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Chapter 14: Problem 96

Using your own words, write a definition of \(n !\)

Short Answer

Expert verified
The factorial of a number \\ n \\, denoted \\ n! \\\, is the product of all positive integers less than or equal to \\ n \\, and \\ 0! = 1 \\.

Step by step solution

01

Understanding the Factorial Notation

The factorial notation, represented by an exclamation mark (!), refers to the product of all positive integers up to a specific number. For example, the expression \( n! \) means multiplying all integers from 1 to \ n \.
02

Breaking Down the Factorial

To grasp the concept of a factorial, consider the number 4. The factorial of 4, written as \( 4! \), is calculated as 1 * 2 * 3 * 4. Factorials grow rapidly since they imply not just addition, but multiplication of increasingly larger numbers.
03

Writing the General Formula

The general formula to define \( n! \) is \ n! = n \times (n-1) \times (n-2) \times \ldots \times 2 \times 1 \. This formula works for any positive integer \( n \). Special case: \( 0! \) is defined as 1, which is crucial for consistency in mathematical formulas.

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Key Concepts

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

Positive Integers
Factorials are a fascinating mathematical concept closely linked with positive integers. When we talk about positive integers in the context of factorials, we are discussing whole numbers greater than zero. These integers are the building blocks of factorial operations because the factorial notation, represented with an exclamation mark (!), requires multiplying a series of consecutive positive integers.
To illustrate:
  • The factorial of 3, written as 3!, equals 3 * 2 * 1 = 6.
  • The factorial of 5, depicted as 5!, results in 5 * 4 * 3 * 2 * 1 = 120.
As you can see, each factorial is a product of the number itself and all positive integers that come before it.
General Formula
The general formula for factorials provides a consistent way to calculate the factorial of any positive integer. Written mathematically, this formula is expressed as: \[ n! = n \times (n-1) \times (n-2) \times \ldots \times 3 \times 2 \times 1 \] This formula allows us to accurately determine the factorial of a given number, ensuring that we multiply all positive integers in sequence from the number itself down to one.
The importance of this formula cannot be overstated as it is the fundamental principle that governs calculations involving factorials in many areas of mathematics and statistical analyses.
Special Case
Factorials have a special case that often surprises people not familiar with its mathematical rationale: the factorial of zero. Conventionally, factorials involve multiplication across a range of positive integers, but when it comes to zero, there are no positive integers to multiply.
Even so, by mathematical definition, we set:
  • \( 0! = 1 \)
This might look puzzling initially, but this definition is crucial for maintaining consistency across mathematical expressions and equations.
For instance, in the expansions of series or in combinatorial mathematics, having \( 0! = 1 \) ensures that formulas work correctly even at their most basic limits.

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Most popular questions from this chapter

On its first swing to the right, a pendulum swings through an arc of 96 inches. Each successive swing, the pendulum travels \(\frac{99}{100}\) as far as on the previous swing. Determine the total distance that the pendulum will travel by the time it comes to rest.

Find the first term of the geometric sequence with a common ratio \(-3\) and an eighth term \(-81\).

Find the first six terms of the arithmetic sequence if the common difference is -3 and the ninth term is 10 .

Find the 40th term of the arithmetic sequence with a second term of 6 and a fourth term of 16.

Write the series associated with each summation. $$ \sum_{k=1}^{4}(3 k) $$
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