Implementations of Scheme are not required to implement the whole tower of subtypes (see Numerical types), but they must implement a coherent subset consistent with both the purposes of the implementation and the spirit of the Scheme language. For example, an implementation in which all numbers are real may still be quite useful.1
Implementations may also support only a limited range of numbers of any type, subject to the requirements of this section. The supported range for exact numbers of any type may be different from the supported range for inexact numbers of that type. For example, an implementation that uses flonums to represent all its inexact real numbers may support a practically unbounded range of exact integers and rationals while limiting the range of inexact reals (and therefore the range of inexact integers and rationals) to the dynamic range of the flonum format. Furthermore the gaps between the representable inexact integers and rationals are likely to be very large in such an implementation as the limits of this range are approached.
An implementation of Scheme must support exact integers throughout the
range of numbers that may be used for indexes of lists, vectors, and
strings or that may result from computing the length of a list, vector,
or string. The
string-length procedures must return an exact integer, and it is
an error to use anything but an exact integer as an index. Furthermore
any integer constant within the index range, if expressed by an exact
integer syntax, will indeed be read as an exact integer, regardless of
any implementation restrictions that may apply outside this range.
Finally, the procedures listed below will always return an exact integer
result provided all their arguments are exact integers and the
mathematically expected result is representable as an exact integer
within the implementation:
* gcd modulo + imag-part numerator - inexact->exact quotient abs lcm rationalize angle magnitude real-part ceiling make-polar remainder denominator make-rectangular round expt max truncate floor min
Implementations are encouraged, but not required, to support exact
integers and exact rationals of practically unlimited size and
precision, and to implement the above procedures and the
procedure in such a way that they always return exact results when given
exact arguments. If one of these procedures is unable to deliver an
exact result when given exact arguments, then it may either report a
violation of an implementation restriction or it may silently coerce its
result to an inexact number. Such a coercion may cause an error
An implementation may use floating point and other approximate representation strategies for inexact numbers. This report recommends, but does not require, that the ieee 32-bit and 64-bit floating point standards be followed by implementations that use flonum representations, and that implementations using other representations should match or exceed the precision achievable using these floating point standards.
In particular, implementations that use flonum representations must
follow these rules: A flonum result must be represented with at least as
much precision as is used to express any of the inexact arguments to
that operation. It is desirable (but not required) for potentially
inexact operations such as
sqrt, when applied to exact arguments,
to produce exact answers whenever possible (for example the square root
of an exact 4 ought to be an exact 2). If, however, an exact number is
operated upon so as to produce an inexact result (as by
and if the result is represented as a flonum, then the most precise
flonum format available must be used; but if the result is represented
in some other way then the representation must have at least as much
precision as the most precise flonum format available.
Although Scheme allows a variety of written notations for numbers, any particular implementation may support only some of them.2 For example, an implementation in which all numbers are real need not support the rectangular and polar notations for complex numbers. If an implementation encounters an exact numerical constant that it cannot represent as an exact number, then it may either report a violation of an implementation restriction or it may silently represent the constant by an inexact number.
 MIT/GNU Scheme implements the whole tower of numerical types. It has unlimited-precision exact integers and exact rationals. Flonums are used to implement all inexact reals; on machines that support ieee floating-point arithmetic these are double-precision floating-point numbers.
 MIT/GNU Scheme implements all of the written notations for numbers.