Calc has a set of functions for accessing the current declarations in a convenient manner. These functions return 1 if the argument can be shown to have the specified property, or 0 if the argument can be shown not to have that property; otherwise they are left unevaluated. These functions are suitable for use with rewrite rules (see section Conditional Rewrite Rules) or programming constructs (see section Conditionals in Keyboard Macros). They can be entered only using algebraic notation. See section Logical Operations, for functions that perform other tests not related to declarations.
For example, `dint(17)' returns 1 because 17 is an integer, as
do `dint(n)' and `dint(2 n - 3)' if n has been declared
int, but `dint(2.5)' and `dint(n + 0.5)' return 0.
Calc consults knowledge of its own built-in functions as well as your
own declarations: `dint(floor(x))' returns 1.
The dint function checks if its argument is an integer.
The dnatnum function checks if its argument is a natural
number, i.e., a nonnegative integer. The dnumint function
checks if its argument is numerically an integer, i.e., either an
integer or an integer-valued float. Note that these and the other
data type functions also accept vectors or matrices composed of
suitable elements, and that real infinities `inf' and `-inf'
are considered to be integers for the purposes of these functions.
The drat function checks if its argument is rational, i.e.,
an integer or fraction. Infinities count as rational, but intervals
and error forms do not.
The dreal function checks if its argument is real. This
includes integers, fractions, floats, real error forms, and intervals.
The dimag function checks if its argument is imaginary,
i.e., is mathematically equal to a real number times i.
The dpos function checks for positive (but nonzero) reals.
The dneg function checks for negative reals. The dnonneg
function checks for nonnegative reals, i.e., reals greater than or
equal to zero. Note that the a s command can simplify an
expression like x > 0 to 1 or 0 using dpos, and that
a s is effectively applied to all conditions in rewrite rules,
so the actual functions dpos, dneg, and dnonneg
are rarely necessary.
The dnonzero function checks that its argument is nonzero.
This includes all nonzero real or complex numbers, all intervals that
do not include zero, all nonzero modulo forms, vectors all of whose
elements are nonzero, and variables or formulas whose values can be
deduced to be nonzero. It does not include error forms, since they
represent values which could be anything including zero. (This is
also the set of objects considered "true" in conditional contexts.)
The deven function returns 1 if its argument is known to be
an even integer (or integer-valued float); it returns 0 if its argument
is known not to be even (because it is known to be odd or a non-integer).
The a s command uses this to simplify a test of the form
`x % 2 = 0'. There is also an analogous dodd function.
The drange function returns a set (an interval or a vector
of intervals and/or numbers; see section Set Operations using Vectors) that describes
the set of possible values of its argument. If the argument is
a variable or a function with a declaration, the range is copied
from the declaration. Otherwise, the possible signs of the
expression are determined using a method similar to dpos,
etc., and a suitable set like `[0 .. inf]' is returned. If
the expression is not provably real, the drange function
remains unevaluated.
The dscalar function returns 1 if its argument is provably
scalar, or 0 if its argument is provably non-scalar. It is left
unevaluated if this cannot be determined. (If matrix mode or scalar
mode are in effect, this function returns 1 or 0, respectively,
if it has no other information.) When Calc interprets a condition
(say, in a rewrite rule) it considers an unevaluated formula to be
"false." Thus, `dscalar(a)' is "true" only if a is
provably scalar, and `!dscalar(a)' is "true" only if a
is provably non-scalar; both are "false" if there is insufficient
information to tell.