Can someone pleeeease explain this for me. Throughout all of my math classes, mostly Algebra, I was always told that zero factorial (0!) is equal to 1. Now I understand how 3! is the same as 6, and 5! is the same as 120, and so forth, but I cannot figure out the 0! mystery. Any help would be appreciated.
Don’t you think this should be in GQ?
(and sorry, can’t really help you with the math there)
Well, I really have no clue, but I’ve learned something about math. If it doens’t follow the rule then…theres a special rule!! or it is by definition :D. 0!=1 is probably the latter.
[Moderator Hat ON]
Yeah, I think this should go to the mathematically-inclined in General Questions. Here ya go, Manny, Nickrz…
[Moderator Hat OFF]
The most natural way is to think of it as an “empty product”. n! is defined to be the product of all positive integers less than or equal to n. That’s an empty set when n=0, so you’re multiplying over an empty product (there’s nothing left to be multiplied). Empty products are generally defined to be equal to 1, because that’s the multiplicative identity (the multiplicative “nothingness”, I guess you could say). Similarly, empty sums are generally defined to be 0, since that’s the additive identity.
no, no, no.
0! is defined as 1 because that follows the pattern. if you go backward
you can accept that 4! = 24
then the pattern is this:
3! = (4!)/4 = 6
2! = (3!)/3 = 2
1! = (2!)/2 = 1
0! = (1!)/1 = 1
(-1)! = 0!/0 = undefined
in fact you can define a “factorial” for non-integers by the same basic relation.
that is to say, define a function such that
f(x) == x*f(x-1)
f(1) == 1.
this is basicly how the gamma function is defined ( the gamma function is actually defined slighly differently so that gamma(n-1)=n! if i remember correctly).
good luck with your math.
-Luckie
Yes, I agree with Luckie. There are several functions, similar to the factorial function, that are actually defined as a similar build-up.
Factorial or gamma function is defined by:
f(1) = 1 and
f(n) = n * f(n-1)
Similarly, the Fibonnacci sequence is defined as:
f(1) = 1 and
f(n) = n + f(n-1)
The concept of iterative functions is a complicated one, and leads to interesting stuff in chaos theory, where functions are defined as cumulating:
f(n) = f (f(n-1) )
My understanding is similar to that which has been posted already. The factorial function is defined only for the natural numbers, and 0 is nto a natural number. But, the factorial function is a special case of the gamma function, and gamma(1) is defined. Since gamma(n) = (n-1)! for the natural numbers, it is reasonable to decide that 0! = gamma(1) just to kep the correspondence between the two functions. See X Factorial and the Gamma Function (which includes another reason for deciding that 0! = 1) and What is the Gamma Function?.
jrf
It’s also defined that way to make the formulas work.
What is the formula for the number of ways that you can choose M objects from N objects? Now, plug in M=3 and N=3. Obviously, the answer is 1, but the formula reduces to 1/(3-3)! (I’m not excited about this, that’s a factoral sign) or 1/0!
In order for that to give the appropriate answer of 1, we are limited in our choice of definitions for 0!
rocks
(First of all, am I correct that “natural numbers” = “positive integers” ? If I’m wrong, the following won’t make sense, so please correct me.)
Jonf wrote
That makes sense, but is it really true? Intuitively, exponents should also be limited to natural numbers: You can square a number, or cube a number, or raise it to any other integral power, but how do you multiply a number by itself two and a half times? or a negative number of times? Yet, we know that any real number can be used as an exponent. Maybe any real number can have a factorial too?
Actually, any nonnegative number x does in effect have a factorial, and that factorial is gamma(x+1). (Check out Jon’s links.) The factorial function n!, as we generally know it, is a restriction of the gamma function to integer inputs.
That allows us to replace the definition in terms of integrals with the more common (and easier to use) definition in terms of multiplication. But it makes the definition of 0! look arbitrary when, actually, it’s not.
“Living in this complex world of the future is not unlike having bees live inside your head.” - F. Scott Firesign
Actually, there’s some disagreement whether or not zero is a natural number. See Do Natural Numbers Start with Zero or One. There are some links at the end to several discussions.
One might say that, but there’s a parallel to the factorial and gamma functions. If you define exponentiation as multiplying a number by itself the number of times that is specified by the exponent, then only integer exponents make sense. However, it you define exponentiation more generally as:
F(x) = antilog(log(x)*exponent) (x>=0)
F(x) = antilog(log(x)/exponent) (x<0)
then you get a much more useful and general function, that reduces to the other function for integer exponents.
jrf
JonF: Actually, there’s some disagreement whether or not zero is a natural number. See Do Natural Numbers Start with Zero or One. There are some links at the end to several discussions.
That Dr. Derwood 
is flipped! Natural numbers are 1, 2, 3, 4… Whole numbers are 0, 1, 2, 3, 4… These fit his description:
When I first learned about natural numbers, I was told to think of them as anything you could use to fill in the blank of the following sentence: I have ____ bowls of ice cream (or pieces of cake, or piles of dirt, etc.).
substituting “natural” for “whole”, the above paragraph works.
Wrong thinking is punished, right thinking is just as swiftly rewarded. You’ll find it an effective combination.
Just a small correction on what CKDextHavn said:
Factorial is defined as
0! = 1
n! = (n-1)! * n
Fibonnacci sequence is defined as
f(1) = 1
f(2) = 1
f(n) = f(n-1) + f(n-2)
Virtually yours,
DrMatrix
These words are mine and they are true - Chief Meninock