**Common Lisp the Language, 2nd Edition**

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( a b)` is equivalent to

#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,i

The 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.

[email protected]