Introduction to MathObjects

MathObjects are programing objects which behave much as you would expect their true mathematical counterparts to behave. For example a+b means one thing if a and b are vectors or matrices, another if they are real numbers and a third if a and b are complex numbers. Likewise multiplication: a b or a*b mean different things depending on the mathematical object (although of course all of the versions of multiplication have certain similarities.)

How to create a MathObject

$a = Real(3.5);
$b = Complex(3, 4);
$b = Complex("3 +4i");

$a represents a real number 3.5 and $b (defined by either method) represents a complex number.

Which MathObject types (classes) can be created?

These classes are listed and made available for writing problems in pg/macros/Value.pl. It is loaded automatically when you load MathObjects.pl.

Standard types

• Real: Behave like real numbers
• Infinity: The positive infinity of the extended reals. Can be negated, but can't be added to real numbers.
• Complex: Behave like complex numbers. The interpretations of + and * are those standardly used for mathematical complex numbers.

List types

List objects are math objects whose description involves delimiters (parentheses) of some type. For example points (4, 5) or vectors <2,5> . Here are examples of the construction of the List Objects.

• Point: $a = Point("(4,5)");
• Vector: $b = Vector("<3,5,6!>");
• Matrix: $c = Matrix("[[1,0],[0,1]]");
• List: $d = List("3, 7, 3+2i");

Types that represent some subset of the real numbers

• Interval: $I = Interval("[0,1)");
• Set (a finite collections of points): $S = Set("{3,5,6,8}");
• Union (of intervals and sets): $U = Union(""I U J"); (I union J)

The String type

String is a special purpose type which allows comparison to an arbitrary string.

String("DNE")

The Formula type

A Formula object represents a functions whose output is one of the MathObject types defined above. Every Formula contains a parse tree which allows you to calculate output values from given input values.

$f = Formula('2x^2+3x-5');

How to invoke a method of a MathObject

Use the standard Perl method call syntax:

$obj->method;
$obj->method($arg1,$arg2);

For example:

ANS($a->cmp);

This compares the student's answer with $a. If $a is Real then this comparison will be "fuzzy" which means that equality is checked to a tolerance defined by the current Context.

Methods shared by all MathObjects

• cmp: Returns an answer checker for the Value. All of the answer checkers are defined in the file lib/Value/AnswerChecker.pm.
• perl -- Returns a string which represents the object as Perl source code.
• perlFunction -- Returns a Perl subroutine which represents the object. (Only available for Formula objects.)
• value -- Returns the value of the object.
• TeX -- Returns a string which represents the object as a TeX math expression.
• string -- Returns a string similar to that used to create the object. May include extra parentheses.
• stringify -- Produces the output of the object when inside quotes. Depending on context this is either a TeX string or a regular string. (This is called automatically by Perl when when an object is used in string context, and should not need to be called explicitly by the problem author.)
• getFlag("flag name") -- Returns the value of one of the object's internal flags. For example: $a->getFlag("tolerance");

The MathObjects Parser

The parser works "behind the scenes" to create formula. It's purpose is to parse a string representing a formula and turn it into a parse tree. Objects containing a parse tree are of the Formula class and have these additional methods.

Parser methods include:

• eval
• reduce
• perl
• TeX

The parser is defined in the file pg/lib/Parser.pm and the files in the pg/lib/Parser directory. Even though the subdirectory names under pg/lib/Parser are similar to those under pg/lib/Value they refer to different although related concepts. Under pg/lib/Parser the files refer to tokens in a string that is to be parsed, while the files under pg/lib/Value refer to MathObjects.

The Context

This is essentially a table of values that provides default values for the MathObjects and for the Parser. As a quick example: in Numeric context the answer (4,5) is interpreted as a point in the two dimensional plane. in Interval context it is interpreted as the real values x satisfying 4 < x < 5.

• Define context using: Context("Numeric");
• To obtain the current context: $context = Context();
• Context names: defined in pg/lib/Parser/Context/Default.pm
  • Numeric: no Matrix, Complex or Vectors (or intervals) are allowed.
  • Complex: no Matrix or Vector, can't use "less than".
  • Point: really the same as the Vector context below
  • Vector: i, j, and k are defined as unit Vectors, no Complex numbers are allowed.
  • Vector2D: i and j are defined as unit Vectors, no Complex numbers are allowed.
  • Matrix: square brackets define Matrix instead of Point or Interval
  • Interval: similar to Numeric context, but (,) and [,] create Real Intervals rather than Lists. {,} creates finite sets of Reals.
  • Full: For internal use. This context is used to seed the others.
    • pi is defined
    • i is square root of minus one, but j and k are unit Vectors
    • Matrix, Vector and Complex are all defined.
    • x is a variable