Scheme Programming/Abstractions with Data

Introduction to Complex Numbers

In order to show how abstractions with data can be built, we're going to go through making a complex number package. A complex number is one that has 2 parts, a real part, and an imaginary part. They are often written in one of two ways, in rectangular form:

$a+bi$

And in polar form:

$Re^{i\theta }$

Now, we can can do all of the usual arithmetic with complex numbers, addition, subtraction, multiplication and division. There are simple formulae for this;

Addition:

$(a+bi)+(x+yi)=(a+x)+((b+y)i)$

Subtraction:

$(a+bi)-(x+yi)=(a-x)+((b-y)i)$

Multiplication:

$Re^{i\theta }\cdot Pe^{i\phi }=R\cdot Pe^{i(\theta +\phi )}$

Division:

${\frac {Re^{i\theta }}{Pe^{i\phi }}}={\frac {R}{P}}e^{i(\theta -\phi )}$

Note how multiplication and division are best expressed in polar form, while addition and subtraction are best expressed in rectangular form. This raises an interesting question: How does one best go about computing these? Do we have one internal representation? If so, which do we choose? There are a large amount of questions. These can be answered by trying to implement a new type of data: the complex number type.

Creating our Generic 'Typed' Variable

Firstly, we shall create a generic 'Typed' variable:

(define typed-variable
  (lambda (type value)
    (cons 'Typed (list type value))
  )
)

We now need a way to tell if a given variable has a type:

(define typed?
  (lambda (var)
    (and (list? var) (= 'Typed (car var)))
  )
)

Now, we've introduced two important concepts here, a 'Predicate' and a 'Constructor'. The first is a construct to find if some data is of the correct form, and the second is a procedure that builds our data structure for us.

We must have a way of extracting our data (in this case, the type) from this structure, a way of 'selecting' it:

(define type-of
  (lambda (var)
    (if (typed? var)
      (car (cdr var)
    )
  )
)

Creating our Complex Number Data Type

Building our Constructors

Using this typed value, we can go on to form a more detailed data structure for out complex number:

(define complex-rect
  (lambda (a b)
    (typed-variable 'Rect-Complex (list a b))
  )
)

Now let's continue, and create a complex-polar:

(define complex-polar
  (lambda (r thet)
    (typed-variable 'Polar-Complex (list r thet))
  )
)

(define complex
  (lambda (type first-var second-var)
    (if (equal? 'type Polar)
        (cons (complex-polar first-var second-var) 
              (complex-rect (sqrt (+ (expt first-var 2) 
                                     (expt second-var 2)
                                  )
                            ) 
                            0
              )
        )  ;; Change second half to be the calculated values.
        (cons (complex-polar 0 0) (complex-rect first-var second-var))
    )
  )
)

Building our Predicates

We have our constructors, now we need our predicates:

(define is-complex?
  (lambda (var)
    (and (typed? (car var)) 
         (or (= 'Rect-Complex (type-of (car var))) 
             (= 'Polar-Complex (type-of (car var)))
         )
    )
  )
)

Now we can define our arithmetic in terms of these procedures.