How factorial works

Number Theory Level 2

$\Large (-1)! = \, ?$

Notation : $!$ denotes the factorial notation.

Undefined 0 -1 1 None of the above

1 solution

Pelob Chakraborti
Mar 9, 2016

If you use your calculator, you will get the answer -1. But, actually what happens, calculator just calculates for the number. It doesn't consider the negative sign. You may use calculator for the value of -2!, -5!, -11! .... you will get the same.

On the other hand, we know the formula for the factorial of any number, n! = $\frac{(n+1)!}{(n+1)}$

Such as, 5! = $\frac{6!}{6}$ = 120, then 4! = $\frac{5!}{5}$ = 24 ; 3! = $\frac{4!}{4}$ = 6 ; 2! = $\frac{3!}{3}$ = 2 ; 1! = $\frac{2!}{2}$ = 1 ; 0! = $\frac{1!}{1}$ = 1

In the same manner, -1! = $\frac{0!}{0}$ = Undefined.

$n!=\frac { \left( n+1 \right) ! }{ n+1 }$ is not an ultimate definition for a factorial and there are many ways to define it. Definitions of factorials can be extended, for eg, $\left( -\frac { 1 }{ 2 } \right) !=\sqrt { \pi }$ .

Arulx Z - 5 years, 3 months ago

Would you please show how it works?

Pelob Chakraborti - 5 years, 3 months ago

You can try reading about Gamma Function

Arulx Z - 5 years, 3 months ago

@Arulx Z – Anyway Gamma Function still does not define (-1)!

展豪張 - 5 years, 3 months ago

@展豪張 – That's true. I meant to say that this cannot be generalized for every negative number.

Arulx Z - 5 years, 3 months ago

@Arulx Z – You are right. The gamma function is the analytic continuation of factorial.

展豪張 - 5 years, 3 months ago

How factorial works

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