Recursion on Numbers

• Exercises
  • Exercise 0: Preparation
  • Exercise 1: Power of two
  • Exercise 2: Counting down
  • Exercise 3: Filling lists
  • Exercise 4: Counting To
  • Exercise 5: Counting Between
  • Exercise 6: Counting To, Revisited
  • Exercise 7: How Many Digits?
• Notes

Exercises

Exercise 0: Preparation

You may wish to scan through the reading on recrusion and the second reading on recursion.

Exercise 1: Power of two

Using natural-number recursion, define and test a recursive Scheme procedure named power-of-two that takes a natural number as its argument and returns the result of raising 2 to the power of that number. (For instance, the value of (power-of-two 3) should be 2³, or 8.)

It's possible to define this procedure non-recursively, using Scheme's primitive expt procedure, but the point of the exercise is to use recursion.

Exercise 2: Counting down

Define and test a Scheme procedure named count-down that takes a natural number as argument and returns a list of all the natural numbers less than or equal to that number, in descending order:

> (count-down 5)
(5 4 3 2 1 0)
> (count-down 0)
(0)

Exercise 3: Filling lists

Define and test a Scheme procedure named fill-list that takes two arguments, the second of which is a natural number, and returns a list consisting of the specified number of repetitions of the first argument:

> (fill-list 'sample 5)
(sample sample sample sample sample)
> (fill-list (list 'left 'right) 3)
((left right) (left right) (left right))
> (fill-list null 1)
(())

Exercise 4: Counting To

Define and test a recursive Scheme procedure that takes a natural number as argument and returns a list of all the natural numbers that are strictly less than the argument, in ascending order. (The traditional name for this procedure is iota -- another Greek.letter.)

Exercise 5: Counting Between

What is the value of the call (count-from -10 10)?

a. Write down what you think that it should be.

b. Copy the definition of count-from into DrScheme and use it to find out what the call actually returns.

Exercise 6: Counting To, Revisited

Using count-from, define and test a Scheme procedure that takes a natural number as argument and returns a list of all the natural numbers that are strictly less than the argument, in ascending order. Note that your procedure should use count-from as a helper.

Exercise 7: How Many Digits?

Here is the definition of a procedure that computes the number of digits in the decimal representation of number:

(define number-of-decimal-digits
  (lambda (number)
    (if (< number 10)
        1
        (+ (number-of-decimal-digits (quotient number 10)) 1))))

a. Test this procedure.

The definition of number-of-decimal-digits uses direct recursion.

b. Describe the base case of this recursion.

c. Identify and describe the way in which a simpler instance of the problem is created for the recursive call.

d. Explain how the procedure correctly determines that the decimal numeral for the number 2000 contains four digits.

e. What preconditions does number-of-decimal-digits impose on its argument?

Notes