Question

How do you simplify $\left( 1.5 \right)!?$

Answer 1

Hint: A factorize is a number multiplied by all of the integers below it.
Example “Four factorial” $=4!=4\times 3\times 2\times 1=24$
The factorial of a number $'n'$ is denoted as $'n!'$ This is the product of all numbers from $1$ to $n.$

Complete step by step solution:
As per the given problem,
You have to simplify $\left( 1.5 \right)!$
Here, the factorial of a fraction number is defined by the gamma function is defined by the gamma function as follows.
$n!=n\times \left( n-1 \right)!$
$\Gamma \left( n \right)=\left( n-1 \right)!$
$n!=n.\Gamma (n)$
And
$\Gamma \left( \dfrac{1}{2} \right)=\sqrt{\pi }...(i)$
Hence, simplify you can solve $\left( 1.5 \right)!$
$\left( 1.5 \right)!=\left( \dfrac{3}{2} \right)!=\left( \dfrac{3}{2} \right).\left( \dfrac{1}{2} \right)!$
$=\left( \dfrac{3}{2} \right)\left( \dfrac{1}{2} \right)\Gamma \left( \dfrac{1}{2} \right)$
$=\dfrac{3}{4}\sqrt{\pi }$ as from equation $(i)$
$\Gamma \left( \dfrac{1}{2} \right)=\sqrt{\pi }$

Note: You can either use gamma function identity either gauss’s duplication formula. For simplifying the given factorial. Note that for using the Gauss’s duplication formula put $n=1$ and simplify.
You can also solve $1.5!$ by using gauss’s duplication formula which is defined as below.
$\left( n+\dfrac{1}{2} \right)!=\dfrac{\sqrt{\pi }\left( 2n+2 \right)!}{{{4}^{n+1}}\left( n+1 \right)!}$
This allows you to express a fractional number in terms of factorials of integer.
Now, for $n=1$ we get from the above formula.
$\left( 1+\dfrac{1}{2} \right)!=\dfrac{\sqrt{\pi }.4!}{{{4}^{2}}.2!}$
$=\dfrac{\sqrt{\pi }.24}{16.2!}$
$=\dfrac{\sqrt{\pi }.24}{32}$
Here, above $24$ and $34$ comes in a table of $'8'$ so you can simplify it.
$\left( 1+\dfrac{1}{2} \right)!=\dfrac{\sqrt{\pi }24}{32}$
$\left( 1+\dfrac{1}{2} \right)!=\dfrac{3\sqrt{\pi }}{4}$
Hence,
The simplification of $1.5!$ is $\dfrac{3\sqrt{\pi }}{4}$