(1/2)! = 1/2 sqrt(pi)

[michel paul]

factorial(1/2) == 1/2*sqrt(pi).

That's fascinating. Can

anyone

 shed any light on this?

This came up in a dialog with someone about the recursive representation of factorial(n).

Something that caught my attention in exploring this

that I think is relevant for secondary math education

- when you express the factorial value of 1/2 you need parentheses: (1/2)!, and this

might

make it clear that you're dealing with a function, but it's postfix notation, and most

secondary

students/teachers

 aren't used to thinking of a function in postfix notation. They usually only think of

it

in prefix notation, and that's why they

also

don't recognize infix operators like '+' and '*' as functions.

Therefore, 

I think most

secondary

students and teachers probably don't think of '!' as a function. They could probably readily agree that it is, but I think they most likely put '!' and 'f(x)' into two different conceptual categories.

T

his is

one of those

issue

s

in math syntax that we tend to take for granted once we think we understand it, but when you examine it from a programming perspective, you see it in a new light.

--

Michel

===================================
"What I cannot create, I do not understand."

- Richard Feynman

===================================
"Computer science is the new mathematics."

- Dr. Christos Papadimitriou
===================================

[Oleg]

Dear Michel,

Gamma function is a way to expand factorials to all the complex numbers excluding the points 0, -1, -2, -3, etc. where it has poles.

Gamma(n+1) = n! for positive integers. Gamma function has a lot of beautiful properties, including the following.

Gamma(x) Gamma(1-x) = pi/sin(pi x)

discovered by Euler. Plugging x = 1/2 in the above formula gives you the formula you wondering about.

The Wikipedia article on Gamma function is pretty exhausting, see

http://en.wikipedia.org/wiki/Gamma_function

Oleg

--
You received this message because you are subscribed to the Google Groups "MathFuture" group.
To unsubscribe from this group and stop receiving emails from it, send an email to mathfuture+...@googlegroups.com.
To post to this group, send email to mathf...@googlegroups.com.
Visit this group at http://groups.google.com/group/mathfuture?hl=en.
For more options, visit https://groups.google.com/groups/opt_out.
 
 

[Christian Baune]

Hi,

Even 2^3 is a function.
Better, exponential is multiplication "condensed". Like multiplication is additions "condensed".

You can play with whole number to figure it out.

Then you can. "Extend" these operators to real, complex and irrational number.

By the way, you may even understand why √2+√3 do not give √n (with n being a whole number).

The npi shows better what is "data" and what's not:

9,√,1,2,+,+,2,/

This is (√9+(1+2))/2

A factorial would be used like this:
3,!
Yes, the same construct as square root:
4,√

Although you can make things a (little) bit harder:
4,1,2,/,^

In most calculators, you can type what you see but:
- they cost more;
- input is slower;
- for students : they do not have to understand .

Often we talk about "calculators, good or bad?"

Good if they are a learning accelerator.
Using NPI one's is a safe move.

In programmers world: "+" and all "flow control  constructs" are called "syntactic sugars".

(Assembly like code)
Move eax,1
Move ebx,1
Add

Kind regards,
Christian