commit 442238a0d5d01a485df2e4c24d0fca98d5cac33c
Author: Emin Arslan <arslanemin2662@gmail.com>
Date:   Thu Jan 2 17:49:30 2025 +0300

    Added my solutions so far

diff --git a/.gitignore b/.gitignore
new file mode 100644
index 0000000..40f4f43
--- /dev/null
+++ b/.gitignore
@@ -0,0 +1,2 @@
+\#*
+*~
diff --git a/ex-1-11.rkt b/ex-1-11.rkt
new file mode 100644
index 0000000..97f7d88
--- /dev/null
+++ b/ex-1-11.rkt
@@ -0,0 +1,43 @@
+#lang racket
+
+;; for n=100
+;; iterative version takes roughly 0.008 milliseconds.
+;; recursive version does not seem like it will finish any time soon.
+;; for n=50
+;; iterative version takes roughly 0.003 milliseconds in drracket
+;; recursive version has not yet finished executing after a couple minutes, and I got bored.
+;; for n=25.
+;; iterative version takes 0.002ms
+;; recursive version takes 46.5 ms
+;; yeah. that's big. okay. this shit's important.
+;; I wonder how it would be with memoization.
+
+
+(define-syntax-rule (meas form)
+  (let ([my-time (current-inexact-milliseconds)])
+    (let ([res form])
+      (- (current-inexact-milliseconds) my-time))))
+
+
+(define (recursive n)
+    (if (< n 3)
+        n
+        (+ (recursive (- n 1))
+           (* 2 (recursive (- n 2)))
+           (* 3 (recursive (- n 3))))))
+(define (iter a b c n)
+  (if (<= n 0)
+      c
+      (iter b c (+ c (* 2 b) (* 3 a)) (- n 1))))
+(define (iter-start n)
+  (iter 0 1 2 (- n 2)))
+
+(define (ex-1-11 n)
+  (let loop ((i n))
+    (if (< i 2)
+        #t
+        (if (= (recursive i) (iter-start i))
+            (loop (- i 1))
+            (begin 
+              (display "UNEQUAL!")
+              (displayln i))))))
diff --git a/ex-1-16.rkt b/ex-1-16.rkt
new file mode 100644
index 0000000..8223638
--- /dev/null
+++ b/ex-1-16.rkt
@@ -0,0 +1,11 @@
+#lang sicp
+
+(define (fast-exp b n)
+  ;(define (even?)); already defined by racket
+  (define (actual b n a)
+    (cond
+      ((= n 0) a)
+      ;((= n 1) a)
+      ((even? n) (actual (* b b) (/ n 2) a))
+      (else (actual b (- n 1) (* a b)))))
+  (actual b n 1))
diff --git a/ex-1-17.rkt b/ex-1-17.rkt
new file mode 100644
index 0000000..f48061e
--- /dev/null
+++ b/ex-1-17.rkt
@@ -0,0 +1,17 @@
+#lang racket
+
+;; we are told to assume these are already defined.
+(define (double x)
+  (+ x x))
+(define (halve x)
+  (/ x 2))
+
+;; multiplication. defined in terms of addition, double and halve.
+;; logarithmic time, constant space.
+(define (mult x y)
+  (define (recc x y a)
+    (cond
+      ((= y 0) a)
+      ((even? y) (recc (double x) (halve y) a))
+      (else (recc x (- y 1) (+ a x)))))
+  (recc x y 0))
diff --git a/ex-1-19.rkt b/ex-1-19.rkt
new file mode 100644
index 0000000..f52c8f4
--- /dev/null
+++ b/ex-1-19.rkt
@@ -0,0 +1,36 @@
+#lang sicp
+
+;; Yeah okay, this one was extremely satisfying.
+;; I just did the calculations on-paper.
