factorial (of an integer)

Factorial (of an Integer) in Math

Definition

Factorial of an integer is a mathematical operation that is used to calculate the product of all positive integers less than or equal to a given integer. It is denoted by the symbol "!". For example, the factorial of 5 is written as 5! and is equal to 5 x 4 x 3 x 2 x 1 = 120.

Knowledge Points

Factorial of an integer involves the following key points:

1. The factorial of a positive integer n is the product of all positive integers less than or equal to n.
2. Factorial is only defined for non-negative integers. The factorial of a negative integer or a non-integer is not defined.
3. The factorial of 0 is defined as 1.

Formula or Equation

The formula for calculating the factorial of an integer n is given by: n! = n x (n-1) x (n-2) x ... x 3 x 2 x 1

Application of the Formula

To calculate the factorial of an integer using the formula, simply multiply the integer by all positive integers less than it, until you reach 1. For example, to find the factorial of 5, you would multiply 5 by 4, then by 3, then by 2, and finally by 1.

Symbol for Factorial

The symbol used to represent factorial is "!". For example, 5! represents the factorial of 5.

Methods for Factorial

There are several methods to calculate the factorial of an integer:

1. Direct multiplication: Multiply the integer by all positive integers less than it, until you reach 1.
2. Recursive function: Define a recursive function that calls itself with a smaller value until it reaches the base case of 1.

Solved Examples

1. Find the factorial of 4. Solution: 4! = 4 x 3 x 2 x 1 = 24

2. Calculate the factorial of 7. Solution: 7! = 7 x 6 x 5 x 4 x 3 x 2 x 1 = 5040

Practice Problems

1. Find the factorial of 6.
2. Calculate the factorial of 10.
3. What is the factorial of 0?

FAQ

Q: What is the factorial of a negative integer? A: The factorial of a negative integer is not defined.

Q: Can factorial be calculated for non-integer values? A: No, factorial is only defined for non-negative integers.

Q: What is the factorial of 1? A: The factorial of 1 is 1, as there are no positive integers less than 1 to multiply.

Q: What is the factorial of 0? A: The factorial of 0 is defined as 1.