Some transformations for simplifying mathematical expressions are valid for both real and complex expressions.  For example,

(√z)²  →  z

is valid whether z is a real or complex expression.  Some transformations are valid only for real expressions.  For example,

√(x²)  →  |x|

is valid only if x is real.  Some transformations are valid only for a restricted interval of real values.  For example,

√(x·y)  →  √(x)·√(y)

is valid for real x and y unless both are negative.  Some transformations are valid only for integer-valued expressions.  For example,

sin(n·ð)  →  0

is valid only if n has an integer value.  Even as basic a transformation as commuting the operands of a product is valid only if the operands are scalars.


When simplifying an expression, Derive exploits a transformation only if it can determine that the transformation is valid.  This determination depends heavily on the declared domain of the variables in the expression.  Use the Author > Variable Domain command8MNT4Y to examine or change the domain declaration of a variable.


Since real-valued variables are the most common, the default domain of undeclared variables is real.  However, if an expression contains variables that only assume integer values or a subset of real values, you can restrict the domain of these variables.  On the other hand, if an expression contains variables that can be complex or even a vector, set, or logical truth-value, you can expand the domain of these variables.


The transformation

x − x  →  0 · x  →  0

is essential for simplifying expressions.  Since this and other transformations are questionable if x is infinity, Derive does not permit variable domains to include an infinite magnitude.  Thus infinite bounds are always strict.


Although variables cannot be declared infinite, you can substitute an infinite value for a variable in an expression (see the Simplify > Variable Substitution commandGP2.L1).  However, you risk an invalid result if the expression was derived using transformations, such as the above, which are questionable for infinite values.


As might be expected, Derive simplifies both x/x and x^0 to 1, even if the domain of the variable x includes 0.  For consistency, Derive also simplifies 0^0 to 1, since this is just a special case of the above x^0 simplification.


The domains of variables are used during simplification of fractional powers, absolute values, logarithms, and some other irrational functions.  They do not restrict the values that can be substituted for variables (see the Simplify > Variable Substitution commandGP2.L1).  Thus, you can substitute a complex value for a real variable or a negative value for a positive variable.  Such substitutions are made at your own risk.

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