52 Factorial

dowie said:
well if you look at it in terms of sets of n numbers - number of ways to arrange them....

{1} can be just {1} so 1
{1,2} can be {1,2} or {2,1} so 2

etc...

how many ways can you order the empty set? {}

just 1: {}
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but there is NOTHING in an empty set, surely you "cant" order something that is not there, not even once
 
ilcp said:
but there is NOTHING in an empty set, surely you "cant" order something that is not there, not even once
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You can't reorganise nothing as it will only occur in one way. Hence, one. :)

OH MATHS, I WILL FOREVER HATE YOU! :p
 
estebanrey said:
It is 1 but even top mathematicians admit it is slightly philosophical (4:20), maybe the first 'pattern' way sits better with you.

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Ignore the philosophy stuff. 0! = 1 because 'what a factorial is' has been defined in a way which results in that.
 
Burnsy2023 said:
It's not arbitrary though. This works to explain certain physics functions.
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Very little in maths is arbitrary. ;)

Remember though, that physics is only a set of mathematical laws used to model observations. It is entirely distinct from philosophy/reality (by definition).

Of course people may choose to base their ontology on physics, but that is personal preference.
 
AGD said:
Ignore the philosophy stuff. 0! = 1 because 'what a factorial is' has been defined in a way which results in that.
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Well no, factorial is defined as....

"...the product of all positive integers less than or equal to n"

There are no positive integers lees or equal to 0, therefore by 'defintion' 0 factorial should be either 0 or invalid (like how dividing by zero is).
 
estebanrey said:
Well no, factorial is defined as....

"...the product of all positive integers less than or equal to n"

There are no positive integers lees or equal to 0, therefore by 'defintion' 0 factorial should be either 0 or invalid (like how dividing by zero is).
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Well in modern mathematics they use the continuous integral form to define factorials, which is why you can evaluate say 3.456! even though it wouldn't make sense under your definition. But yes it depends on which definition you choose.
 
estebanrey said:
Well no, factorial is defined as....

"...the product of all positive integers less than or equal to n"

There are no positive integers lees or equal to 0, therefore by 'defintion' 0 factorial should be either 0 or invalid (like how dividing by zero is).
Click to expand...

Well no, any complete definition of the factorial operation includes the case of 0! = 1, so by definition 0! = 1.
 
Liampope said:
Well no, any complete definition of the factorial operation includes the case of 0! = 1, so by definition 0! = 1.
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It includes it as a 'special case' that doesn't fit the definition that applies to every other number.

I'm not arguing that the bods in mathematics have decided 0!= 1 and they explain that in books, just that it isn't consistent and therefore leads to definition that is not always true.

It would like just deciding that 0 is an odd number and then using the common definition of what an even number is and just suffixing '..oh except for 0 which is odd just because it is'.
 
You can calculate 0! if you use a recursive definition of the factorial function:

x! = (x+1)!/(x+1)

Starting from 4! (which is 24)

3! = 4!/4 = 24/4 = 6
2! = 3!/3 = 6/3 = 2
1! = 2!/2 = 2/2 = 1
0! = 1!/1 = 1/1 = 1

and then

-1! = 0!/0 = 1/0 = undefined
etc.
 
estebanrey said:
It includes it as a 'special case' that doesn't fit the definition that applies to every other number.

I'm not arguing that the bods in mathematics have decided 0!= 1 and they explain that in books, just that it isn't consistent and therefore leads to definition that is not always true.

It would like just deciding that 0 is an odd number and then using the common definition of what an even number is and just suffixing '..oh except for 0 which is odd just because it is'.
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Now getting pedantic, but special case or not it is still part of the definition, which is why I responded to correct your post which said by 'definition' 0! should be 0 or invalid - it shouldn't.

Besides, it's not really accurate to say it's just appended to the definition as an 'oh but 0! is special' extra statement. Even when you define factorial as a product of integers, the case of 0! is 'automatically' covered by the rule that a product of no terms is always 1. Other definitions also allow you to evaluate factorials and get the right answer for 0! without doing anything 'special'. E.g. Tuppy's example above, and also the power rule definition...

n! = D^n x^n

So it's not really even a flawed or internally inconsistent definition.
 
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