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Chapter 11: Problem 7

Evaluate. $$0 !$$

Short Answer

Expert verified
0! = 1

Step by step solution

01

Understand the Factorial Definition

The factorial of a non-negative integer is the product of all positive integers less than or equal to that integer. It is denoted by an exclamation mark (!).
02

Special Case Identification

Recognize that 0! is a special case in the factorial operation. By definition, the factorial of 0 is considered to be 1.
03

Write the Expression

Using the definition, write down the expression 0! = 1.

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Key Concepts

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

Factorial Definition
The concept of a factorial is essential in mathematics. Factorial, denoted by an exclamation mark (!), represents the product of all positive integers up to a given number. For example, the factorial of 5 (written as 5!) is calculated as 5 脳 4 脳 3 脳 2 脳 1, which equals 120.

Factorials are commonly used in permutations, combinations, and other areas of mathematics and computer science. When you see the exclamation mark (!), it tells you to multiply all whole numbers from the given number down to 1. Understanding the factorial is crucial for solving many mathematical problems.
Special Cases in Mathematics
In mathematics, special cases can often simplify problems or clarify unique scenarios. For factorials, some special cases include:
  • 1! = 1: This is because there is only one number to multiply, which is 1 itself.
  • 2! = 2: This means 2 脳 1, which equals 2.
  • 0!: Known as the factorial of zero, this special case is defined differently from other factorials.

Special cases, like 0!, play a significant role in simplifying mathematical theories and ensuring consistency across various equations and formulas.
Factorial of Zero
One of the most intriguing special cases in factorials is the factorial of zero, written as 0!. Unlike other numbers, 0! is defined to be equal to 1.

This might seem confusing at first, but defining 0! = 1 helps maintain consistency in mathematical equations, especially in permutations and combinations. For instance, the number of ways to arrange zero objects is one way: do nothing.

Therefore, while the factorial of zero may appear to break the pattern of decreasing products found in other factorial values, it is an essential definition that helps in various types of mathematical calculations and maintains the integrity of mathematical rules.

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