Factorial
In mathematics, the factorial is a function that maps a non-negative integer to the product of all positive integers less than or equal to that integer. It is denoted by the symbol "!" and is pronounced as "factorial". For example, the factorial of 4 is denoted as 4! and is equal to the product of all positive integers less than or equal to 4, which is 4 x 3 x 2 x 1 = 24.
Notation
The factorial function is denoted by the symbol "!" and is written after the non-negative integer that is being factorialized. For example, 4! denotes the factorial of 4.
Properties
1. Factorial of 0
The factorial of 0 is defined to be 1. This is because the product of all positive integers less than or equal to 0 is 1.
2. Factorial of 1
The factorial of 1 is defined to be 1. This is because the product of all positive integers less than or equal to 1 is 1.
3. Recursive Definition
The factorial function can be defined recursively as follows:
n! = n x (n-1)!
This means that the factorial of a number n is equal to n times the factorial of (n-1). For example, to calculate the factorial of 4, we can use the recursive definition as follows:
4! = 4 x 3 x 2 x 1
= 4 x 3!
= 4 x 3 x 2!
= 4 x 3 x 2 x 1!
= 4 x 3 x 2 x 1
4. Factorial of a Negative Number
The factorial of a negative number is not defined. This is because the product of all positive integers less than or equal to a negative number does not exist.
5. Factorial of a Fractional Number
The factorial of a fractional number is not defined. This is because the product of all positive integers less than or equal to a fractional number does not exist.
Applications
Factorial is used in many areas of mathematics, including combinatorics, probability theory, and calculus. Some common applications of factorial include:
1. Permutations
In combinatorics, the number of permutations of n objects is given by n!. This formula is used to calculate the total number of possible arrangements of n objects.
2. Combinations
In combinatorics, the number of combinations of n objects taken r at a time is given by n!/r!(n-r)!. This formula is used to calculate the total number of possible selections of r objects from a set of n objects.
3. Probability
In probability theory, the factorial is used to calculate the number of possible outcomes in a sample space. For example, if we have n objects and we want to calculate the probability of selecting r objects without replacement, we use the formula n!/(r!(n-r)!).
4. Calculus
In calculus, the factorial is used to define the gamma function, which is a generalization of the factorial function to complex numbers.
Conclusion
In conclusion, the factorial function is a fundamental concept in mathematics that is used in many areas of mathematics, including combinatorics, probability theory, and calculus. It is defined as the product of all positive integers less than or equal to a non-negative integer and is denoted by the symbol "!". The factorial function has several important properties, including a recursive definition and the fact that the factorial of 0 and 1 are both equal to 1.