Welcome to the factorial calculator, which computes the factorial of any integer between 0 and 170. In addition to calculating, for example, the 0-factorial or 5-factorial…, we will teach you how to utilize the exclamation point in arithmetic and give information about the n-factorial formula and its applications. Using the factorial definition, we will answer the question “what is a factorial?” after this. Finally, we’ll look at the mathematics behind it to see how we can use the gamma function to incorporate more than just positive integers.

## What is a factorial?

In mathematics, factorial is the product of all positive integers less than or equal to a particular positive integer, characterized by that integer plus an exclamation point. For example, factorial seven is written as 8!, which means 1 2 3 4 5 6 7 8.

## Getting mathematical: factorial definition and factorial formula

A number’s factorial is the function that multiplies the number by each natural number below it. Factorial can be represented symbolically as “!”. As a result, n factorial is the product of the first n natural numbers and is denoted by n!

The formula for n factorial is: n! = n * (n - 1)

## Basic values of factorials

A number’s factorial is the function that multiplies the number by each natural number below it. Factorial can be represented symbolically as “!”. As a result, n factorial is the product of the first n natural numbers and is denoted by n!

We will use the basic factorials in the examples below.

0!=1\newline 1! = 1 = 1 \newline 2! = 2 * 1 = 2 \newline 3! = 3 * 2 * 1 = 6 \newline 4! = 4 * 3 * 2*1 = 24 \newline 5! = 5 * 4 * 3 * 2 * 1 = 120 \newline !6 = 6 * 5 * 4 * 3 * 2 * 1 = 720 n! = n(n-1)(n-2)...

## Double factorial

The sum of all the numbers from 1 to n with the same parity (odd or even) as n is the double factorial of a non-negative integer n. It’s also known as the semi-factorial number, represented by the symbol!!. For example, the double factorial of 9 is 9*7*5*3*1 = 945. It’s worth noting that 0!! = 1 is a result of this definition.

The formula for even n

n!! = n(n – 2)(n – 4)…4 * 2

and for odd n double factorials

n!! = n(n – 2)(n – 4)…3 * 1

## 0-factorial and why it is so special

The factorial definition is incomplete without the 0-factorial. To see why it’s so crucial, consider the difficulties we run into while trying to compute it using the factorial formula above:

0! = (0 – 1) *0!

It appears that regardless of the value of (0-1)! is, the outcome should always be 0! = 0, but math is more sophisticated than that. As we saw in the last section, the n-factorial is only defined for n > 0. Therefore we have an issue here. (0-1)! is what mathematicians refer to as an undefined expression, which indicates that the expression is incorrect and thus has no math meaning. This is the same problem as division by 0. It’s not that we can’t figure it out. The issue is that the terms are meaningless and incomprehensible.

This is why defining 0! as a convention value is so significant. Setting it to 0! = 0 is not a smart idea since it will result in n! = 0 for any value of n. By using the factorial formula, you can understand why. On the other hand, set zero-factorial to 1, and keep the anticipated values for n-factorial while having a simple convention for the value of 0!

We’ll see another justification for setting zero-factorial to 1 in the next section, this time with a bit more mathematical explanation.

## Factorial of a negative number

The factorial for an integer n higher than or equal to one is the product of all numbers less than or equal to n but greater than or equal to one. As a result, calculating the factorial of a negative number is impossible. Real negative integer factorials have an imaginary portion equal to 0 and are hence real numbers.