How to perform factorial operation

[On Radioactive Waves]

I don't know if this is the right place for this question, but here it goes.

Could someone explain how I would go about solving for x of x=(5/6)!

Thanks

 

[StephenPrivitera]

I believe that factorials aren't defined for nonintegers. Perhaps the intent is 5!/6!, which would be 1/6.

 

[jcsd]

Factorials only work for natural numbers, so 5/6 does not have a factorial.

 

[Njorl]

There is a function that is equivalent to taking factorials that works on real numbers (maybe just positive reals, it's been a while) as well. I believe it is the Gamma function.
Njorl

 

[selfAdjoint]

The Gamma function is defined by Γ(z) = ∫₀^oo t^z-1dt

It has the property that n! = Γ(n+1).

This isn't going to be a lot of help in solving the equation though.

 

[jcsd]

You can only take the factorial of a postive integer, thus Γ(n+1) = n! is only true when n is a postive integer.

 

[jcsd]

Actually just done some research, using the gamma function you can find values for half-integrals.

 

[ahrkron]

Originally posted by selfAdjoint
The Gamma function is defined by Γ(z) = ∫₀^oo t^z-1dt

There is a small, yet important omission here. The definition includes a negative exponential:

Γ(z) = ∫₀^oo t^z-1e^-tdt

The exponential is important because it makes the integral converge for almost all values of z (the exponential goes to zero much faster than the growth of t^z-1).

In particular, for what you want, you can obtain the value as

(5/6)! = Γ(5/6 + 1) = ∫₀^oot^(5/6+1)-1e^-tdt

Mathworld has a nice entry for the Gamma function. In the plot, you can see that the value for Γ(1+5/6) = Γ(1.833) is slightly less than one.

 

[selfAdjoint]

Woops! Sorry. You are actually right. BTW the z in the definition is a complex variable, and the Γ function is meromorphic (it has poles at 0 and negative integers, but is otherwise analytic). And with that proviso, we have the recurrence relation Γ(z+1) = zΓ(z).

In the half plane of complex numbers with real part > 1, it can be defined by Γ(z-1) = π/(Γ(z)sin(πz)) = πz/(Γ(1+z)sin(πz)).

 

[On Radioactive Waves]

Oops, my bad.

I've looked at the mathworld site, many time actually.

It turns out this problem stems back a few years, and I just realized its not factorial I'm actually wondering about (tried it on my TI-89)

Okay, here is there real problem then. My calculator will give sums for negative numbers and fractions, and I was confused about that.

This is actually a lot less confusing than i thought. Thanks for the input everyone. I remembered wrong but hey its been a while.