+;; it's some simple algebra anyway, but I'll type it here:
+;; assuming Tpq on (a0, b0):
+;; a1 = b0q + a0 * ( p + q)
+;; b1 = b0p + a0q
+;; a2 = (b0p + a0q) * p + (b0q + a0 * ( p + q)) * (p + q)
+;; b2 = (b0p + a0q) * p + (b0q + a0 * ( p + q)) * q
+;;
+;; rearrange a2 and b2 into a similar form to the definition of a1 and b1:
+;; (i.e. define a2 and b2 in terms of a0 and b0 and p and q)
+;; a2 = b0 * (q^2 + 2*p*q) + a0 * (2*q^2 + 2*p*q + p^2)
+;; b2 = b0 * (p^2 + q^2) + a0 * (q^2 + 2*p*q)
+;;
+;; from here we can see that p'= p^2 + q^2, and q'= q^2 + 2 * p * q
+;; and as a result, we have logarithmic fibonacci!
+
+;; printing the resulting number takes longer than calculating it lmao.
+
+(define (fib n)
+  (fib-iter 1 0 0 1 n))
+(define (fib-iter a b p q count)
+  (cond ((= count 0) b)
+        ((even? count)
+         (fib-iter a
+                   b
+                   (+ (* p p) (* q q)) ; compute p′
+                   (+ (* q q) (* 2 (* p q))) ; compute q′
+                   (/ count 2)))
+        (else (fib-iter (+ (* b q) (* a q) (* a p))
+                        (+ (* b p) (* a q))
+                        p
+                        q
+                        (- count 1)))))
diff --git a/ex-1-29.rkt b/ex-1-29.rkt
new file mode 100644
index 0000000..3416c34
--- /dev/null
+++ b/ex-1-29.rkt
@@ -0,0 +1,57 @@
+#lang racket
+
+
+(display "hell yrah")
+
+
+(define (sum term a next b)
+  (define (iter a result)
+    (if (>= a b)
+        result
+        (iter (next a) (+ result (term a)))))
+  (iter a 0))
+
+(define (integral f a b n)
+  (define h (/ (- b a) n))
+  (define (apply-f k)
+    (f (+ a (* k h))))
+  (define (term k)
+    (+ (* 4 (apply-f k))
+       (* 2 (apply-f (+ k 1)))))
+  (* (/ h 3)
+     (+ (apply-f 0) (- (apply-f n))
+        (sum term 1 (λ (x) (+ x 2)) n))))
+
+;;; HOLY SHIT. This has nothing to do with the integral example, I just
+;;; found out while reading section 1.3.2 that let is/can be implemented
+;;; using just lambdas.
+;;; So of course, I wrote a macro to do it.
+;;; now I'm wondering what are the absolute minimum amount of primitives
+;;; I would need to implement an entire scheme from the ground up.
+;;; (Ideally to write a compiler in itself, for itself!)
+;;; I guess this kind of thing really puts it into perspective huh.
+(define-syntax my-let
+  (syntax-rules ()
+    [(_ () body ...) (begin body ...)]
+    [(_ ([var expr] binding ...) body ...)
+     ((lambda (var) (my-let (binding ...) body ...))
+      expr) ]))
+
+;;; god. I thought Common Lisp was nice. Scheme and Racket are just something else.
+
+;; some stuff from the next exercise
+(define (pn x) (display x) (newline) x)
+(define tolerance 0.00001)
+(define (fixed-point f first-guess)
+  (define (close-enough? v1 v2)
+    (< (abs (- v1 v2))
+       tolerance))
+  (define (try guess)
+    (let ((next (f guess)))
+      (if (close-enough? guess next)
+          (pn next)
+          (try (pn next))
+          )))
+  (try first-guess))
+; value of phi i think?
+(fixed-point (lambda (x) (+ 1 (/ 1 x))) 1)
diff --git a/ex-1-37.rkt b/ex-1-37.rkt
new file mode 100644
index 0000000..550f7bb
--- /dev/null
+++ b/ex-1-37.rkt
@@ -0,0 +1,42 @@
+#lang racket
+
+(define (cont-frac nf df k)
+  (define (recurse i)
+    (if (>= i k)
+        0
+        (/ (nf i) (+ (df i) (recurse (+ 1 i))))))
+  (recurse 0))
+
+; produces the same result as cont-frac, but
+; generates an iterative process.
