(This post is inspired by the book The cult of Pythagoras: Math and Myths which I recently
read and reviewed. See here for my review.)
STUDENT: The factorial function is only defined on the natural numbers. Is there some way to extend it to all the reals? For example, what is (1/2)! ?
BILL: Actually (1/2)! is sqrt(π)/2
STUDENT: Oh well, ask a stupid question, get a stupid answer.
BILL: No, I'm serious, (1/2)! is sqrt(π)/2.
STUDENT: C'mon, be serious. If you don't know or if its not known just tell me.
The Student has a point. (1/2)! = sqrt(π)/2 is stupid even though its true. So I ask--- is there some other way that factorial could be expanded to all the reals that is as well motivated as the Gamma function? Since 0!=1 and 1!=1, perhaps (1/2)! should be 1.
Is there a combinatorial interpretation for (1/2)!=sqrt(π) /2?
If one defined n! by piecewies linear interpolation that works but is it useful? interesting?
For that matter is the Gamma function useful? Interesting?
ANOTHER CONVENTION: We say that 0^0 is undefined. But I think it should be 1.
Here is why:
d/dx x^n = nx^{n-1} is true except at 1. Lets make it ALSO true at 1 by saying that x^0=1 ALWAYS
and that includes at 0.
A SECOND LOOK AT A CONVENTION: (-3)(4) = -12 makes sense since if I owe my bookie
3 dollars 4 times than I owe him 12 dollars. But what about (-3)(-4)=12. This makes certain
other laws of arithmetic extend to the negatives, which is well and good, but we should not
mistake this convention for a discovered truth. IF there was an application where definiting
NEG*NEG = NEG then that would be a nice alternative system, much like the diff geometries.
I COULD TALK ABOUT a^{1/2} = sqrt(a) also being a convention to make a rule work out
however (1) my point is made, and (2) I think I blogged about that a while back.
So what is my point- we adapt certain conventions which are fine and good, but should not
mistake them for eternal truths. This may also play into the question of is math invented or
discovered.
Actually, it's Gamma(1/2) = sqrt(pi), so (-1/2)! = sqrt(pi). Remember that (x)!=Gamma(x+1)=x Gamma(x). Off by one error... ;)
ReplyDeleteSo we have (1/2)! = Gamma(3/2) = sqrt(pi) / 2.
I understand your student: (1/2)! is NOT sqrt(pi). It is sqrt(pi)/2... OK, it is maybe not much more intuitive, but still!
ReplyDeleteThe convention 0^0 = 1 is MUCH more used than 0^0 = undefined I think.
For NEG * NEG, I think the point is that we like to have associative laws: If you take the convention NEG * POS = NEG and associativity, you have (1 + (-3)) * (2 + (-6)) = (-2)*(-4) = ± 8, and it is also equal to 1*2 + (-3)*2 + 1*(-6) + (-3)*(-6) = ± 18 - 10. Then only possibility here is to get NEG * NEG = POS as convention.
More generally, of course you could define an operation which is as the multiplication but with NEG * NEG = NEG. The point is that is does not have good properties. So to my mind, it is NOT ONLY a question of applications, and it wouldn't be a NICE alternative system.
P.S.: I hope I respect GASARCH's typographic rules! ;-)
gamma(3/2) = "(1/2)!" = sqrt(pi)/2, not sqrt(pi)
ReplyDeleteThe volume of n-balls gives a pretty good geometric reason why (m+1/2)! is defined the way it is.
ReplyDeleteExcellent, this is JUST the kind of thing I could tell STUDENT.
DeleteFor those who don't know, the volume of an n--ball of radius R is
( pi^{n/2}/Gamma(n/2 + 1))R
for n even you would use factorial on something- I won't say to avoid being off-by-one again, but for n odd you need to use the actual Gamma Function!
actual Gamma function.
Thanks for the correction. on the off-by-one thing.
ReplyDeleteAs for viloating assoc las- YES, and that is indeed why we have NEG*NEG=POS and
I don't disagree with it as a convention.
But realize that when the Quaternions wehre invnted they were NOT COMMUATIVE!
but they were USEFUL!
I wonder if the same might happen for some system with NEG*NEG=POS.
While we are onto conventions making things work out...
ReplyDeleteWhat about 1+2+3+4 + ....= -1/12?
I definitely think that 0^0 is 1, not undefined.
ReplyDeleteWhat is the sum of the weights of all unicorns? Most people would agree that it is 0.
What is the product of the weights of all unicorns? It should be 1, to make the identity
(product of unicorns)(product of people) = (product of people and unicorns)
correct. Why would this argument change if the unicorns were individually weightless?
For that matter, the concatenation of the names of all the unicorns is the empty string.
neg*neg = pos is not a convention, it is true since it follows the laws of arithmetic, essentially following cancellation law. Consider (-a)*(b-b) = 0 => (-a)* b+ (-a)*(-b) = 0, then adding a*b to both sides yields (-a)*(-b)= ab. Did I miss something there?
ReplyDeleteActually, neg * pos = neg and neg*neg=pos are not conventions and can be derived easily from the basic laws of arithmetic, eg.; (a-a)*b = 0 and then rearranging and simplifying as needed.
ReplyDeleteHi, Dave---I gave further argument for your instances at https://rjlipton.wordpress.com/2015/02/23/the-right-stuff-of-emptiness/
ReplyDelete