Problem 6 - Beginning and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

1

Step 1: 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 × 2 × 1 = 6.

Step 2: 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.

Step 3: Apply the definition

Using the definition from Step 2, the value of 0! is 1.

Key Concepts

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 × 4 × 3 × 2 × 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 × 3 × 2 × 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.