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 can represent both exact and inexact rational numbers, but it
cannot represent precise finite irrational numbers. Exact rationals are
represented by storing the numerator and denominator as two exact
integers. Inexact rationals are stored as floating point numbers using
the C type
Exact rationals are written as a fraction of integers. There must be no whitespace around the slash:
Even though the actual encoding of inexact rationals 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 finite “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
real? predicates are equivalent for finite
Dividing by an exact zero leads to a error message, as one might expect. However, dividing by an inexact zero does not produce an error. Instead, the result of the division is either plus or minus infinity, depending on the sign of the divided number and the sign of the zero divisor (some platforms support signed zeroes ‘-0.0’ and ‘+0.0’; ‘0.0’ is the same as ‘+0.0’).
Dividing zero by an inexact zero yields a NaN (‘not a number’)
value, although they are actually considered numbers by Scheme.
Attempts to compare a NaN value with any number (including
#f. Although a NaN value is not
= to itself, it is both
equal? to itself
and other NaN values. However, the preferred way to test for
them is by using
The real NaN values and infinities are written ‘+nan.0’,
‘+inf.0’ and ‘-inf.0’. This syntax is also recognized by
read as an extension to the usual Scheme syntax. These special
values are considered by Scheme to be inexact real numbers but not
rational. Note that non-real complex numbers may also contain
infinities or NaN values in their real or imaginary parts. To
test a real number to see if it is infinite, a NaN value, or
Every real number in Scheme belongs to precisely one of those three
On platforms that follow IEEE 754 for their floating point
arithmetic, the ‘+inf.0’, ‘-inf.0’, and ‘+nan.0’ values
are implemented using the corresponding IEEE 754 values.
They behave in arithmetic operations like IEEE 754 describes
(= +nan.0 +nan.0) ⇒
#t if obj is a real number, else
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.
#t if x is a rational number,
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.
Returns the simplest rational number differing from x by no more than eps.
As required by R5RS,
rationalize only returns an
exact result when both its arguments are exact. Thus, you might need
inexact->exact on the arguments.
(rationalize (inexact->exact 1.2) 1/100) ⇒ 6/5
#t if the real number x is ‘+inf.0’ or
‘-inf.0’. Otherwise return
#t if the real number x is ‘+nan.0’, or
#t if the real number x is neither infinite nor a
Return the numerator of the rational number x.
Return the denominator of the rational number x.
scm_is_true (scm_real_p (val)) and
scm_is_true (scm_rational_p (val)), respectively.
Returns the number closest to val that is representable as a
double. Returns infinity for a val that is too large in
magnitude. The argument val must be a real number.
SCM value that represents val. The returned
value is inexact according to the predicate
inexact?, but it
will be exactly equal to val.