Difference between revisions of "Introduction to MathObjects"
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Revision as of 20:27, 15 February 2008
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.)
Contents
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+3x5');
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.
 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
, andk
are defined as unit Vectors, no Complex numbers are allowed.  Vector2D:
i
andj
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 definedi
is square root of minus one, butj
andk
are unit Vectors Matrix, Vector and Complex are all defined.
x
is a variable