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Why factorial of 0 equals 1?

Updated: 4/28/2022
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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.

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Q: Why factorial of 0 equals 1?
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How is 0 factorial equals 1?

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.


Factorial notation in mathematics?

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


What is the rationale for defining 0 factorial to be 1?

What is the rationale for defining 0 factorial to be 1?AnswerThe 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 acceptby 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 sidesby n one has: n!/n (n-1)! or (n-1)! = n!/nwhen n=2 one has (2-1)! = 2!/2 or 1! = 2x1/2 or 1! = 1when 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 WordsThat Zero Factorial Is Equal To One.... ... ...Now the expression 0! = 1 is already a proof, not need a definitionnor a convention. So the defining 0 factorial to be 1 is unvaluable.The proof "without words" abovethat zero factorial is equal to one is a New that:*One has not to accept by convention 0!=1 anymore.*Zero factorial is not an empty product.*This Schema leads to a Law of Factorial.Note that the above schema is true but should not be used in a formal proof for 0!=1.The problem arises when you simplify the pattern formed by this schema into a MacLauren Series, which is the mathematical basis for it in the first place. Upon doing so you arrive with, . 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.


The expression 3 factorial equals what?

3! = 3×2×1 = 6


Given 5 zeros using any mathematical operations can you make a total of 120?

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

Related questions

What is the value of 0 factorial?

Zero factorial, written as 0!, equals 1. This is a simple math equation.


Zero factorial equal to one factorial then if we cancel the factorials on both side then the answer becomes zero equals one. do u accepts this?

0!=1! 1=1 The factorial of 0 is 1, not 0


What is factorial of 0?

Factorial(0), or 0! = 1.


Use 5 0s and any mathematical operators to get 120?

The trick is that zero factorial (0!) equals 1[1], so (0!+0!+0!+0!+0!)! = 5! = 120


What is the value of abc if abc equals a factorial plus b factorial plus c factorial?

145 1! = 1 4! = 24 5! = 120


How is 0 factorial equals 1?

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.


Factorial notation in mathematics?

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


Is factorial of zero is 1?

yes, 0!=1 default.


What is 25 Factorial?

25 factorial equals 15,511,210,043,330,985,984,000,000


What is the factorial of 0?

A recursive formula for the factorial is n! = n(n - 1)!. Rearranging gives (n - 1)! = n!/n, Substituting 'n - 1' as 0 -- i.e. n = 1 -- then 0! = 1!/1, which is 1/1 = 1.


Why we write 1 of the factorial of 0?

simply, any number divided by 0 is 0.


What is the rationale for defining 0 factorial to be 1?

What is the rationale for defining 0 factorial to be 1?AnswerThe 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 acceptby 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 sidesby n one has: n!/n (n-1)! or (n-1)! = n!/nwhen n=2 one has (2-1)! = 2!/2 or 1! = 2x1/2 or 1! = 1when 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 WordsThat Zero Factorial Is Equal To One.... ... ...Now the expression 0! = 1 is already a proof, not need a definitionnor a convention. So the defining 0 factorial to be 1 is unvaluable.The proof "without words" abovethat zero factorial is equal to one is a New that:*One has not to accept by convention 0!=1 anymore.*Zero factorial is not an empty product.*This Schema leads to a Law of Factorial.Note that the above schema is true but should not be used in a formal proof for 0!=1.The problem arises when you simplify the pattern formed by this schema into a MacLauren Series, which is the mathematical basis for it in the first place. Upon doing so you arrive with, . 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.