Question

Write a recursive definition of \(x^{n}\), where \(n \geq 0\). Hint: How do you compute \(x^{n}\) from \(x^{n-1}\) ? How does the recursion terminate?

Short answer

\(x^{0} = 1; x^{n} = x \times x^{n-1}\) for \(n > 0\).

Step-by-step solution

Understanding the Base Case

For any power of a number, there is a base condition where the exponent is zero. For any number \(x\), \(x^{0}\) is 1 by definition. This gives us our base case for recursion.

Recursive Case Analysis

To calculate \(x^{n}\) for \(n > 0\), we recognize that it can be derived from \(x^{n-1}\) by multiplying \(x\). Therefore, the recursive step can be defined as \(x^{n} = x \times x^{n-1}\).

Writing the Recursive Formula

Using the base case and the recursive step, we define the recursive formula as:1. \(x^{0} = 1\) (base case)2. \(x^{n} = x \times x^{n-1}\) for \(n > 0\).

Termination Condition

The recursion will terminate when \(n\) reaches 0, at which point \(x^{0} = 1\). This ensures the recursive process has an end point.

Key concepts

Base Case

The base case is a crucial aspect of any recursive function. It's the simplest instance of the problem and acts as a stopping point for the recursion. In the context of exponentiation, the base case is when the exponent is zero.

According to mathematical definition, any number raised to the power of zero equals one, i.e., \( x^{0} = 1 \). This is a universally accepted rule and forms the foundation for our recursive approach.

• The base case provides a solution for the smallest value of the input. Without it, a recursive function would run infinitely.
• It is analogous to the lowest step in a staircase. Once you reach this step, you can no longer go downward.
• In programming a recursive function for exponentiation, identifying and coding the base case is the first step.

Recursive Case

The recursive case is where the magic happens in a recursive function. This is the part of the function that breaks down a complex problem into smaller, more manageable ones of the same type. For our exponentiation problem, the recursive case is based on how \( x^{n} \) can be computed from \( x^{n-1} \).

Suppose you want to calculate \( x^{3} \). Instead of multiplying \( x \) three times in a loop, think of \( x^{3} \) as \( x \times x^{2} \). This relationship continues until you reach the base case.

• The recursive step for exponentiation is formulated as \( x^{n} = x \times x^{n-1} \).
• Each recursive call reduces the exponent \( n \) by 1, making the problem smaller and smaller.
• This step depends on reaching the base case to eventually halt.

The recursive case works in a stack-like manner; each call waits for the result of the next until reaching the base case. Then, the results roll back up the stack, multiplying \( x \) back into the expression at each level.

Exponentiation

Exponentiation is a mathematical operation involving two numbers, the base \( x \) and the exponent \( n \). It signifies multiplying the base by itself \( n \) times. In the recursion context, exponentiation can be efficiently handled as a sequence of multiplications, leveraging the computer's ability to recall previous calculations.

The recursive definition of exponentiation allows for a clean and elegant solution to what might otherwise require iterative loops.

• Exponentiation by recursion optimizes memory and execution time, especially for larger values of \( n \).
• The simple and clear rule \( x^{n} = x \times x^{n-1} \) naturally aligns with how recursion reduces problems in size.

For concrete applications, such as computing powers in algorithms, efficient handling through recursion proves beneficial, provided base cases and termination conditions are well-defined.