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Chapter 5: Problem 10

Find the value of each factorial. \(1 !\)

Short Answer

Expert verified
1! = 1

Step by step solution

01

Understanding Factorials

A factorial, represented by the symbol (!), indicates the product of all positive integers up to that number. For example, (3!) is calculated as 3 * 2 * 1.
02

Identify the Factorial to Find

The problem asks to find the value of (1!). According to the definition, (!), the series needs to include all positive integers from 1 down to 1.
03

Calculate 1!

By definition, (1!) equals 1 because there is only one integer (1) involved in the product.

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Key Concepts

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

Factorial Definition
Factorials are a fundamental concept in mathematics, especially in fields like combinatorics, algebra, and calculus. A factorial of a number is the product of all positive integers up to and including that number. It is denoted by an exclamation mark after the number, like this: n!. For example, 5! (read as 'five factorial') is calculated as 5 脳 4 脳 3 脳 2 脳 1. Factorials grow very quickly with larger numbers.
Factorial Calculation
Calculating a factorial involves multiplying a series of descending positive integers. Let's break down how you would calculate the factorial of various numbers:
  • 3! = 3 脳 2 脳 1 = 6
  • 4! = 4 脳 3 脳 2 脳 1 = 24
  • 5! = 5 脳 4 脳 3 脳 2 脳 1 = 120
Notice that each step involves multiplying the current number by the factorial of the number that comes after it. This can be formally expressed as: n! = n 脳 (n-1)!. For instance, 4! = 4 脳 3! and since 3! = 6, we get 4! = 4 脳 6 = 24. The calculation follows a recursive pattern, proving helpful in complex problems and computer algorithms.
Positive Integers
Understanding positive integers is crucial for grasping the concept of factorials. Positive integers are the set of all whole numbers greater than zero. This set includes numbers like 1, 2, 3, 4, and so on, extending infinitely. These numbers do not include fractions, decimals, or negative values. When calculating factorials, only positive integers are considered. For example, for calculating 1!, you only include the integer 1. Similarly, for 5!, you use the integers 1, 2, 3, 4, and 5. Thus, the factorial of a positive integer n is the product of all positive integers less than or equal to n.

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Most popular questions from this chapter

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