Factorial of 0, Proof and Examples

Factorial of 0 Proof:

since we know that

n!=n(n-1)(n-2)(n-3)…….(n-n+2)(n-n+1)(-n-n)
Factorial of 0
n!=n(n-1)(n-2)(n-3)…….3.2.1

where n belongs to positive integer

Examples.

2!=2(2-1)=2(1)=2

3!=3(3-1)(3-2)=3.2.1=6

Now we will prove that 0!=1 is equal to one.

As

n!=n(n-1)(n-2)(n-3)!
Factorial of 0
n!=n(n-1)!

Put n=1

1!=1(1-1)!

1!=1(0)!

1!=(0)!

\boxed {1=0!}

Hence factorial of 0 is equal to 1

Why it is not possible to have a negative factorial ?

Is factorial define for nagative numbers?

why can not have a negative factorial?

why no negative factorial?

n!=n(n-1)(n-2)(n-3)…….3.2.1

for negative integers

put n=-m\in Z

(-m)!=-m(-m-1)(-m-2)(-m-3)…….(-m-m)(-m-m-1).........

(-m)!=(negative).(negative).(negative).(negative)…………(negative).(negative)……

(-m)!=-\infty

Hence prove that negative factorial does not exist.

Power zero is equal to ?

power zero is equal to 1

what does zero power mean

Anything power zero is equal to

x^{n+1}=x^n.x^1.....(a)

put n=0 in eq(a)

x^{0+1}=x^0.x^1

x^{1}=x^0.x^1

dividing x on both sides.

\frac xx=x^0.\frac xx

1=x^0.1

\boxed{1=x^0}

hence any number power zero is equal to 1

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Azhar Ali

Azhar Ali

Mathematician and Blogger.

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