What is the rationale for defining 0 factorial to be 1?
Answer
The defining 0 factorial to be 1 is not a rationale.
"Why is zero factorial equal to one?" is a problem that one has to prove.
When 0 factorial to be 1 to be proved,
the defining 0 factorial to be 1 is unvaluable.
One has only one general primitive definition of a factorial number:
n! = n x (n-1) x (n-2) x (n-3) x ... x 2 x 1.
After that zero factorial denoted 0! is a problem that one has to accept
by convention 0!=1 as a part of definition.
One has to prove zero factorial to be one.
Only from the definition of a factorial number and by dividing both sides
by n one has: n!/n (n-1)! or (n-1)! = n!/n
when n=2 one has (2-1)! = 2!/2 or 1! = 2x1/2 or 1! = 1
when n=1 one has (1-1)! = 1!/1 or 0! = 1/1 or 0! = 1. =
This is a proof that zero factorial is equal to one to be known.
But a new proof is:
A Schema Proof Without Words
That Zero Factorial Is Equal To One.
. This representation illustrates that upon solving it you use 0!.
In proofs you cannot define something by using that which you are defining in the definition. (ie) 0! can't be used when solving a problem within a proof of 0!.
For clarification, the above series will represent the drawn out solution for the factorial of a number, i. (ie) 1×76 -6×66 +15×56 -20×46 +15×36 -6×26 +1×16 , where i=6.
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The simple answer is that it is defined to be 1. But there is reason behind the decision.As you know, the factorial of a number (n) is equal to:n! = n * (n-1) * (n-2) ... * 1Another way of writing this is:n! = n * (n-1)!Suppose n=1:1! = 1 * 0!or1 = 1 * 0!or1 = 0!So by defining 0! as 1, formula involving factorials will work for all integers, including 0.
Zero factorial is one because n! = n-1! X n. For example: 4! = (4-1) X 4. If zero factorial was zero, that would mean 1! =(1-1) X 1 = 0 X 1=0. Then if 1!=0, then even 999! would equal zero. Therefore, zero factorial equals 1.
Definition of FactorialLet n be a positive integer. n factorial, written n!, is defined by n! = 1 * 2 * 3 * ... (n - 1) * nThe special case when n = 0, 0 factorial is given by: 0! = 1
AnswerAnswer: ( 0! + 0! + 0! + 0! + 0! ) ! = 120 Explanation: Here we have used operator called " factorial ". As you know that 0! = 1 so, = ( 0! + 0! + 0! + 0! + 0! ) ! = ( 1 + 1 + 1 + 1 + 1 ) ! = (5 )! = 120 : ( 0! + 0! + 0! + 0! + 0! ) ! = 120 Explanation: Here we have used operator called " factorial ". As you know that 0! = 1 so, = ( 0! + 0! + 0! + 0! + 0! ) ! = ( 1 + 1 + 1 + 1 + 1 ) ! = (5 )! = 120
It is not except when n = 1.