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]
[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]
[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