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12.2: Simple Math Recursion

  • Page ID
    117600
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    Learning Objectives

    By the end of this section you should be able to

    • Identify a recursive case and a base case in a recursive algorithm.
    • Demonstrate how to compute a recursive solution for the factorial function.

    Calculating a factorial

    The factorial of a positive integer is defined as the product of the integer and the positive integers less than the integer.

    Ex: 5! = 5 * 4 * 3 * 2 * 1

    Written as a general equation for a positive integer n: n! = n * (n - 1) * (n - 2) * . . . * 1

    The above formula for the factorial of n results in a recursive formula: n! = n * (n - 1)!

    Thus, the factorial of n depends upon the value of the factorial at n - 1. The factorial of n can be found by repeating the factorial of n - 1 until (n - 1)! = 1! (we know that 1! = 1). This result can be used to build the overall solution as seen in the animation below.

    Checkpoint: Finding the factorial of 5
    Concepts in Practice: Recognizing recursion

    Can the following algorithms be written recursively?

    1.
    The summation of 1 + 2 + 3 + . . . + (n - 1) + n where n is a positive integer.
    2.
    Listing the odd numbers greater than 0 and less than a given number n.
    3.
    Listing the primary numbers (prime numbers) greater than 3.
    4.
    Listing the Fibonacci sequence of numbers.

    Defining a recursive function

    Recursive algorithms are written in Python as functions. In a recursive function different actions are performed according to the input parameter value. A critical part of a recursive function is that the function must call itself.

    A value for which the recursion applies is called the recursive case. In the recursive case, the function calls itself with a smaller portion of the input parameter. Ex: In the recursive function factorial(), the initial parameter is an integer n. In the function's recursive case, the argument passed to factorial() is n - 1, which is smaller than n.

    A value of n for which the solution is known is called the base case. The base case stops the recursion. A recursive algorithm must include a base case; otherwise, the algorithm may result in an infinite computation.

    To calculate a factorial, a recursive function, factorial() is defined with an integer input parameter, n. When n > 1, the recursive case applies. The factorial() calls itself with a smaller argument, n - 1. When n == 1, the solution is known because 1! is 1; therefore, n == 1 is a base case.

    Note: 0! is defined to be 1; therefore, n == 0 is a second base case for factorial(). When n < 1, an error is returned.

    Checkpoint: Factorial using recursion
    Concepts in Practice: Programming recursion

    For the questions below, the function rec_fact() is another recursive function that calculates a factorial. What is the result of each definition of rec_fact() if n = 17 is the initial input parameter?

    5.
    def rec_fact(n):
      return n * rec_fact(n - 1)
    1. 355687428096000
  • 0
  • no result / infinite computation
  • 6.
    def rec_fact(n):
      if n < 0:
        print("error")
      elif n == 0:
        return n
      else:
        return n * rec_fact(n - 1)
    1. 355687428096000
    2. 0
    3. no result / infinite computation
    7.
    def rec_fact(n):
      if n < 0:
        print("error")
      elif n == 0:
        return 1
      else:
        return n * rec_fact(n - 1)
    1. 355687428096000
    2. 0
    3. no result / infinite computation
    8.
    def rec_fact(n):
      if n < 0:
        return -1
      else:
        return n * rec_fact(n - 1)
    1. 355687428096000
    2. 0
    3. no result / infinite computation
    Try It: Recursive summation

    Write a program that uses a recursive function to calculate the summation of numbers from 0 to a user specified positive integer n.

    Try It: Digits

    Write a program that computes the sum of the digits of a positive integer using recursion.

    Ex: The sum of the digits of 6721 is 16.

    Hint: There are 10 base cases, which can be checked easily with the right condition.


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