本节，有一半题目在查别人的答案，索然无味……cons那段还是很带劲的～

练习 2.1
(define (make-rat n d)
  (let ((g (gcd (abs n) (abs d))))
    (if (< d 0)
        (cons (/ (- n) g) (/ (- d) g))
        (cons (/ n g) (/ d g)))))

原题不改其实也是符合要求的，只是钻了个空子。科学严谨点，还是加上abs吧～

练习 2.2
(define (make-point x y)
  (cons x y))
(define (x-point x)
  (car x))
(define (y-point x)
  (cdr x))
(define (print-point p)
  (display "(")
  (display (x-point p))
  (display ", ")
  (display (y-point p))
  (display ")")
  (newline))

(define (make-segment s e)
  (cons s e))
(define (start-segment x)
  (car x))
(define (end-segment x)
  (cdr x))
(define (midpoint-segment l)
  (make-point (average (x-point (start-segment l)) (x-point (end-segment l)))
              (average (y-point (start-segment l)) (y-point (end-segment l)))))

练习 2.3
使用正交边 two orthogonal segments
(define (make-rect top right)
  (cons top right))

(define (width-line r)
  (car r))
(define (height-line r)
  (cdr r))
用一条对角线记录 a diagonal line
(define (make-rect diagonal-line) diagonal-line)

(define (width-line r)
  r)
(define (height-line r)
  r)
;; 公共部分
(define (get-length f r)
    (abs (- (f (start-segment r))
            (f (end-segment r)))))

(define (perimeter r)
  (+ (* (get-length x-point (width-line r)) 2)
     (* (get-length y-point (height-line r)) 2)))

(define (area r)
  (* (get-length x-point (width-line r))
     (get-length y-point (height-line r))))

练习 2.4
(define (cons x y)
  (lambda (m) (m x y)))
(define (car z)
  (z (lambda (p q) p)))
(define (cdr z)
  (z (lambda (p q) q)))

(car (cons 1 2)) =>
((cons 1 2) (lambda (p q) p)) =>
((lambda (m) (m 1 2)) (lambda (p q) p)) =>
((lambda (p q) p) 1 2) =>
1

练习 2.5
(define (even? x)
  (if (= (remainder 2) 0)
      #t #f))
(define (cons x y)
  (* (expt 2 x) (expt 3 y)))
(define (get x n)
  (define (iter a b)
    (let ((v (cons a b)))
      (cond ((= v x) (if (= n 2) a b))
            ((> v x) (iter (+ a 1) 1))
            (else (iter a (+ b 1)))
      )))
  (iter 1 1)
  )
(define (car x)
  (get x 2))
(define (cdr x)
  (get x 3))
算法太糟糕了，还是这样的美
(define (count-powers n d)
  (define (iter i pow)
    (if (zero? (remainder i d))
        (iter (/ i d) (+ pow 1))
        pow))
  (iter n 0))

练习 2.6
(add-1 zero) 代换结果
(add-1 (lambda (f) (lambda (x) x)))
(lambda (f) (lambda (x) (f (((lambda (f) (lambda (x) x)) f) x))))
(lambda (f) (lambda (x) (f ((lambda (x) x) x))))
(lambda (f) (lambda (x) (f x)))

(define one
  (lambda (f) (lambda (x) (f x))))
(define two
  (lambda (f) (lambda (x) (f (f x)))))

(define (plus a b)
  (lambda (f) (lambda (x) ((a f) ((b f) x)))))
抽象，研究了老半天，最后把(add-1 zero)代换出来，才明白啥意思…

练习 2.7
(define (upper-bound x) (cdr x))
(define (lower-bound x) (car x))

练习 2.8
(define (sub-interval x y)
 (make-interval (- (lower-bound x) (upper-bound y))
                (- (upper-bound x) (lower-bound y))))

练习 2.9
不大明白其意思，就是写段代码验证么？
copy来的
> (mul-interval (make-interval 1 3) (make-interval 1 3))
(1 . 9)
> (mul-interval (make-interval 0 2) (make-interval 1 3))
(0 . 6)

Here widths of arguments is the same (1 and 2 for first and second arguments) but width of result differs (4 in the first case, 3 - in the second).

练习 2.10
我郁闷了，横跨0点或者是宽度为0，我不理解啥叫“意义不清楚”，翻了别人的答案居然是这样。OTL，是我想太多了么。
(define (div-interval x y)
  (if (and (<= (lower-bound y) 0) (>= (upper-bound y) 0))
    (error "divided by zero!")
    (mul-interval x
                  (make-interval (/ 1.0 (upper-bound y))
                                 (/ 1.0 (lower-bound y)))))

练习 2.11
累，写了些草稿后，决定还是——抄袭
(define (mul-interval x y)
  (let ((xl (lower-bound x))
        (xu (upper-bound x))
        (yl (lower-bound y))
        (yu (upper-bound y)))
       (cond ((and (>= xl 0) (>= xu 0) (>= yl 0) (>= yu 0))
              (make-interval (* xl yl) (* xu yu)))
             ((and (>= xl 0) (>= xu 0) (<= yl 0) (>= yu 0))
              (make-interval (* xu yl) (* xu yu)))
             ((and (>= xl 0) (>= xu 0) (<= yl 0) (<= yu 0))
              (make-interval (* xu yl) (* xl yu)))
             ((and (<= xl 0) (>= xu 0) (>= yl 0) (>= yu 0))
              (make-interval (* xl yu) (* xu yu)))
             ((and (<= xl 0) (>= xu 0) (<= yl 0) (>= yu 0))
              (make-interval (min (* xl yu) (* xu yl)) (max (* xl yl) (* xu yu))))
             ((and (<= xl 0) (>= xu 0) (<= yl 0) (<= yu 0))
              (make-interval (* xu yl) (* xl yl)))
             ((and (<= xl 0) (<= xu 0) (>= yl 0) (>= yu 0))
              (make-interval (* xl yu) (* xu yl)))
             ((and (<= xl 0) (<= xu 0) (<= yl 0) (>= yu 0))
              (make-interval (* xl yu) (* xl yl)))
             ((and (<= xl 0) (<= xu 0) (<= yl 0) (<= yu 0))
              (make-interval (* xu yu) (* xl yl))))))

练习 2.12
(define (make-center-percent x per)
  (make-interval (- x (* x per)) (+ x (* x per))))

(define (center i)
  (/ (+ (lower-bound i) (upper-bound i)) 2))

(define (percent i)
  ((lambda (x) (/ (- (upper-bound i) x) x)) (center i)))

练习 2.13
(c1, t1) (c2, t2)
c1 - c1*t1 ~ c1 + c1*t1
c2 - c2*t2 ~ c2 + c2*t2
c1 * c2 * (1 - t1) * (1 - t2) ~ c1 * c2 * (1 + t1) * (1 + t2)
center = c1 * c2 * (1 + t1 + t2 + t1*t2 + 1 - t1 - t2 + t1*t2) / 2
       = c1 * c2 * (1 + t1*t2)
width = c1 * c2 * (t1 + t2)
percent = width / center
        = (t1 + t2) / (1 + t1*t2)
        ≈t1 + t2
t1/t2 非常小，t1*t2忽略不计

练习 2.14 2.15 2.16
对于par1，会造成上下r1和r2取值不同。而par2，只用到一次r1和r2，因此不存在这个问题。当宽度越小时，这种误差越小。