Question

Write a recursive algorithm to multiply two positive integers m and n using repeated addition. Specify the base case and the recursive case.

Short answer

A recursive algorithm for multiplying two integers using addition involves reducing to zero additively.

Step-by-step solution

Identify the Problem

We need to multiply two positive integers, \(m\) and \(n\), using a recursive algorithm that relies on repeated addition.

Understand the Base Case

The base case is the condition where the recursion will stop. In multiplication through repeated addition, when \(n = 0\), the result should be 0, because any number multiplied by zero is zero.

Define the Recursive Case

The recursive case addresses how to reduce the problem into a smaller instance. When \(n > 0\), we can express the multiplication \(m \times n\) as \(m + (m \times (n-1))\). This reduces the problem by decrementing \(n\) each time, while accumulating the sum.

Implement the Base Case in the Algorithm

The algorithm checks if \(n = 0\). If true, it returns 0 because this signifies that multiplying by zero results in zero, which halts further recursion.

Implement the Recursive Case in the Algorithm

If \(n > 0\), the algorithm calls itself with the parameters \(m\) and \(n-1\) and adds \(m\) to the result of this recursive call. This step effectively adds \(m\) to itself \(n\) times.

Construct the Full Algorithm

The full recursive algorithm is as follows:```function Multiply(m, n): if n == 0: return 0 else: return m + Multiply(m, n-1)```This function effectively calculates \(m \times n\) by adding \(m\) to itself \(n\) times using recursion.

Key concepts

Base Case in Recursion

When dealing with recursive algorithms, understanding the base case is crucial. The base case is the simplest instance of the problem—a scenario where further recursive calls are unnecessary. Without it, the algorithm would end up in an infinite loop, as it would keep calling itself indefinitely. In the context of multiplying two integers through recursion, the base case occurs when the multiplier, \(n\), reaches zero.

Why is this so important? At this point, the multiplication should yield zero because any number multiplied by zero equals zero. Thus, the base case for our recursive multiplication function is specified as: if \(n = 0\), return 0. This base case forms the foundation of our algorithm, ensuring that the recursion stops correctly and produces meaningful results. In short, the base case prevents endless function calls and establishes the condition for terminating the recursion.

Recursive Case in Algorithms

The recursive case is another fundamental element of a recursive algorithm. After establishing the base case of when to stop recursion, the recursive case defines how to divide the problem into smaller parts, inching closer to that base case with each step.

For the problem of multiplying two numbers using recursion, the recursive case is where the actual repeated addition occurs. Here, the multiplication of \(m\) by \(n\) can be broken down into:

• \( m \times n = m + (m \times (n-1)) \)

This statement essentially translates to you adding \(m\) to the result of multiplying \(m\) by \(n-1\). Each recursive call reduces \(n\) by 1, effectively performing addition \(n\) times.

The recursive case drives the computation forward until the base case condition is met. This segment of your algorithm decides how the problem is diminished with every new call and consequently does most of the computational work.

Using Recursion for Multiplication

Using recursion to multiply through repeated addition is an insightful exercise to grasp the mechanics of recursive algorithms. Recursion breaks down a problem into similar, smaller instances of itself until a base condition halts the chain.

Why use recursion for multiplication? While not the most efficient method, it's excellent for demonstrating how processes can be simplified using basic operations—in this case, addition. The recursive algorithm calculates the product of two numbers by cumulatively adding one of the numbers (\(m\)) to itself \(n\) times.

The recursion process can be visualized as follows:

• Start with \(n\), the number of times \(m\) will be added.
• Subtract 1 from \(n\) with every call, adding \(m\) each time.
• Continue until \(n\) becomes zero.

This technique showcases recursion's potential to solve problems through elegantly structured repeated operations, fostering a deeper understanding of both recursive thinking and the fundamental workings of multiplication.