+(define (cont-frac-iterative nf df k)
+  (define (iterate i numer denom)
+    (if (<= i 0)
+        (/ numer denom)
+        (iterate (- i 1) (nf (- i 1)) (+ (df (- i 1)) (/ numer denom)))))
+  (iterate (- k 1) (nf k) (df k)))
+
+(define (golden-ratio k)
+  "cont-frac generates 1/golden ratio, so we just inverse it to get the
+   real thing. call with k=1000 or something to get an accurate result"
+  (/ 1.0 (cont-frac-iterative (λ (i) 1.0) (λ (i) 1.0) k)))
+
+;; example 1-38
+(define (e-helper i)
+  (let ((r (remainder (- i 1) 3)))
+    (if (= 0 r)
+        (* 2.0 (/ (+ i 2) 3))
+        1.0)))
+(define (eulers-constant k)
+  "Finds euler's constant. the continued fraction gives e - 2, so we add 2."
+  (+ 2 (cont-frac-iterative (λ (i) 1.0) e-helper k)))
+
+;; example 1-39
+(define (tan-cf x k)
+  "Finds an approximation of the tangent function. The first numerator is not negative,
+   even though our numerator function calculates them all as negative, so we need to negate
+   the result at the end."
+  (-
+   (cont-frac-iterative (λ (i) (- (if (<= i 0) x (* x x))))
+                        (λ (i) (- (* 2 (+ i 1)) 1))
+                        k)))
\ No newline at end of file
diff --git a/ex-1-40-thru-45.rkt b/ex-1-40-thru-45.rkt
new file mode 100644
index 0000000..7ffe908
--- /dev/null
+++ b/ex-1-40-thru-45.rkt
@@ -0,0 +1,99 @@
+#lang racket
+
+; definitions from some earlier chapters.
+(define (fixed-point f first-guess)
+  (define tolerance 0.01)
+  (define (close-enough? v1 v2)
+    (< (abs (- v1 v2))
+       tolerance))
+  (define (try guess)
+    (let ((next (f guess)))
+      (if (close-enough? guess next)
+          next
+          (try next)
+          )))
+  (try first-guess))
+
+; some definitions: these are given by the book.
+(define dx 0.00001)
+(define (deriv g)
+  (lambda (x) (/ (- (g (+ x dx)) (g x)) dx)))
+
+(define (newton-transform g)
+  (lambda (x) (- x (/ (g x) ((deriv g) x)))))
+(define (newtons-method g guess)
+  (fixed-point (newton-transform g) guess))
+
+; example 1-40
+(define (cubic a b c)
+  (λ (x) (+ (expt x 3)
+            (* a (expt x 2))
+            (* b (expt x 1))
+            c)))
+
+; example 1-41
+; NOTE: inc isn't defined in racket, so I just defined that too
+(define (inc x) (+ 1 x))
+(define (double f)
+  (λ (x) (f (f x))))
+
+(display (((double (double double)) inc) 5))
+(newline)
+; => 21
+
+; example 1-42
+(define (square x) (* x x))
+(define (compose f g) (λ (x) (f (g x))))
+
+(display ((compose square inc) 6))
+; => 49
+
+; example 1-43
+(define (repeated f n)
+  "This procedure generates an iterative process."
+  (define (iter ret i)
+    (if (<= i 1)
+        ret
+        (iter (compose f ret) (- i 1))))
+  (iter f n))
+
+; example 1-44
+; procedure average is defined here for convenience
+(define (average a b c)
+  (/ (+ a b c) 3))
+(define (smooth f dx)
+  (λ (x)
+    (average (f (- x dx))
+             (f x)
+             (f (+ x dx)))))
+; example asks us to "show" how n-fold smoothing would be done.
+; here it is, with the procedure `repeated`
+; note we need to pass an anonymous function to do it because the way
+; I defined it, `smooth` accepts two arguments, not one.
+; thankfully one of the arguments will be the same for all calls,
+; so we can just curry it up a little.
+(define (some-func x) x) ; assume there is some function we want to smooth.
