Question

How do you find the factorial of negative numbers?

Answer 1

To solve the given problem, we will first see the definition of a factorial function. The factorial function is a special type of function that multiplies a number by every number below it, and gives their product as the output. The factorial function mainly takes only positive integers as well as zero as the domain. It should be noted that the definition does not hold for 0, as the value of $0!$ equals 1.

Complete step by step solution:
We are asked to find the way to evaluate the factorials of negative numbers. As we already know that, a factorial function is a special type of function that multiplies a number by every number below it, and gives their product as the output. This function mainly takes non-negative integers as its input. So, to find the factorials of negative numbers, we have to extend its definition. We will do this as follows,
First let’s see some values of factorials of non-negative integers. Refer to the following table.

Integer (n)	Factorial ($n!$)
0	1
1	1
2	2
3	6
4	24

To evaluate the value of $n!$, here n is a negative integer. First find the value of $\left| n \right|!$, then multiply by ${{1}^{-n}}$.

Note: Now that we know $n!$, n is a negative integer can be evaluated as ${{1}^{-n}}\left( -n \right)!$. Using the above method, we can prepare a table showing the values of factorials of negative numbers, as follows

Integer (n)	Factorial ($n!$)
0	1
-1	-1
-2	2
-3	-6
-4	24