Why does factorial above 170 return infinite?

Question

I made this php code to calculate the factor of positive integer numbers , however , for numbers from 170 the result is always infinite and I would like to understand why this.

Ex3

 $numero_usuario = isset($_POST['a'])?$_POST['a']:0;
 $fatorial = $numero_usuario;

 if($numero_usuario == 0){
   echo $numero_usuario;
 }
 if($numero_usuario > 0){
   while($fatorial!=1){
       $fatorial--;
       $numero_usuario *= $fatorial;

   }
   echo $numero_usuario;
 }
 ?>

Answer 1

In the Soen you can find an answer on that, which I adapted and added some things I found researching:

What happens is that you are exceeding the precision of a double in a 64bit system. This precision varies between 10^−308 and 10^308. In approximate values:

• Factor of 170: 7.25741562 * 10^307 (within the limit of accuracy)
• Factor of 171: 1.24101807 * 10^309 (outside the limit of precision, then "infinite")

Then it is recommended to use an extension capable of doing this kind of multiplication. One of the possibilities is the Bcmath (Bcmath Arbitrary Precision Mathematics), which was the one I found the staff using for multiplication of large values (with the function bcmul).

Adapting your code:

while($fatorial!=1){
    $fatorial--;
    $numero_usuario = bcmul($numero_usuario, $fatorial);
}

---

Interesting fact: search by factor of 170 in Google and then by the factor of 171:

Answer 2

The factorial with a low value already returns a very high value, see

10! = 3.628.800

170 is exceeding the allocation memory limit stipulated by the Programming Language.

Take the example of Language C: en.wikibooks.org/wiki/Program_em_c/Type_de_data

The other languages follow the same way, but each with its specific allocation size.