+(define n 2)
+((repeated (λ (f) (smooth f 0.001)) n) some-func)
+
+; example 1-45
+; this one looks long, I'm skipping for now
+
+; example 1-46
+(define (iterative-improve good-enough? improve)
+  "I use a define here because the returned procedure will be recursive
+   (an iterative process, but recursive function)
+   I guess we could use `do` for this but whatever"
+  (define (ret x)
+    (if (good-enough? x)
+        x
+        (ret (improve x))))
+  ret)
+
+(define (sqrt n)
+  "Careful: we're calling iterative-improve, which creates a procedure,
+   then we're calling the returned procedure with 1 as a parameter."
+  ((iterative-improve
+    (λ (x) (< (abs (- (* x x) n)) 0.0000001))
+    (λ (x) (/ (+ x (/ n x)) 2.0)))
+   2.0))
diff --git a/ex-2-20.rkt b/ex-2-20.rkt
new file mode 100644
index 0000000..23071a4
--- /dev/null
+++ b/ex-2-20.rkt
@@ -0,0 +1,167 @@
+#lang sicp
+;; 2.17 : while I was having fun with this,
+;; I came up with the following monstrosity
+;; yes, it works, and is completely equivalent
+;; the second, less horrible implementation
+(define (last-pair l)
+  (or (and (or (null? l) (null? (cdr l))) l)
+      (last-pair (cdr l))))
+(define (last-pair2 l)
+  (cond
+    ((null? l) l)
+    ((null? (cdr l)) l)
+    (#t (last-pair2 (cdr l)))))
+
+;; 2.18
+(define (reverse l)
+  (define (iter acc l)
+    (if (null? l)
+        acc
+        (iter (cons (car l) acc)
+              (cdr l))))
+  (iter '() l))
+
+(equal? '(4 3 2 1) (reverse '(1 2 3 4)))
+
+;; 2.19
+
+(define (cc amount coin-values)
+  (define (first-denomination cv)
+    (car cv))
+  (define (except-first-denomination cv)
+    (cdr cv))
+  (define no-more? null?)
+  (cond ((= amount 0) 1)
+        ((or (< amount 0) (no-more? coin-values)) 0)
+        (else
+         (+ (cc amount
+                (except-first-denomination
+                 coin-values))
+            (cc (- amount
+                   (first-denomination
+                    coin-values))
+                coin-values)))))
+
+;; 2.20
+(define (filter f l)
+  (cond
+    ((null? l) l)
+    ((f (car l)) (cons (car l) (filter f (cdr l))))
+    (#t (filter f (cdr l)))))
+(define (same-parity a . b)
+  (cons a
+        (filter (if (even? a) even? odd?)
+                b)))
+(equal? (same-parity 1 2 3 4 5 6) '(1 3 5))
+
+;; 2.21
+(define (square x) (* x x))
+(define (square-list-direct items)
+  (if (null? items)
+      '()
+      (cons (square (car items)) (square-list-direct (cdr items)))))
+(define (square-list-map items)
+  (map square items))
+
+;; 2.22 this exercise asks for an explanation on why the
+;; accumulated result is in reverse order.
+;; This is because, at each iteration, the newest item
+;; is added to the head of the list - always.
+;; this means that the first processed item will be added
+;; to the head, then the second, then the third. So
+;; the list '(1 2 3) will be processed as such:
+
+;; '(1 2 3) '()
+;; '(2 3) '(1)
+;; '(3) '(4 1)
+;; '() '(9 4 1)
+
+;; So the lists behave like stacks, i.e. FILO ADTs.
+
+;; the "solution" proposed by Louis Reasoner
+;; does not work, because that doesn't change the order in
+;; which the items are processed or where the items are
+;; inserted. It only changes the positions of car and cdr,
+;; and in practice this will ruin the structure of the list,
+;; nothing more.
+
+;; 2.23
+;; a trivial implementation: (just use map and discard result)
+(define (trivial-for-each f l)
+  (map f l)
+  #t)
+;; but this isn't much useful, as the memory is still allocated
+;; (if promptly recollected by the GC) for map, defeating the purpose.
