1. Computing Square Roots
= the y such that
and y2=x
The mathematical definition does not describe a procedure.
Declarative : describing properties of things
Imperative: describing how to do things
(define (sqrt x)
(the y (and (>= y 0)
(= (square y) x))))

2. Newton’s Method Successive approximation
Guess y for the square root of a number x
Get a better guess by averaging y with x/y
Guess Quotient Average
1 2/1=2 (1+2)/2=1.5
1.5 2/1.5= ( )/2=1.4167
/1.4167= ( )=1.4142
/1.4142=...

3. (define (sqrt-iter guess x)
(if (good-enough? guess x)
guess
(sqrt-iter (improve guess x)
x)))
(define (improve guess x)
(average guess (/ x guess)))
(define (average x y)
(/ (+ x y) 2))
(define (good-enough? guess x)
(< (abs (- (square guess) x)) 0.001))
(define (sqrt x)
(sqrt-iter 1.0 x))

4. Decomposition of sqrt sqrt Sqrt-iter improve Good-enough? square abs
average

5. Procedural Abstraction
Procedures as “black boxes”
Each procedure accomplishes an identifiable task.
The details can be suppressed at the moment.
;; implementation #1
(define (square x) (* x x))
;; implementation #2
(define (square x)
(exp (double (log x))))
(define (double x) (+ x x))

6. Formal Parameters
The meaning of a procedure should be independent of the parameter names (bound variables) used.
The meaning of a procedure definition is unchanged if a bound variable is consistently renamed throughout the definition.
If a variable is not bound, we say that it is free.
In a procedure definition, the formal parameters have the body of the procedure as their scope.
(define (square x) (* x x))
(define (square y) (* y y))