------------------------------
   # Atomic/Molecular/Formula weight
   #
   # Units: "Atomic/molecular/formula weight" is strictly measured in AMU
   #        (Atomic Mass Units) but is used here and in practice as a
   #        dimensionless number.
   #----------------------------------------------------------------------
   # 1 proton   = 1.007276  AMU
   # 1 neutron  = 1.008665  AMU
   # 1 electron = 0.0005486 AMU
   #----------------------------------------------------------------------


Atomic weight x atoms = 1.007276 x protons + 1.008665 x neutrons + 0.0005486 x electrons


####################
#                  #
# Pricing formulas #
#                  #
####################

Cost is price plus tax

   #-------------------------------
   # Tax rate is user-definable...
   #-------------------------------

Taxes are _TAX_RATE_% of the price

   #-------------------------------
   # Tip rate is user-definable...
   #-------------------------------

Tips are _TIP_RATE_% of the cost


#############################
#                           #
# Information Storage units #
#                           #
#############################

There are four bits in a nibble
There are eight bits in a byte
There are two bytes in a word
There are four bytes in a longword

There are 1024 bytes in a kilobyte
There are 1024 kilobytes in a megabyte
There are 1024 megabytes in a gigabyte
There are 1024 gigabytes in a terabyte
There are 1024 terabytes in a petabyte

There are a thousand bits in a kilobit
There are a million bits in a megabit
There are a billion bits in a gigabit


###############
#             #
# Wave Motion #
#             #
###############

   #-------------------------------------
   # Period
   #
   # Units: time (e.g. s, minutes, days)
   #-------------------------------------

Frequency is the inverse of period

   #------------------------------------------------------------------------
   # Wave speed
   #
   # Units: length / time (e.g. mph, kph, feet per sec)
   #------------------------------------------------------------------------
   # For sound waves, "wave speed" would be the speed of sound (or Mach 1).
   #
   # For light waves (or other electromagnetic radiation), "wave speed"
   #    would be the speed of light (or C).
   #------------------------------------------------------------------------

Wave speed is wavelength times frequency

   #--------------------------------------------------------------
   # Standing waves
   #
   # Units: ("harmonic" and "overtone" are dimensionless numbers)
   #--------------------------------------------------------------

Length is the harmonic times wavelength over two
The overtone is one less than the harmonic

   #--------------------------------------------------------------------------
   # Doppler effect
   #
   # "wave speed" - the speed of the wave through the medium (e.g. Mach 1, C)
   # "frequency" - the wave's frequency (e.g. 500[MHz], Eb, F#)
   # "doppler freq" - the wave's altered frequency due to the relative motion
   # "relative speed" - the difference in speed between source and receiver
   #--------------------------------------------------------------------------
   # In this equation, "relative speed" is:
   #    positive when the source and receiver's positions are diverging
   #    negative when the source and receiver's positions are converging
   #--------------------------------------------------------------------------

Doppler frequency = frequency x wave speed / ( wave speed + relative speed )

      #-------------------------------------------------------------
      # In this equation, "doppler shift" is:
      #    positive when the doppler effect increases the frequency
      #    negative when the doppler effect decreases the frequency
      #-------------------------------------------------------------

Doppler shift is the difference between doppler frequency and frequency


###############
#             #
# Electricity #
#             #
###############

   #------------------
   # Electrical units
   #------------------

Amperes = Coulombs per second
There are a thousand milliamperes in an ampere

Volts = Joules / coulomb

Ohms = volts / amperes

Farads = Coulombs per volt
There are a million microfarads in a farad
There are a trillion picofarads in a farad

   #---------------------
   # Electrical concepts
   #---------------------

Current is charge over time

Voltage is current times resistance

Capacitance is charge over voltage

Electrical energy is charge times voltage

      #---------------------------------------------------------------------
      # The assumptions for kilgrams and meters allows reduction down to SI
      # units without causing too much complexity for the general strategy.
      #---------------------------------------------------------------------

Power is current times voltage. Assume kilograms=1; meters=1


##################
#                #
# Thermodynamics #
#                #
##################

   #----------------
   # Pressure units
   #----------------

There are 14.7 PSI in an atmosphere
There are 760 Torr in an atmosphere
There are 2.036254 inHg per PSI
There are 101325 Pascals in an atmosphere
There are a hundred thousand Pascals per bar
There are a thousand millibars in a bar
There are a thousand Pascals in a kilopascal

   #---------------
   # Ideal Gas Law
   #---------------

Pressure in [atm] x volume in [L] = moles x 0.0821 x temperature in [Kelvin]

   #---------------------------------------------------------------------
   # Heat/Thermal Energy of an Ideal Gas
   #
   # Units: energy (e.g. J, calories, kilocalories, BTU's)
   #---------------------------------------------------------------------
   # The assumptions for seconds and meters allows reduction down to SI
   # units without causing too much complexity for the general strategy.
   #---------------------------------------------------------------------

Heat energy in [Joules] is 3/2 x moles x 8.314 x temperature in [Kelvin]. Assume seconds=1; meters=1

   #---------------------------------------------------------------------
   # Heat/Thermal Energy and Specific Heat (any substance)
   #
   # Units:
   #    Heat Energy: energy (e.g. J, calories, kilocalories, BTU's)
   #    Specific Heat: energy / mass (e.g. J/gm, BTU's/lb))
   #---------------------------------------------------------------------
   # "Specific Heat" is calculated for 1 degree C or K.
   #
   # "deltaT" is the (absolute) temperature change in degrees C or K. In
   # this equation, "deltaT" is used as a dimensionless number.
   #
   # The assumptions for seconds and meters allows reduction down to SI
   # units without causing too much complexity for the general strategy.
   #
   # The specific heat of several common substances are built into the
   # vocabulary:
   #    water (H2O), ice, steam, mercury (Hg), alcohol, aluminum (Al),
   #    copper (Cu), glass, iron (Fe), steel, lead, wood, flesh
   #---------------------------------------------------------------------

Heat energy is specific heat times mass times deltaT. Assume seconds=1; meters=1

   #--------------------------------------------------------
   # Heat Capacity
   #
   # Units: energy (e.g. J, calories, kilocalories, BTU's))
   #--------------------------------------------------------
   # "Specific Heat" is calculated for 1 degree C or K.
   #--------------------------------------------------------

Heat capacity is mass times specific heat