The real question is, how come you can't find your keyboard's "!" key?
There are a couple of different reasons why 0! is defined as 1. If you have n different objects, then the number of ways to arrange them in a line is n!. If you have 0 objects, then there's only one way to "arrange" them: in an empty pile. Emptiness is emptiness.
Likewise, setting 0! equal to 1 makes a lot of permutation formulas work out correctly without having to make exceptions.
Factorial may be defined as, for all _positive integers_ n,
n! = n*(n-1)!
1! = 1*(1-1)!
1! = 1*0!
1!/1 = 0!
0! = 1
Although according to Wikipedia, "0! is explicitly defined to be 1".
One application of factorial is rearranging n number of items. What if n=0, how would we interpret that? How many times would we rearrange 0 items? Probably once, is that there's no arrangement. Haha. I know this is lame. But this is only IMHO.
imagine you have a number to a power eg. 2^3 you know it is 2x2x2=8 to get 2^2 you divide the previous by 2, 2x2=4 on the same basis then 2^1 (2x2 then divided by 2) will equal 2. if you keep going then 2^0 will be 2^1 divided by 2, 2 divided by 2 = 1. this is true of any number
Comments
The real question is, how come you can't find your keyboard's "!" key?
There are a couple of different reasons why 0! is defined as 1. If you have n different objects, then the number of ways to arrange them in a line is n!. If you have 0 objects, then there's only one way to "arrange" them: in an empty pile. Emptiness is emptiness.
Likewise, setting 0! equal to 1 makes a lot of permutation formulas work out correctly without having to make exceptions.
This article has some more explanations:
http://mathforum.org/dr.math/faq/faq.0factorial.ht...
Factorial may be defined as, for all _positive integers_ n,
n! = n*(n-1)!
1! = 1*(1-1)!
1! = 1*0!
1!/1 = 0!
0! = 1
Although according to Wikipedia, "0! is explicitly defined to be 1".
One application of factorial is rearranging n number of items. What if n=0, how would we interpret that? How many times would we rearrange 0 items? Probably once, is that there's no arrangement. Haha. I know this is lame. But this is only IMHO.
imagine you have a number to a power eg. 2^3 you know it is 2x2x2=8 to get 2^2 you divide the previous by 2, 2x2=4 on the same basis then 2^1 (2x2 then divided by 2) will equal 2. if you keep going then 2^0 will be 2^1 divided by 2, 2 divided by 2 = 1. this is true of any number
ie.
3^3=27
3^2=9
3^1=3
3^0=1
Don't try to prove this. It is simply a definition that mathematicians came up with in order for everything else to fit.