Question

What is a factorial of $0?$

Answer 1

Hint: The factorial of a natural number $n$ is the product of all the natural numbers less than or equal to $n.$ That is, $1.2.3...\left( n-1 \right)n.$ The product of all the natural numbers from$1$ to $\left( n-1 \right)$ is the factorial of $\left( n-1 \right).$ That is, $1.2.3...\left( n-2 \right)\left( n-1 \right).$

Complete step by step solution:
The factorial of a natural number $n$ is defined as $n!=1.2.3...\left( n-1 \right)n.$
The symbol $!$ is used to denote the factorial. And we call $n!$ as $n$ factorial.
As defined above, the factorial of $n$ is the product of the natural numbers less than or equal to $n.$
According to our definition,
The factorial of $1,$ that is, $1!=1.$
The factorial of $2,$ that is, \[2!=1\times 2=2.\]
The factorial of $3,$ that is $3!=1\times 2\times 3=6.$
The factorial of $4,$ that is $4!=1\times 2\times 3\times 4=24.$
Similarly, we can find the factorial of any natural number.
Consider $2!=1\times 2.$
We have $1!=1.$
Put this value in the factorial of $2.$
We will get $2!=1!\times 2.$
Take $3!=1\times 2\times 3.$
Now we have $2!=1\times 2.$
Let us substitute this in $3!.$
We will get \[3!=2!\times 3.\]
Also, consider the factorial of $4.$
It is found that $4!=1\times 2\times 3\times 4.$
We have already obtained that $1\times 2\times 3=3!$
When we substitute this in the value of $4!,$ we will get $4!=3!\times 4.$
From this we can discover the fact about the factorial which says that the factorial of a natural number $n$ is a product of all the natural numbers less than or equal to itself and thus, the factorial of a natural number $n$ is a product of the factorial of the natural number $\left( n-1 \right),$ which is less than $n$ and greater than every other natural numbers less than $n,$ and $n$ itself.
Therefore, we can write, $n!=1.2.3....\left( n-1 \right)n=\left( n-1 \right)!n$
Also, we can write $n!=\left( n-2 \right)!\left( n-1 \right)n.$
This is how we define the factorial of a natural number.
But we have an already defined fact that the factorial of $0$ is $1.$
That is, $0!=1.$
So, unlike any other cases, $0$ can be put in the denominator of a fraction.
For example, $1=\dfrac{1}{0!}.$

Note: If we just use the fact $n!=\left( n-1 \right)!n,$ we can write $1!=0!\times 1.$
We know that any number multiplied with $1$ gives that number itself.
Also, $1!=1.$
Therefore, $0!=1.$
Although this is not an established way to say that $0!=1,$ we often use this method.