+;; Here's an implementation that simply ignores the results:
+(define (my-for-each f l)
+  (if (null? l)
+      #t
+      (begin (f (car l))
+             (my-for-each f (cdr l)))))
+;; no consing is done, and this generates an iterative process,
+;; so no stack allocations either. Equivalent to a for-loop in
+;; a language like C or C++.
+
+
+;; 2.25 - can also be written as: (car (cdaddr l))
+(car (cdr (car (cdr (cdr '(1 3 (5 7) 9))))))
+(car (car '((7))))
+; third one's too long. I'll be using cadr to mean (car (cdr x))
+(cadr (cadr (cadr (cadr (cadr (cadr '(1 (2 (3 (4 (5 (6 7))))))))))))
+
+;; 2.26 - doesn't require doing anything, just see what they do lol
+
+
+;; 2.27
+;; similar to regular reverse, with the addition that
+;; if the current element is itself a list, we cons
+;; its reverse instead.
+;; O(n) time complexity (where n = leaf count)
+;; O(k) space complexity (where k = branch depth? not sure about the correct terminology)
+(define (deep-reverse l)
+  (define (iter acc l)
+    (cond
+      ((null? l) acc)
+      ((list? (car l)) (iter (cons (deep-reverse (car l)) acc)
+                             (cdr l)))
+      (#t (iter (cons (car l) acc)
+               (cdr l)))))
+  (iter '() l))
+
+;; 2.28
+;; ooh, this one's spicy
+;; okay, I think I'll first use a recursive process for this,
+;; then translate it to an iterative one.
+;; Note: iterative process will require a reverse at the end,
+;; unless mutable conses are in play. That's why I used sicp lang for this.
+(define (fringe l)
+  (cond
+    ((null? l) l)
+    ((list? (car l)) (append (fringe (car l))
+                             (fringe (cdr l))))
+    (#t (cons (car l)
+              (fringe (cdr l))))))
+
+(define (fringe-iter l)
+  "This is mostly the same actually. We just change the mechanism for adding stuff.
+   Efficiency-wise, the only difference is that this impl only goes another level
+   deep for every level of nested lists (instead of going another level deep *at every iteration*)
+   So, better, but still not perfect if you have a tree a billion levels deep.
+
+   The difference here from a typical iterative process is the mutation: instead
+   of using pure functions I just had the iterator keep track of the end of the list."
+  (define (add end x)
+    (let ((c (cons x '())))
+      (set-cdr! end c)
+      c))
+  (define (iter acc-end l)
+    (cond
+      ((null? l) acc-end)
+      ((list? (car l)) (iter (iter acc-end (car l))
+                             (cdr l)))
+      (#t (iter (add acc-end (car l))
+                (cdr l)))))
+  (let ((sentinel (cons #f '())))
+    (iter sentinel l)
+    (cdr sentinel)))
+
diff --git a/ex-2-30.rkt b/ex-2-30.rkt
new file mode 100644
index 0000000..0f793ba
--- /dev/null
+++ b/ex-2-30.rkt
@@ -0,0 +1,61 @@
+#lang racket
+
+(define (accumulate op initial sequence)
+  (if (null? sequence)
+      initial
+      (op (car sequence)
+          (accumulate op initial (cdr sequence)))))
+
+(define (square x) (* x x))
+
+;; 2.30 oooh, we getting cool now are we?
+(define (square-tree l)
+  (cond
+    ((null? l) l)
+    ((list? (car l)) (cons (square-tree (car l))
+                           (square-tree (cdr l))))
+    (#t (cons (square (car l))
+              (square-tree (cdr l))))))
+
+(define (square-tree-map l)
+  (map (λ (x)
+         (if (list? x)
+             (square-tree-map x)
+             (square x)))
+       l))
+
+;; 2.31 hohoohoho nice
+(define (tree-map f l)
+  (map (λ (x)
+         (if (list? x)
+             (tree-map f x)
+             (f x)))
+       l))
+(define (square-tree-final l) (tree-map square l))
+
+;; 2.32 hmm.. cool stuff.
