Why is zero factorial equal to one?

Answer 1

It is defined to be so, presumably for the sake of convenience. Some people will tell you that it can be proven, but if it could be proven, it would be categorized as a theorem, not a definition. The common notion is that, since n!/n = (n-1)!, we can substitute 1 in for n and we can see that 0! is 1. The problem with this is that, if 0! is not assumed to be 1 (which is an assumption mathematicians do make), this rule will only hold for values of n that are equal to or greater than 2. To see why, let's look at the proof that n!/n = (n-1)!:

n!/n = n!/n | reflexive property

(n)(n-1)(n-2).../n = n!/n | definition of factorial

(n-1)(n-2)... = n!/n | cancelling the common factor of n

(n-1)! = n!/n | definition of factorial

Notice that, in order for n! to be described as (n)(n-1)(n-2)... and proceed to be rewritten as (n-1)! after the n's cancel, the natural number n had to be greater than some natural number for (n-1) to be a factor in the factorial. This means that n must be at least 2, because if n were 1, (n-1) would not have been a factor of the factorial, and the proof would fail unless we assume that n is at least 2. So now you know that this rule cannot prove that 0! is 1 because 1 cannot be substituted into the rule because, as it stands, the rule is only valid for values of 2 or greater. The rule is valid for values of 1 or greater if it is assumed that 0! is 1, but that is what you are trying to prove.

Answer 2

n factorial is defined as:

n! = n x (n-1) x (n-2) ... x 1

So,

n! = n x (n-1)!

If we replace "n" with 1:

1! = 1 x 0!

1 = 1 x 0!

1 = 0!

so, 0! is equal to 1. Why Zero Factorial is Equal to One Why should you be concerned that 0!=1? You may not even know what a factorial number is.

Reading the following you may come to understand the idea of a factorial.You may also be able to please your friends and confound your enemies bybeing able to show that 0!=1. Here is an explanation, requiring only knowledge of simple arithmetic tounderstand.

When considering the numbers of different groups which may be formed from acollection of objects one frequently finds the need to make calculations ofthe form, 4 x 3 x 2 x 1 for 4 objects. I suppose I must explain why that is.

Say we have the four objects A, B, C, D.
If I select from that collection to form another group, you will see that Icould choose any one of the four for the first selection. Thiswould leave three objects. I could select any of those three for the second selection,leaving any one of two for the next and so on. The total number of ways I could selectthose objects would be 4 x 3 x 2 x ... , and the series would stop at one when I selectedthe last object. To save repeatedly writing down long strings of such products thenotation 4! is used to represent 4 x 3 x 2 x 1 or 6! for 6 x 5 x 4 x 3 x 2 x 1.The ! is read as factorial. So the examples quoted above are more easily written as 4!and 6!. If you care to calculate their values they are 24 and 720 respectively.

In general any factorial number (call it n!), may be written,
n! = n x (n-1) x (n-2) x (n-3) x ... x 2 x 1
This is the general definition of a factorial number.
If you want it in words; a factorial number is the product of all positive integers from1 to the number under consideration.
The main place it is likely to be encountered is when considering those groups andarrangements of objects mentioned above.

So where does all this 0! stuff fit in?

Nobody has trouble in stating 2! = 2 x 1 , or even that 1! = 1, but 0! appears to make nosense.
It does however, have a value of 1. This is rather counter intuitive but arisesdirectly from our general definition.
n! = n x (n-1) x (n-2) x (n-3) x ... x 2 x 1

Notice this may be written,
n! = n x (n-1)! Still exactly the same definition.

If the left hand side (LHS) = the right hand side (RHS) then dividing both sides by nshould leave them still equal, so it is still true to write,
n!/n = n x (n-1)!/n

The (n-1)! in the RHS is being both multiplied by n and divided by n. These cancelleaving,
n!/n = (n-1)! If you doubt this, try it with real numbers, e.g. 4!/4 = 3! or (4 x 3 x 2 x 1)/4 = 3 x 2 x 1 = 6

The equation we now have is,
n!/n = (n-1)!
It is still our original definition in arearranged form. For convenience I shall write it the other way round.
(n-1)! = n!/n

We also said that our factorial uses the positive integers 1 and above.Try the value of n=2 in our rearranged formula and we get,
(2-1)! = 2!/2 or 1! = 2x1/2
The RHS calculates to 1, so we have the statement 1!=1That is what we guessed intuitively above. It is now confirmed.But look what happens when we substitute the legitimate value of n=1in our formula.
(1-1)! = 1!/1
Evaluating this statement gives
0! = 1!/1
We have just shown 1!=1 so the RHS is 1/1 or 1.

Our rearranged definition of a factorial number gives directly the statement,
0!=1 Counter intuitive perhaps but if the definition is true then this is true.

Why not tell your friends about it? Help dispel the widespread ignorance about 0!