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Chapter 13: Problem 6

Fill in each blank with the correct response. The value of 0! is _____.

Short Answer

Expert verified
The value of 0! is 1.

Step by step solution

01

Understand Factorials

A factorial, denoted by the symbol !, represents the product of all positive integers up to a given number. For example, 4! = 4 脳 3 脳 2 脳 1.
02

Special Case for Zero

By definition, the factorial of zero is a special case. It is defined as equal to 1. This definition is consistent with the properties of factorials and helps maintain the consistency of various mathematical formulas.
03

State the Result

Therefore, the value of 0! is 1.

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Key Concepts

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

Factorial Notation
Factorials are a fundamental mathematical concept often denoted by an exclamation mark (!). When you see a number followed by this symbol, it signifies the product of all positive integers from 1 up to that number. For example, if you come across 5!, it means you must multiply 5, 4, 3, 2, and 1 together. In mathematical terms, 5! equals 5 脳 4 脳 3 脳 2 脳 1, which results in 120.
Understanding factorial notation is essential because it simplifies expressing large products of sequential integers. Factorials are widely used in permutations, combinations, and various algebraic operations.
  • **Example:** 4! equals 4 脳 3 脳 2 脳 1 = 24.
  • **Important Rule:** The factorial of any positive integer n is denoted as n! and is defined as the product of all positive integers up to n.
Zero Factorial
Zero factorial (0!) is an interesting and unique case in factorial notation. By definition, 0! equals 1. This can seem puzzling at first because multiplying no numbers together should intuitively result in zero. However, defining 0! as 1 ensures the consistency of many mathematical formulas and principles.
  • This definition is particularly helpful in combinatorics, where the number of ways to arrange zero objects is exactly one鈥攖he empty arrangement.
  • Additionally, it preserves the recursive relationship of factorials. In general, n! = n 脳 (n-1)!. When n is 1, we get 1! = 1 脳 0!, and to make this true, 0! must be 1.
Understanding why 0! equals 1 stabilizes the use of factorials in various mathematical contexts and provides a robust foundation for more advanced algebraic operations.
Mathematical Definitions
Mathematical definitions lay the groundwork for understanding complex concepts. For factorials, several key definitions and properties help in grasping their application:
  • **Factorial Function:** This is a function that multiplies a series of descending natural numbers. The factorial of a non-negative integer n, denoted as n!, is defined as the product of all positive integers less than or equal to n.
  • **Recursive Property:** This property states that for any positive integer n, n! = n 脳 (n-1)!.
  • **Base Case:** The factorial function's base case is defined as 0! = 1.
These definitions allow for easier manipulation of mathematical expressions involving factorials.
Recognizing and understanding these foundational definitions ensure that one can navigate more complex algebraic and mathematical problems with confidence.

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