+(define (subsets s)
+  (if (null? s)
+      (list '())
+      (let ((rest (subsets (cdr s))))
+        (append rest (map (λ (l) (cons (car s) l)) rest)))))
+;; The reason this works, is because the set of subsets of a set s
+;; can be defined as such:
+;;  - if s is the empty set, the result is a set containing the empty set
+;;  - otherwise, the result is a set containing:
+;;    1. the subsets of s without the first element of s
+;;    2. the subsets of s with the first element of s
+;;    2 can be defined as the first element of s added to every
+;;    subset of s that does not contain the first element of s.
+;; probably not a very rigorous or formal definition, but good
+;; enough for now.
+
+
+;; 2.33
+(define (map p sequence)
+  (accumulate (lambda (x y) ⟨??⟩) nil sequence))
+(define (append seq1 seq2)
+  (accumulate cons ⟨??⟩ ⟨??⟩))
+(define (length sequence)
+  (accumulate ⟨??⟩ 0 sequence))
+
diff --git a/ex-2.1-thru-2.6 b/ex-2.1-thru-2.6
new file mode 100644
index 0000000..c415118
--- /dev/null
+++ b/ex-2.1-thru-2.6
@@ -0,0 +1,93 @@
+#lang racket
+
+;; 2.1
+(define (normalize n d)
+  (if (equal? (< n 0) (< d 0))
+      (cons (abs n) (abs d))
+      (cons (- (abs n)) (abs d))))
+(define (make-rat n d)
+  (let ([g (gcd n d)])
+    ; if they have the same sign, just take absolutes
+    (normalize (/ n g) (/ d g))))
+
+(define (print-rat rat)
+  (display (car rat))
+  (display "/")
+  (display (cdr rat))
+  (newline))
+
+(map print-rat
+     (map (λ (x) (apply make-rat x))
+          '((2 3)
+            (1 2)
+            (-3 4)
+            (-100 -4)
+            (100 -4))))
+
+;; 2.2
+;; we are using racket, so we could also define a structure.
+;; (struct point (x y))
+;; instead we'll keep to the book and use cons cells.
+(define point-x car)
+(define point-y cdr)
+(define point cons)
+;; the above could be replaced with
+; (struct point (x y))
+(define line-p1 car)
+(define line-p2 cdr)
+(define line cons)
+; (struct line (p1 p2))
+
+(define (average a b) (/ (+ a b) 2))
+(define (midpoint-segment ls)
+  (point (average (point-x (line-p1 ls))
+                  (point-x (line-p2 ls)))
+         (average (point-y (line-p1 ls))
+                  (point-y (line-p2 ls)))))
+
+;; 2.3
+;; not sure what the book means here. But I guess we can implement
+;; rectangles as pairs, too, since we only need two corners really.
+;; as "another representation", I guess we could store two line segments?
+;; but either way the rect-p1 and rect-p2
+;; functions can be replaced very easily.
+;; the area and perimeter functions do not care about their implementation.
+(define rect-p1 car)
+(define rect-p2 cdr)
+(define rect cons)
+
+;; further abstraction, actually: rectangle side 1.
+;; returns the length of rect's one side
+(define (rect-side-helper rect fun)
+  (abs (- (fun (rect-p1 rect))
+          (fun (rect-p2 rect)))))
+(define (rect-s1 rect)
+  (rect-side-helper rect point-x))
+(define (rect-s2 rect)
+  (rect-side-helper rect point-y))
+
+(define (area rect)
+  (* (rect-s1 rect) (rect-s2 rect)))
+(define (perimeter rect)
+  (* 2 (+ (rect-s1 rect) (rect-s2 rect))))
+
+; 2.6
+(define zero (lambda (f) (lambda (x) x)))
+(define (add-1 n)
+  (lambda (f) (lambda (x) (f ((n f) x)))))
+(define one (λ (f) (λ (x) (f x)))) ;; applies f once.
