Re: [q-lang-users] Query about Type Tests (Complex and Rational).

[John C. <co...@cc...>]

Rob Hubbard scripsit:

> I think I need clarification on
> (1) When one type should be a subtype of another.
> (2) What the semantics of type test function (such as iscomplex) is
> supposed to be.

Well, FWIW I'll tell you how Scheme does it.  (For Schemers, I'm
oversimplifying some and blurring some distinctions between R5RS and
R6RS-draft.)

In Scheme, the predicates "number?", "complex?", "real?", "rational?", and
"integer?" refer to the mathematical concepts, not the representations.
Thus on "normal" implementations, "number?"  and "complex?" are
the same thing, and "integer?" returns true for any integral value,
even if the underlying representation is floating-point.  "Real?" and
"rational?" are not quite the same, because the former conventionally
includes IEEE infinities whereas the latter does not.

Cutting across this hierarchy or "tower", there are also the predicates
"exact?" and "inexact?".  On "normal" implementations, standard machine
floating-point representations are inexact; machine integers ("fixnums"),
arbitrary-precision integers ("bignums"), and pairs whose numerator and
denominator are integers ("ratnums") are exact.  Note that a complex
number is only real if its imaginary part, expressed or implied, is
an *exact* 0, because 0.0 actually represents every number between the
smallest representable positive float and its negation.

So 1 is exact and integer (and therefore also rational, real, and
complex); 1.0 is inexact and integer (and rational and real and complex);
1.1 is inexact and rational (and real and complex); 11/10 is exact and
rational (and real and complex); 1+2i is exact and complex; 1.0+0i is
inexact and real (and complex); 1.0+0.0i is inexact and complex.

Though not part of the standard, one can write some predicates that
enquire about representations, assuming that the shortest representation
is always used.  It's impossible to write "fixnum?" in portable Scheme,
because it depends on the size of machine integers.  The others can
be written for "normal" Scheme implementations as follows (not Scheme
syntax):

bignum? = integer? and exact? and not fixnum?
ratnum? = rational? and exact? and not integer?
compnum? = complex? and not real?

Some Schemes use different representations for exact and inexact
compnums, where the former is a pair of pointers and the latter is a
pair of unboxed floats.

I hope this is helpful in clarifying everyone's thinking.  Feel free to
ask questions about unclarities.

-- 
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tenebrous ultimate gods -- the blind, voiceless, mindless gargoyles whose soul
is Nyarlathotep. (Lovecraft) John Cowan|co...@cc...|ccil.org/~cowan