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In any recursive definition, the base case is extremely important. It is where the recursion stops.
The base case provides a simple solution for the smallest possible problem. For our given problem of summing an integer and a nonnegative integer, the simplest case is when the nonnegative integer is zero.
For example:
- When adding 0 to any integer, say m, the result is simply m.
We write this in mathematical notation as: \(S_m(0) = m\)
This tells us that whenever the second number (the nonnegative integer) is zero, the sum is just the first number.

The recursive step is the heart of recursion. It reduces a complex problem into simpler versions of itself.
To define the recursive step for our sum function, let's consider what happens when we add a nonnegative integer, n, to another integer, m:
- We can break this down by first thinking of adding (n - 1) to m, and then adding 1 to the result.
This is expressed mathematically as: \(S_m(n) = S_m(n-1) + 1\)
Here, \(S_m(n-1)\) represents the sum of m and (n-1), and we add 1 to this result because n is just one more than (n-1).
This step helps in building the solution by solving simpler instances of the problem until reaching the base case.

Summing integers, especially using recursion, helps in better understanding the concept of breaking down problems.
Here, we are interested in the sum of an integer m and a nonnegative integer n.
Let's illustrate this with an example:
- Suppose m = 5 and n = 3. To find \(S_5(3)\), we follow the recursive steps:
1. \(S_5(3) = S_5(2) + 1\) 2. \(S_5(2) = S_5(1) + 1\) 3. \(S_5(1) = S_5(0) + 1\) 4. And from our base case, \(S_5(0) = 5\)Combining these steps, we can trace back to find the result:
\(S_5(1) = 5 + 1 = 6\)
\(S_5(2) = 6 + 1 = 7\)
\(S_5(3) = 7 + 1 = 8\)
Ultimately, \(S_5(3) = 8\)
This example shows how recursion simplifies the process through breaking down and adding incrementally.