Factorial of 0, Proof and Examples

Factorial of 0 Proof:

since we know that

[katex]n!=n(n-1)(n-2)(n-3)…….(n-n+2)(n-n+1)(-n-n)[/katex]

Factorial of 0

[katex]n!=n(n-1)(n-2)(n-3)…….3.2.1[/katex]

where n belongs to positive integer

Examples.

[katex]2!=2(2-1)=2(1)=2[/katex]

[katex]3!=3(3-1)(3-2)=3.2.1=6[/katex]

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

As

[katex]n!=n(n-1)(n-2)(n-3)![/katex]

Factorial of 0

[katex]n!=n(n-1)![/katex]

Put [katex]n=1[/katex]

[katex]1!=1(1-1)![/katex]

[katex]1!=1(0)![/katex]

[katex]1!=(0)![/katex]

[katex]\boxed {1=0!}[/katex]

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?

[katex]n!=n(n-1)(n-2)(n-3)…….3.2.1[/katex]

for negative integers

put [katex]n=-m\in Z[/katex]

[katex](-m)!=-m(-m-1)(-m-2)(-m-3)…….(-m-m)(-m-m-1)………[/katex]

[katex](-m)!=[/katex](negative).(negative).(negative).(negative)…………(negative).(negative)……

[katex](-m)!=-\infty[/katex]

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

[katex]x^{n+1}=x^n.x^1…..(a)[/katex]

put [katex]n=0[/katex] in eq(a)

[katex]x^{0+1}=x^0.x^1[/katex]

[katex]x^{1}=x^0.x^1[/katex]

dividing [katex]x[/katex] on both sides.

[katex]\frac xx=x^0.\frac xx[/katex]

[katex]1=x^0.1[/katex]

[katex]\boxed{1=x^0}[/katex]

hence any number power zero is equal to 1

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