SICP Sections 2.1.1-2.1.2

Exercise 2.1

#lang racket
;the greatest common divisor of the two integers
(define (gcd a b)
  (if (= b 0)
      a
      (gcd b (remainder a b))))
;reduce the numerator and the denominator to the lowest terms
(define (make-rat-lowest a b)
  (let ((g (abs (gcd a b))))
    (cons (/ a g) (/ b g))))
;get the answer satisfied the require
(define (make-rat n d)
  (cond ((and (< n 0) (< d 0))
         (make-rat-lowest (- n) (- d)))
        ((and (< d 0) (> n 0))
         (make-rat-lowest (- n) (- d)))
        (else
         (make-rat-lowest n d))))

Exercise 2.2

#lang racket
;segment constructor and selectors
(define (make-segment start end)
  (cons start end))
(define (start-segment segment)
  (car segment))
(define (end-segment segment)
  (cdr segment))
;point constructor and selectors
(define make-point cons)
(define x-point car)
(define y-point cdr)
;middle-point of the designate segment
(define (average x y)
  (/ (+ x y) 2))
(define (midpoint-segment segment)
  (make-point
   (average (x-point (start-segment segment)) (x-point (end-segment segment)))
   (average (y-point (start-segment segment)) (y-point (end-segment segment)))))
;output to the console
(define (print-point p)
  (newline)
  (display "(")
  (display (x-point p))
  (display ",")
  (display (y-point p))
  (display ")"))

(print-point (midpoint-segment (make-segment (make-point 1.0 2.0) (make-point 2.0 2.0))))

Exercise 2.3

#lang racket
;compute the perimeter and area of the rectangle 
;no matter how rect-width and rect-height implemented
(define (rect-perimeter rect)
  (+ (* 2 (rect-width rect))
     (* 2 (rect-height rect))))
(define (rect-area rect)
  (* (rect-width rect)
     (rect-height rect)))
;rectangle constructor and selectors
(define (make-rect p1 p2)
  (make-segment p1 p2))
(define (rect-width rect)
  (abs (- (x-point (start-segment rect))
          (x-point (end-segment rect)))))
(define (rect-height rect)
  (abs (- (y-point (start-segment rect))
          (y-point (end-segment rect)))))

Summarize

1. wishful thinking是一种很好的编写抽象程序的指导方法，可以使我们编写“期望”的方法，而忽略底层实现。
2. 数据抽象提供了一种 abstraction barriers，通过提供constructor和selector接口，使我们可以直接访问数据结构而忽略其底层实现，也就是说，底层实现可以任意的组织。