+(define two (λ (f) (λ (x) (f (f x)))))
+;(add-1 zero) ; is the same as
+;(lambda (f) (lambda (x) (f ((zero f) x))))
+;(lambda (f) (lambda (x) ((add-1 zero) (((add-1 zero) f) x))))
+
+(define (addition n1 n2)
+  "Returns a function that takes a function f, returns:
+    A function that takes a parameter x, applies f to x n1 + n2 times
+    (n1 and n2 being church numerals, i.e. λfλx forms.)"
+  (lambda (f) (lambda (x)
+                ((n2 f) ((n1 f) x))
+                )))
+
+(define (de-churchify n)
+  ((n (λ (x) (+ x 1))) 0))
diff --git a/ex-2.rkt b/ex-2.rkt
new file mode 100644
index 0000000..ec25d8b
--- /dev/null
+++ b/ex-2.rkt
@@ -0,0 +1,88 @@
+#lang racket
+
+(define (mul-interval x y)
+  (let ((p1 (* (lower-bound x) (lower-bound y)))
+        (p2 (* (lower-bound x) (upper-bound y)))
+        (p3 (* (upper-bound x) (lower-bound y)))
+        (p4 (* (upper-bound x) (upper-bound y))))
+    (make-interval (min p1 p2 p3 p4)
+                   (max p1 p2 p3 p4))))
+
+(define (make-interval a b) (cons a b))
+
+(define (add-interval a b)
+  (make-interval (+ (lower-bound a) (lower-bound b))
+                 (+ (upper-bound a) (upper-bound b))))
+
+;; 2.7
+;; simple definitions.
+(define upper-bound cdr)
+(define lower-bound car)
+
+
+;; 2.8
+(define (interval-diff f a b)
+  (make-interval (- (lower-bound a) (upper-bound b))
+                 (- (upper-bound a) (lower-bound b))))
+
+
+;; 2.9 this one's just about reasoning, no programming, and I feel too lazy to explain
+;; sorry.
+
+;; 2.10 : simple 'nuff.
+(define (div-interval x y)
+  (if (= (lower-bound y) (upper-bound y))
+      (error "WHY? WHY MUST YOU TORTURE ME SO?")
+      (mul-interval
+       x
+       (make-interval (/ 1.0 (upper-bound y))
+                      (/ 1.0 (lower-bound y))))))
+
+;; 2.11 : This one's also somewhat simple.
+;; we ONLY need more than two multiplications
+;; if BOTH intervals have different signs on their upper
+;; and lower bounds. otherwise, naive logic works fine.
+
+(define (mul-interval2 x y)
+  (define (different? n)
+    (and (< (lower-bound n) 0) (< (upper-bound n) 0)))
+  (if (and (different? x) (different? y))
+      (let ((p1 (* (lower-bound x) (lower-bound y)))
+            (p2 (* (lower-bound x) (upper-bound y)))
+            (p3 (* (upper-bound x) (lower-bound y)))
+            (p4 (* (upper-bound x) (upper-bound y))))
+        (make-interval (min p1 p2 p3 p4)
+                       (max p1 p2 p3 p4)))
+      (make-interval (* (lower-bound x) (lower-bound y))
+                     (* (upper-bound x) (upper-bound y)))))
+
+;; 2.12 : simple nuff.
+(define (make-center-width c w)
+  (make-interval (- c w) (+ c w)))
+(define (center i)
+  (/ (+ (lower-bound i) (upper-bound i)) 2))
+(define (width i)
+  (/ (- (upper-bound i) (lower-bound i)) 2))
+(define (make-center-percent c p)
+  (make-center-width c (* c (/ p 100))))
+(define (percent i)
+  (* (/ (width i) 2) 100))
+
+;; 2.13 & 2.14
+(define (par1 r1 r2)
+  (div-interval (mul-interval r1 r2)
+                (add-interval r1 r2)))
+(define (par2 r1 r2)
+  (let ((one (make-interval 1 1)))
+    (div-interval
+     one (add-interval (div-interval one r1)
+                       (div-interval one r2)))))
+
+(define A (make-interval 100 101))
+(define B (make-interval 200 201))
+
+(div-interval A A)
+(div-interval (make-interval 1 1) A)
+(div-interval B A)
+
+