Question

What is the factorial of $\pi$?

Answer 1

We know that the definition of factorial is only for the non negative integers. So, we must use the Gamma function, defined as $\Gamma \left( n \right)=\int\limits_{0}^{\infty }{{{x}^{n-1}}{{e}^{-x}}dx}$. We can use the property of the Gamma function that $\Gamma \left( n \right)=\left( n-1\right)!$ to find the factorial of $\pi$.

Complete step-by-step solution:
We know that the factorial is defined only for non negative integers. For non negative integers, factorial is calculated by the repeated multiplication from 1 to that number, for example,
factorial of 5 = $5\times 4\times 3\times 2\times 1$.
We know that factorial of $x$ is represented as $x!$ or $\left| \\!{\underline {\, x \,}} \right.$.
Here, we need to find the factorial of $\pi$.
We know that the value of $\pi$ is 3.141 which is not an integer. So, we cannot find the value of $\pi !$ using the above definition.
We can expand the factorial function for non negative real numbers using the Gamma function.
The Gamma function of n is represented as $\Gamma \left( n \right)$.
We can define the Gamma function for variable n as, $\Gamma \left( n \right)=\int\limits_{0}^{\infty }{{{x}^{n-1}}{{e}^{-x}}dx}$.
If we try to integrate the above formula using integration by parts, we will get a very useful property, which is $\Gamma \left( n \right)=\left( n-1 \right)!$.
With the help of this property, we can write
$\pi !=\Gamma \left( \pi +1 \right)$.
So, by using the same definition, we can find the Gamma of $\left( \pi +1 \right)$.
Thus, we have $\Gamma \left( \pi +1 \right)\approx 7.18808272898$.
Hence, the approximate value of pi factorial is 7.18808272898.

Note: We must pay attention that the integration in the definition of Gamma function is definite from $x$ = 0 to positive infinity, with respect to $x$ and not n. We must also understand that the Gamma function is just a mathematical tool, like the factorial function.