create an ACL2 number
Major Section:  ACL2-BUILT-INS

(complex x 3) ; x + 3i, where i is the principal square root of -1
(complex x y) ; x + yi
(complex x 0) ; same as x, for rational numbers x

The function complex takes two rational number arguments and returns an ACL2 number. This number will be of type (complex rational) [as defined in the Common Lisp language], except that if the second argument is zero, then complex returns its first argument. The function complex-rationalp is a recognizer for complex rational numbers, i.e. for ACL2 numbers that are not rational numbers.

The reader macro #C (which is the same as #c) provides a convenient way for typing in complex numbers. For explicit rational numbers x and y, #C(x y) is read to the same value as (complex x y).

The functions realpart and imagpart return the real and imaginary parts (respectively) of a complex (possibly rational) number. So for example, (realpart #C(3 4)) = 3, (imagpart #C(3 4)) = 4, (realpart 3/4) = 3/4, and (imagpart 3/4) = 0.

The following built-in axiom may be useful for reasoning about complex numbers.

(defaxiom complex-definition
  (implies (and (real/rationalp x)
                (real/rationalp y))
           (equal (complex x y)
                  (+ x (* #c(0 1) y))))
  :rule-classes nil)

A completion axiom that shows what complex returns on arguments violating its guard (which says that both arguments are rational numbers) is the following, named completion-of-complex.

(equal (complex x y)
       (complex (if (rationalp x) x 0)
                (if (rationalp y) y 0)))