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(guile.info)Reals and Rationals

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Real and Rational Numbers
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Mathematically, the real numbers are the set of numbers that describe
all possible points along a continuous, infinite, one-dimensional line.
The rational numbers are the set of all numbers that can be written as
fractions P/Q, where P and Q are integers.  All rational numbers are
also real, but there are real numbers that are not rational, for example
the square root of 2, and pi.

Guile represents both real and rational numbers approximately using a
floating point encoding with limited precision.  Even though the actual
encoding is in binary, it may be helpful to think of it as a decimal
number with a limited number of significant figures and a decimal point
somewhere, since this corresponds to the standard notation for non-whole
numbers.  For example:

     0.34
     -0.00000142857931198
     -5648394822220000000000.0
     4.0

The limited precision of Guile's encoding means that any "real" number
in Guile can be written in a rational form, by multiplying and then
dividing by sufficient powers of 10 (or in fact, 2).  For example,
`-0.00000142857931198' is the same as `142857931198' divided by
`100000000000000000'.  In Guile's current incarnation, therefore, the
`rational?' and `real?' predicates are equivalent.

Another aspect of this equivalence is that Guile currently does not
preserve the exactness that is possible with rational arithmetic.  If
such exactness is needed, it is of course possible to implement exact
rational arithmetic at the Scheme level using Guile's arbitrary size
integers.

A planned future revision of Guile's numerical tower will make it
possible to implement exact representations and arithmetic for both
rational numbers and real irrational numbers such as square roots, and
in such a way that the new kinds of number integrate seamlessly with
those that are already implemented.

 - primitive: real? obj
     Return `#t' if OBJ is a real number, `#f' else.  Note that the
     sets of integer and rational values form subsets of the set of
     real numbers, so the predicate will also be fulfilled if OBJ is an
     integer number or a rational number.

 - primitive: rational? obj
     Return `#t' if OBJ is a rational number, `#f' else.  Note that the
     set of integer values forms a subset of the set of rational
     numbers, so the predicate will also be fulfilled if OBJ is an
     integer number.