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Common Lisp the Language, 2nd Edition
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Complex numbers (type complex) are represented in
Cartesian form, with a real part and an imaginary part, each of which is
a non-complex number (integer, ratio, or floating-point number). It
should be emphasized that the parts of a complex number are not
necessarily floating-point numbers; in this, Common Lisp is like PL/I
and differs from Fortran. However, both parts must be of the same type:
either both are rational, or both are of the same floating-point
format.
Complex numbers may be notated by writing the characters
#C followed by a list of the real and imaginary parts. If
the two parts as notated are not of the same type, then they are
converted according to the rules of floating-point contagion as
described in chapter 12. (Indeed,
#C(ab)
is equivalent to
#,(complexab);
see the description of the function complex.) For
example:
#C(3.0s1 2.0s-1) ;Real and imaginary parts are short format
#C(5 -3) ;A Gaussian integer
#C(5/3 7.0) ;Will be converted internally to #C(1.66666 7.0)
#C(0 1) ;The imaginary unit, that is, iThe type of a specific complex number is indicated by a list of the
word complex and the type of the components; for example, a
specialized representation for complex numbers with short floating-point
parts would be of type (complex short-float). The type
complex encompasses all complex representations.
A complex number of type (complex rational), that is,
one whose components are rational, can never have a zero imaginary part.
If the result of a computation would be a complex rational with a zero
imaginary part, the result is immediately converted to a non-complex
rational number by taking the real part. This is called the rule of
complex canonicalization. This rule does not apply to
floating-point complex numbers; #C(5.0 0.0) and
5.0 are different.
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