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5.5.3 Example

In this section we shall continue to define operations on the <complex> class defined in Figure 2. Suppose that we want to use it to implement complex numbers completely. For instance a definition for the addition of two complexes could be

(define-method (new-+ (a <complex>) (b <complex>))
  (make-rectangular (+ (real-part a) (real-part b))
                    (+ (imag-part a) (imag-part b))))

To be sure that the + used in the method new-+ is the standard addition we can do:

(define-generic new-+)

(let ((+ +))
  (define-method (new-+ (a <complex>) (b <complex>))
    (make-rectangular (+ (real-part a) (real-part b))
                      (+ (imag-part a) (imag-part b)))))

The define-generic ensures here that new-+ will be defined in the global environment. Once this is done, we can add methods to the generic function new-+ which make a closure on the + symbol. A complete writing of the new-+ methods is shown in Figure 3.

(define-generic new-+)

(let ((+ +))

  (define-method (new-+ (a <real>) (b <real>)) (+ a b))

  (define-method (new-+ (a <real>) (b <complex>))
    (make-rectangular (+ a (real-part b)) (imag-part b)))

  (define-method (new-+ (a <complex>) (b <real>))
    (make-rectangular (+ (real-part a) b) (imag-part a)))

  (define-method (new-+ (a <complex>) (b <complex>))
    (make-rectangular (+ (real-part a) (real-part b))
                      (+ (imag-part a) (imag-part b))))

  (define-method (new-+ (a <number>))  a)

  (define-method (new-+) 0)

  (define-method (new-+ . args)
    (new-+ (car args)
      (apply new-+ (cdr args)))))

(set! + new-+)


Fig 3: Extending + for dealing with complex numbers

We use here the fact that generic function are not obliged to have the same number of parameters, contrarily to CLOS. The four first methods implement the dyadic addition. The fifth method says that the addition of a single element is this element itself. The sixth method says that using the addition with no parameter always return 0. The last method takes an arbitrary number of parameters¹. This method acts as a kind of reduce: it calls the dyadic addition on the car of the list and on the result of applying it on its rest. To finish, the set! permits to redefine the + symbol to our extended addition.

To terminate our implementation (integration?) of complex numbers, we can redefine standard Scheme predicates in the following manner:

(define-method (complex? c <complex>) #t)
(define-method (complex? c)           #f)

(define-method (number? n <number>) #t)
(define-method (number? n)          #f)
...
...

Standard primitives in which complex numbers are involved could also be redefined in the same manner.

Footnotes

1. The parameter list for a define-method follows the conventions used for Scheme procedures. In particular it can use the dot notation or a symbol to denote an arbitrary number of parameters