Problem 36 - Algebra for College Students

Question

Evaluate expression. \(0 !\)

Step-by-Step Solution

Answer

0! = 1

Step 1: Understanding Factorial

A factorial, denoted by an exclamation mark (!), is the product of all positive integers up to a given number. For any non-negative integer n, it is represented as: \[ n! = n \times (n-1) \times (n-2) \times \rightarrow 3 \times 2 \times 1 \]

Step 2: Special Case: Zero Factorial

By definition, \(0!\) is a special case. In mathematics, the factorial of zero is defined to be 1.

Step 3: Final Evaluation

Therefore, \(0! = 1\)

Key Concepts

zero factorial

The concept of zero factorial (0!) can be confusing at first, but it's a fascinating topic in mathematics. Let's break it down:
Zero factorial is a special case where the factorial of zero is defined as one, i.e., 0! = 1. This might seem strange at first glance because multiplying nothing together should result in zero, right?
Well, not exactly. This definition is critical for various mathematical principles to hold true. For example, in combinatorics, the number of ways to arrange zero objects has to be one for the formulas to work correctly.
So, although it seems a bit counter-intuitive, defining 0! as 1 helps maintain consistency.

factorial definition

A factorial, denoted by an exclamation mark (!), is a mathematical operation that multiplies a series of descending positive integers. The definition is:
\begin\[\begin{equation} n! = n \times (n-1) \times (n-2) \times \rightarrow 3 \times 2 \times 1 \right.onumber \end{equation}\]where \right. n \right. is a non-negative integer.
For example:

• 5! = 5 × 4 × 3 × 2 × 1 = 120
• 3! = 3 × 2 × 1 = 6

Factorials are mainly used in permutations, combinations, and other areas of algebra and calculus. They are useful for determining the number of ways to arrange or select items.

special cases in mathematics

Mathematics is full of special cases like zero factorial, which are important to understand for a solid foundation in the subject. Some particular cases include:

• Zero exponential: Any non-zero number raised to the power of zero equals one. For example, \begin]5^0 = 1\right = 1. This simplifies many mathematical expressions.
• Division by Zero: Dividing by zero is undefined because it doesn't produce a consistent or meaningful result. It leads to undefined behavior in mathematics.
• Imaginary numbers: When dealing with square roots of negative numbers, we use the imaginary unit \((i). For example, the square root of \begin\right(-1)\right is \)-(i), allowing us to solve equations that don't have real solutions.Understanding these special cases helps provide a more comprehensive view of mathematical principles and their applications.