Recursive Cases and Base Cases in Elementary Math Functions

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What is recursion?

Recursion is a problem-solving approach in which a function calls itself, typically on a smaller or simpler input.

How do the recursive case and base case differ?

The recursive case makes progress by calling the function again on a reduced problem. The base case returns a known answer and ends the calls.

What is the recursive definition of factorial?

For $n > 1$, $n! = n \times (n - 1)!$. The base values include $0! = 1$ and $1! = 1$.

How can a recursive function sum integers from 0 through n?

Use $S(0) = 0$ as the base case and, for $n > 0$, return $n + S(n - 1)$.

How does recursion sum the digits of a nonnegative integer?

Add the last digit, found with `n % 10`, to the digit sum of the remaining number, found with `n // 10`. Stop when the number has one digit.

Quiz questions

1. What is the role of a base case in a recursive function?Answer: It returns a known result and ends further recursive calls.
2. What value does a function that sums integers from 0 through n return for n = 4?Answer: 10
3. What does `digit_sum(6721)` return when the function adds the digits of a nonnegative integer?Answer: 16
4. For a nonnegative integer, what does `n // 10` do in a decimal digit-sum function?Answer: Removes the final decimal digit from the integer.
5. When a recursive factorial calculation reaches its base case, how are the pending results combined?Answer: Each returning call multiplies its input by the smaller factorial result.