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Chapter 5: Problem 9

Find the value of each factorial. \(0 !\)

Short Answer

Expert verified
The value of 0! is 1.

Step by step solution

01

Understand the definition of a factorial

Begin by understanding that the factorial of a number is the product of all positive integers from 1 up to that number. It is denoted by an exclamation mark (!). For example, the factorial of 5 is written as 5! and is equal to 5 脳 4 脳 3 脳 2 脳 1.
02

Note the special case of 0!

For the number 0, there is a special rule in mathematics which states that the factorial of 0 is equal to 1. This is a defined convention.
03

Write the solution

By definition, we have: 0! = 1

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factorial Definition
To solve the given exercise, we need to start with understanding what a factorial is. A factorial is a mathematical operation that multiplies a given number by every positive integer less than itself. This operation is denoted by an exclamation mark (!). For instance, to calculate the factorial of 5, written as 5!, you would multiply 5 by 4, then 3, then 2, and finally 1. So, the calculation would look like this: 5 脳 4 脳 3 脳 2 脳 1 = 120.

In general, the factorial of a positive integer n is defined as:
.
This makes it a recursive function, where each step involves the product of the number and the factorial of the preceding number.
Special Case
In understanding factorials, it is important to note there are special cases. One very important special case is when the number is 0. The rule for 0 is unique because, unlike other numbers where you multiply down to 1, the factorial of 0 is defined to be exactly 1.

This might seem surprising at first, but it fits well within the broader definition of factorials and ensures consistency in various mathematical contexts. This special rule is essential for many equations and mathematical functions to work correctly. So remember, whenever you see 0!, it is always equal to 1 by definition.
0 Factorial
To delve deeper into understanding why 0! = 1, let鈥檚 consider how factorial is used in combinations and permutations. In combinatorial mathematics, the number of ways to arrange 0 objects, which is 1 way (doing nothing), supports the convention that 0! should equal 1.

Another way to understand this is through the property of recursive functions. For any positive integer n, the factorial can be expressed as: .
To maintain this property for n = 1, we must have:

    1! = 1 脳 0!
    Therefore, 1 = 1 脳 0!, so 0! must be 1.
Understanding these principles ensures that you can solve not just simple problems, but also more complex ones that involve factorial calculations.

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