1. ## 0!

Why is 0! = 1?

2. it is specially defined.
Since 1!=1*0!.

3. A silly poem I wrote while in college:

Man has wondered
Since time immemorial
Why 1 is the value
Of 0!.

4. Hello 0! is a convention without demonstration.
Same$\displaystyle 2^0 = 1$$\displaystyle$

5. Originally Posted by pacman
Why is 0! = 1?
We can show this is the case using calculus (if this is a bit advanced for you, please bear with me )

The Gamma Function is defined by $\displaystyle \Gamma\left(x\right)=\int_0^{\infty}e^{-t}t^{x-1}\,dt$ for $\displaystyle x>0$

If $\displaystyle n\in\mathbb{N}$, the Gamma Function has a special property: $\displaystyle \Gamma\left(n\right)=\left(n-1\right)!$.

So it follows that $\displaystyle {\color{blue}\Gamma\left(1\right)}=\left(1-1\right)!=\color{red}\boxed{0!}$

By definition, $\displaystyle {\color{blue}\Gamma\left(1\right)}=\int_0^{\infty} e^{-t}t^{1-1}\,dt=\int_0^{\infty}e^{-t}\,dt=\lim_{b\to\infty}\left.\left[-e^{-t}\right]\right|_0^{b}$ $\displaystyle =-\lim_{b\to\infty}e^{-b}-\left(-e^{-0}\right)=0-(-1)=\color{red}\boxed{1}$.

Thus, $\displaystyle {\color{blue}\Gamma\left(1\right)}=\color{red}\box ed{0!=1}$.

6. I tried figuring out the Gamma function from Wikipedia a few weeks ago...but didn't understand it that well. Do you know any website where I can find a proper introduction to it? Or should I just wait until they cover it in one of my future college courses....once I start college?

Thanks.

7. Originally Posted by pacman
Why is 0! = 1?
Hi pacman,

I posted this in another thread, but I'll repost it here.

See if this helps.

n! is defined as 1 * 2 * 3 * . . . * n

And with a little manipulation, we can show that 0! = 1 by demonstrating that...

1! = 1
2! = 1! * 2
3! = 2! * 3
4! = 3! * 4
etc.

Reversing that, we can achieve...

3! = 4!/4
2! = 3!/3
1! = 2!/2
0! = 1!/1 = 1

8. Originally Posted by RobLikesBrunch
I tried figuring out the Gamma function from Wikipedia a few weeks ago...but didn't understand it that well. Do you know any website where I can find a proper introduction to it? Or should I just wait until they cover it in one of my future college courses....once I start college?

Thanks.
See if this helps you: The Gamma Function

9. Let's say that 0! is equal to something else, say 2.

then the taylor series of $\displaystyle \cos x$ expanded about $\displaystyle x=0$ would be $\displaystyle \frac{1}{2} - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} + \frac{x^{6}}{6!} + ...$

which would imply that $\displaystyle \cos (0) = \frac{1}{2}$

and all hell would break loose

10. random variable, can you elaborate further? thanks!

11. Originally Posted by pacman
random variable, can you elaborate further? thanks!
When you evaluate the taylor expansion of cosine at zero, you get $\displaystyle \cos\left(0\right)=\frac{1}{0!}$.

When he said all hell would break loose, he was commenting on the fact that if $\displaystyle 0!$ was some other value than 1, then $\displaystyle \cos\left(0\right)=\frac{1}{\text{other value}}$, which is absurd since we know from trig that $\displaystyle \cos\left(0\right)={\color{red}1}$. This would then force $\displaystyle 0!=1$.

12. ahhh, that is much CLEARER now. Thanks Cris

13. So many different ways! I like this.

The way I was taught was using the fact that that there are $\displaystyle ^{n}C_k$ ways of picking k items from a set of n, where

$\displaystyle ^{n}C_k = \frac{n!}{(n-k)!k!}$ .

If we think about it, it should be that there is only 1 way of choosing no items from a set of n items, therefore $\displaystyle ^{n}C_0 = 1$ and so,

$\displaystyle \frac{n!}{(n-0)!0!} = 1$ $\displaystyle \Rightarrow$ $\displaystyle \frac{1}{0!} = 1$ $\displaystyle \Rightarrow$ $\displaystyle 0! = 1$

Some of the other explanations are nicer I think, but there's my two cents!

14. Pomp, i like your way of obtaining 0! = 1 through Combination.

15. If 0! was equal to 0, then n! would = 0,
since we'd have to assume that ! represents 0*1*2......*n

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