Question

The exponentiation algorithms in this section are based on performing exponentiation by means of repeated multiplication. In a similar way, one can perform integer multiplication by means of repeated addition. The following multiplication procedure (in which it is assumed that our language can only add, not multiply) is analogous to the expt procedure: (define (* a b) (if \((=b \quad 0)\) 0 \((+a(* a(-b \quad 1)))))\)

Short answer

The procedure defines integer multiplication using repeated addition, and works recursively.

Step-by-step solution

Understanding the Problem

The task is to understand the recursive procedure defined for multiplying two integers, \(a\) and \(b\), using repeated addition. The function checks if \(b\) equals 0 (base case) and returns 0 in that case. Otherwise, it adds \(a\) to the result of multiplying \(a\) with \(b-1\).

Identifying the Base Case

The base case of the recursion is when \(b = 0\). The function returns 0 in this case because multiplying any number by 0 results in 0.

Recursive Call Analysis

If \(b\) is not 0, the function calculates \(a\) times \(b\) by adding \(a\) repeatedly. Specifically, it computes \((+ a (* a (- b 1)))\), which means it adds \(a\) to the result of recursively calling the same function with \(b\) decreased by 1.

Example Calculation

Let's consider an example to solidify our understanding. Suppose we want to multiply \(a = 3\) and \(b = 2\). Initially, \(b\) is not 0, so it adds 3 to the recursive call \(* 3 (2-1)\). This will now calculate \(* 3 1\). In the next recursive step, \(* 3 1\) becomes \((+ 3 (* 3 0))\), and since \(b = 0\), it returns 0. So, \(* 3 1\) results in 3, making \(* 3 2\) result in 6.

Summarizing the Procedure

The multiplication function effectively sums the number \(a\), \(b\) times through recursive calls. Each recursion reduces \(b\) by 1 until it reaches the base case. The recursive nature of the procedure builds up the multiplication through addition.

Key concepts

Exponentiation

Exponentiation is the mathematical operation involving raising a base number to the power of an exponent. In simple terms, it's multiplying a number by itself a certain number of times. For example, if we have a base number 2 and an exponent 3, it means we multiply 2 by itself three times, resulting in 8. In terms of algorithms, exponentiation is similar to recursive procedures that help simplify complex calculations into smaller, more manageable steps.

Using recursive algorithms for exponentiation often involves both a base and a recursive case. They break down the task of multiplying the base number by itself repeatedly, reducing the problem size until it hits the base case (e.g., when the exponent equals 0). This systematic approach allows programmers to implement powerful algorithms efficiently, making use of simple operations like multiplication.

Integer Multiplication

Integer multiplication can be achieved through repeated addition, much like how exponentiation involves repeated multiplication. When we multiply two integers using recursive procedures, we repeatedly add one integer, a specified number of times. For instance, the multiplication of 3 and 4 is equivalent to adding 3, four times as in 3 + 3 + 3 + 3.

When an algorithm can only perform addition, integer multiplication is transformed into a repeated addition problem. This showcases how simple operations can build up to solve more complex problems like multiplication. Recursion helps here by decomposing the task of addition into smaller parts, easily manageable by the computer or the human mind.

Base Case

In a recursive algorithm, the base case is the condition that provides a straightforward answer to the problem, effectively stopping further recursion. It serves as an anchor that ensures the recursive algorithm doesn't run infinitely. For example, in integer multiplication by repeated addition, the base case is when the integer being decremented (usually the second operand) reaches 0.

The function checks this base case first. If it is true, it returns a simple value, often 0 or 1, depending on the problem. This base case resolves the smallest instance of the problem and allows for the cumulative build-up of the result as the recursion "unwinds.", leading to a complete solution.

Recursive Procedures

Recursive procedures are algorithms that solve complex problems by breaking them into smaller sub-problems that have the same structure. They rely on two key ideas: a base case and a recursive call that moves the problem towards the base case.

A recursive call is made within the procedure itself. This self-referencing ensures that each step of the recursion acts as a mini-loop, processing the variables until the base case is met. For integer multiplication, the recursive procedure repeatedly calls itself with one operand reduced by 1 until it reaches 0.

Recursive algorithms are especially efficient because they reduce repetitive coding and simplify the process of solving complex mathematical equations into a series of simpler, repeated steps.