### 14` `Bignums, Rationals, and Complex Numbers

Racket supports integers of an arbitrary magnitude; when an integer
cannot be represented as a fixnum (i.e., 30 or 62 bits plus a sign
bit), then it is represented by the Racket type
scheme_bignum_type. There is no overlap in integer values
represented by fixnums and bignums.

Rationals are implemented by the type scheme_rational_type,
composed of a numerator and a denominator.
The numerator and denominator fixnums or bignums (possibly mixed).

Complex numbers are implemented by the types
scheme_complex_type and scheme_complex_izi_type,
composed of a real and imaginary part. The real and imaginary parts
will either be both flonums, both exact numbers (fixnums, bignums, and
rationals can be mixed in any way), or one part will be exact 0 and
the other part will be a flonum. If the inexact part is inexact 0, the
type is scheme_complex_izi_type, otherwise the type is
scheme_complex_type; this distinction make it easy to test
whether a complex number should be treated as a real number.

Returns 1 if n is an exact number, 0 otherwise
(n need not be a number).

Returns 1 if n is an inexact number, 0 otherwise
(n need not be a number).

Creates a bignum representing the integer

v. This can create a
bignum that otherwise fits into a fixnum. This must only be used to
create temporary values for use with the

bignum functions. Final
results can be normalized with

scheme_bignum_normalize. Only
normalized numbers can be used with procedures that are not specific
to bignums.

Converts a bignum to a floating-point number, with reasonable but
unspecified accuracy.

If Racket is not compiled with single-precision floats, this procedure
is actually a macro alias for

scheme_bignum_to_double.

Creates a bignum that is close in magnitude to the floating-point
number d. The conversion accuracy is reasonable but unspecified.

If Racket is not compiled with single-precision floats, this procedure
is actually a macro alias for

scheme_bignum_from_double.

Writes a bignum into a newly allocated byte string.

Reads a bignum from a

mzchar string, starting from position

offset in

str. If the string does not represent an
integer, then

NULL will be returned. If the string represents a
number that fits in 31 bits, then a

scheme_integer_type
object will be returned.

If

n fits in 31 bits, then a

scheme_integer_type object
will be returned. Otherwise,

n is returned.

Creates a rational from a numerator and denominator. The n and
d parameters must be fixnums or bignums (possibly mixed). The
resulting will be normalized (thus, a bignum or fixnum might be returned).

Converts the rational n to a double.

If Racket is not compiled with single-precision floats, this procedure
is actually a macro alias for

scheme_rational_to_double.

Returns the numerator of the rational n.

Returns the denominator of the rational n.

Converts the given double into a maximally-precise rational.

Creates a complex number from real and imaginary parts. The r
and i arguments must be fixnums, bignums, flonums, or rationals
(possibly mixed). The resulting number will be normalized (thus, a real
number might be returned).

Returns the real part of the complex number n.

Returns the imaginary part of the complex number n.