( Reals and Rationals

Info Catalog ( Integers ( Numbers ( Complex Numbers
 21.2.3 Real and Rational Numbers
 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:
    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
    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.
  -- Scheme Procedure: real? obj
  -- C Function: scm_real_p (obj)
      Return `#t' if OBJ is a real number, else `#f'.  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.
  -- Scheme Procedure: rational? x
  -- C Function: scm_real_p (x)
      Return `#t' if X is a rational number, `#f' otherwise.  Note that
      the set of integer values forms a subset of the set of rational
      numbers, i. e. the predicate will also be fulfilled if X is an
      integer number.  Real numbers will also satisfy this predicate,
      because of their limited precision.
Info Catalog ( Integers ( Numbers ( Complex Numbers
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