# Mathematical Expressions

In terms of word-problem solving, requests for calculation and simple questions translate into mathematical expressions.  A mathematical expression is some arithmetic combination of numbers, operators (excluding "="), and variables.

As AutoMathic interprets its input, it attempts to translate it into valid expressions or equations.  In fact, if the result of any translation is not a valid expression or equation, AutoMathic will conclude that it did not understand the input!  When this happens, it will simply say, "I didn't understand" followed by your input.  You should read this as an opportunity to paraphrase the confusing input and let AutoMathic try again.

In AutoMathic, mathematical expressions consist of:

As seen in the previous sections, AutoMathic allows many ways of expressing each of these components, resulting in a wide variety of ways to request calculations and phrase simple questions.

Regardless of their complexity, expressions are the building blocks of equations.

Request for Calculation

A Request for Calculation could be as simple as "1+1" or as complicated as "Convert Mach 5 into units meters per second".  The common feature of requests is that they get translated into mathematical "expressions".  A mathematical expression NEVER contains an equals (=) sign.  In general, requests for calculation can be:

• algebraic expressions ("1+1", "Pi over two", etc.)
• noun-centric sentence fragments ("the speed of light", "half the mass", etc.)
• verb-centric commands ("solve for time", "find 2x the diameter", etc.)

AutoMathic keeps track of requests and responds to them as soon as it has enough information to do so.  If no additional information is needed, it will answer immediately.  If the request has any outstanding unknowns, it will be stored and (hopefully) answered later.

Trivial Question

A trivial question is really just a request for calculation that is written to sound like a normal, English question, usually using a form of the verb "to be" such as "is", "are", "was", "would usually have been", etc.  Trivial questions get translated into expressions, and are handled just like a Request for Calculation:

 Question Example What is ___ ? What is the ratio of pi to e? How much is ___ ? How much is pi over 2? How many is ___ ? How many is 12 dozen? There is how much ___ ? There is how much mass per sample? There are how many ___ ? There are how many feet per mile?

Trivial questions can be phrased the opposite way also:

 Question Example ___ is what? The ratio of pi to e is what? ___ is how much? Pi over 2 is how much? ___ is how many? 12 dozen is how many? How much ___ is there? How much mass per sample is there? How many ___ are there? How many feet per mile are there?

Question fragments that end with a verb are also accepted:

 Question Example ___ is? 3 added to 12 is?

However, questions that begin with a verb are not accepted, since they ask a "yes or no" question.  AutoMathic does NOT handle "yes or no" questions, so it cannot answer about the truth or falsity of a logical statement!  (e.g.  "Is the mass 5 unit kilograms?", "Is the width less than half of the length?")

Trivial questions can also produce complicated expressions that include mathematical constants, units, variables, etc.:

`> What's the speed of light in terms of miles per hour? Working from "THE SPEED OF LIGHT IN TERMS OF MILES PER HOUR"...   Let 'K' stand for "KILOMETERS"   Let 'S' stand for "SECOND"   Let 'M' stand for "MILES"   Let 'H' stand for "HOUR" * Simplifying...            299792.5              -------- M               K                     -                     S                  ----------                H      * Converting to standard form...            299792.5              -------- M               K                     -                     S                  ----------                H      * Simplifying division by a fraction...           299792.5 S             ---------- M               K                  ------------                H       * Collecting factors over a denominator...           299792.5 S M           ------------                K                 ------------                H       * Converting to standard form...           299792.5 M S           ------------                K                 ------------                H       * Simplifying division of a fraction...           299792.5 M S           ------------               K H      * Converting to standard form...           299792.5 M S           ------------               H K        (Find 299792.5MS/(HK))`