Question

Implement the factorial function. The factorial of \(n\), written as \(n !\), is defined as $$ n !=n(n-1)(n-2) \cdots 2 \cdot 1 $$ with the special cases $$ 1 !=1, \quad 0 !=1 $$ For example, \(4 !=4 \cdot 3 \cdot 2 \cdot 1=24\), and \(2 !=2 \cdot 1=2\). Write a function fact \((n)\) that returns \(n !\). Return 1 immediately if \(x\) is 1 or 0 , otherwise use a loop to compute \(n !\). Name of program file: fact.py. Remark. You can import a ready-made factorial function by >> from math import factorial >>> factorial (4) 24

Short answer

Create a function `fact(n)` in a file called `fact.py` to compute factorial iteratively; handle 0 and 1 by returning 1 immediately.

Step-by-step solution

Understanding Factorial Definition

Factorial of a non-negative integer is the product of all positive integers less than or equal to that number. For example: \( 4! = 4 \times 3 \times 2 \times 1 = 24 \). Also note the special cases: \( 0! = 1 \) and \( 1! = 1 \).

Creating Function Structure

Define a function named `fact` that takes one parameter, \( n \). Start with the function signature: `def fact(n):`. This function will compute and return the factorial of \( n \).

Handling Special Cases

Inside the `fact` function, check if \( n \) is either 0 or 1 using an if statement: `if n == 0 or n == 1:`. If true, return 1 immediately since these cases have predefined results of 1.

Implementing Iterative Computation

If \( n \) is greater than 1, initialize a variable `result` to 1. Use a loop to multiply `result` by each integer from 2 to \( n \) and update `result` with the calculated product.

Returning the Result

After the loop completes, return the value of `result` from the function. This value will be the factorial of \( n \).

Writing the Full Code

Combine all previous steps into the following complete code for `fact.py`:```pythondef fact(n): if n == 0 or n == 1: return 1 result = 1 for i in range(2, n + 1): result *= i return result```This function now correctly calculates and returns the factorial of any non-negative integer \( n \).

Key concepts

Iterative Computation

Iterative computation is crucial when dealing with tasks such as calculating factorials. An iteration is the repeated execution of a set of instructions, leading to a gradual build-up of the complete computation. For the factorial function, it's all about multiplying a series of descending natural numbers starting from the given number itself, down to 1.

In the iterative method for factorial computation, you start with a base case, often initializing a result variable to 1. You then iterate over a sequence of numbers generated by the `range` function. In Python, the function call `range(2, n+1)` produces a sequence of numbers from 2 to n. During each iteration of the loop, the result is multiplied by the current number, thereby building up the product step by step.

Using loops is efficient in reducing the potential for human error that might arise from manual calculation. It also allows the function to accommodate any non-negative integer within the computational limits of the environment.

Special Cases in Mathematics

In mathematics, special cases are often exceptions to generalized rules, and they need to be handled carefully to avoid erroneous results. When it comes to factorials, two noteworthy special cases are \(0!\) and \(1!\), both of which are defined to be 1.

Why these definitions? For mathematical consistency purposes, particularly when working with permutations and combinations. Having \(0!\equiv 1!\equiv 1\) allows for uniform mathematical expressions that include factorial operations and simplifies them significantly.

• 0! = 1: By defining the empty product to be 1, it resolves many counting scenarios elegantly.
• 1! = 1: Since there is only one way to arrange a single item, the factorial logically results in 1.

Including these special cases in your code ensures that the program doesn't break or return an error when presented with these inputs. It is crucial to include an `if` condition to handle these exceptions before proceeding to other computations.

Function Definition in Python

Defining functions in Python is a fundamental skill which enables code reuse, abstraction, and modular design. With functions, you can break down complex problems into smaller, manageable parts.

The factorial function in Python is defined using the `def` keyword, followed by the function name and parentheses enclosing any parameters, like `def fact(n):`. This definition sets up the structure that allows us to encapsulate the factorial logic and to use it repeatedly by calling `fact(n)` with different values.

A function in Python includes:

• Function Signature: Defines the function's entry structure, typically including the name and any parameters it accepts. For instance, `fact(n)`.
• Docstring (optional): A brief description enclosed in triple quotes at the beginning of the function to explain its purpose.
• Logic: The block of code that executes the intended task, such as computing the factorial using iteration for values beyond the special cases.
• Return Statement: Ends the function execution and returns the result of the computation back to the caller.

Defining functions aptly enables cleaner, more readable, and organized code, which is crucial when programs become complex or involve multiple equations.