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Chapter 14: Problem 6

Fill in each blank with the correct response. The value of \(0 !\) is _______.

Short Answer

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1

Step by step solution

01

Understand the concept of factorial

A factorial, denoted by an exclamation mark (!), is the product of all positive integers up to a given number. For instance, the factorial of 3 (written as 3!) is calculated as follows: 3! = 3 脳 2 脳 1 = 6.
02

Special case of 0 factorial

It is a common convention in mathematics that the factorial of 0, written as 0!, is defined to be 1. This is a special case that has been defined to make various mathematical formulas and functions work correctly.
03

Apply the definition

Using the definition from Step 2, 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 definition
A factorial is a special kind of mathematical operation, symbolized by an exclamation mark (!). When you see a number followed by this symbol, it means you should multiply all positive integers up to that number. For example, if we take the number 5, the factorial of 5 (written as 5!) is calculated as follows: 5! = 5 脳 4 脳 3 脳 2 脳 1. The result of this multiplication is 120. Factorials are used in various areas, such as permutations and combinations, to calculate the number of possible arrangements or selections.
special case of 0 factorial
Understanding the factorial of 0 is unique and important in mathematics. Unlike other numbers, the factorial of 0, written as 0!, is defined to be 1. This might seem a bit strange at first. However, this definition helps to make many mathematical formulas and functions work correctly. For example, in combinations and permutations, having 0! as 1 ensures the formulas hold true even when the numbers involved reach zero. Think of it as a mathematical convenience, making sure everything stays consistent and reliable.
positive integers product
When calculating a factorial, you are essentially finding the product of positive integers. Let's break this down with an example: suppose you need to find the factorial of 4, written as 4!. You will multiply all the positive integers up to 4: 4 脳 3 脳 2 脳 1. The result, 24, is the product of these numbers. It's like a series of multiplications starting from 1 up to the number in question. This fundamental concept helps in understanding more complex mathematical areas and is a building block for further studies in mathematics. Elementary math problems often use this to demonstrate factorial growth and the basics of multiplication sequences.

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Most popular questions from this chapter

Determine an expression for the general term \(a_{n}\) of each sequence \(7,14,21,28, \ldots\)

Evaluate each expression. $$ 4 ! $$

Determine an expression for the general term \(a_{n}\) of each sequence \(-8,-16,-24,-32, \ldots\)

Fill in each blank with the correct response. For the geometric sequence having \(a_{n}=(-2)^{n},\) the term \(a_{5}=\) _____.

Find the indicated term for each sequence. $$ a_{n}=(5 n-2)(3 n+1) ; \quad a_{10} $$
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