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

... ... ...








Now the expression 0! = 1 is already a proof, not need a definition
nor a convention. So the defining 0 factorial to be 1 is unvaluable.
The proof "without words" above
that 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.

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Q: What is the rationale for defining 0 factorial to be 1?
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Related Questions

Why 0 factorial is assumed to be 1?

That is related with the fact that 1 is the identity element (or neutral element) of multiplication - and factorials are defined as multiplications. Defining 0 factorial thus simplifies several formulae.


What is factorial of 0?

Factorial(0), or 0! = 1.


Why and how 0 factorial is 1?

This is related to the fact that 1 is the neutral element for multiplication. Defining the factorial this way makes some equations simpler, making it unnecessary to include additional conditions every time a rule is stated.


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


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 value of 0 factorial?

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


Why factorial of 0 equals 1?

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.


Is factorial of zero is 1?

yes, 0!=1 default.


Algorithm of factorial of a number?

Factorial (n) = n * Factorial (n-1) for all positive values n given Factorial (1) = Factorial (0) = 1. Pseudo-code: Function: factorial, f Argument: positive number, n IF n<=1 THEN RETURN 1 ELSE RETURN n * f(n-1) END IF


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 a program in c to calculate factorial of number?

First of all we will define what factorial is and how to it is calculated.Factional is non negative integer. Notation would be n! It is calculated by multiplying all integers from 1 to n;For example:5! = 1 x 2 x 3 x 4 x 5 = 120.Note: 0! = 1Small C program that illustrates how factorial might be counted:#include int factorial(int num);int main() {int num;printf("Enter number: ");scanf("%d", &num);printf("Factorial: %d\n", factorial(num));return 0;}int factorial(int num) {if (num == 0) {return 1;}return num * factorial(num - 1);}Testing:Enter number: 5Factorial: 120Enter number: 0Factorial: 1


What is 1 factorial?

1 factorial = 1


What is the algorithm to find the factorial of given number?

For any positive integer, n, factorial (n) can be calculated as follows: - if n<2, return 1. - otherwise, return n * factorial (n-1). The algorithm is recursive, where n<2 represents the end-point. Thus for factorial (5) we find the following recursive steps: factorial (5) = 5 * factorial (4) factorial (4) = 4 * factorial (3) factorial (3) = 3 * factorial (2) factorial (2) = 2 * factorial (1) factorial (1) = 1 We've now reached the end-point (1 is less than 2) and the results can now filter back up through the recursions: factorial (2) = 2 * factorial (1) = 2 * 1 = 2 factorial (3) = 3 * factorial (2) = 3 * 2 = 6 factorial (4) = 4 * factorial (3) = 4 * 6 = 24 factorial (5) = 5 * factorial (4) = 5 * 24 = 120 Thus factorial (5) = 120. We can also use a non-recursive algorithm. The factorial of both 0 and 1 is 1 thus we know that the return value will always be at least 1. As such, we can initialise the return value with 1. Then we begin iterations; while 1<n, multiply the return value by n and then subtract 1 from n. We can better represent this algorithm using pseudocode: Function: factorial (n), where n is an integer such that 0<=n. Returns an integer, f. Let f = 1 Repeat while 1<n Let f = f * n Let n = n - 1 End repeat Return f


How to write a program to find the delimiter matching in stacks?

== == using recursions: unsigned int Factorial( unsigned int x) { if(x>0) { return ( x * Factorial(x-1)); } else { return(1); } } factorial: unsigned int Factorial( unsigned int x) { unsigned int u32fact = 1; if( x == 0) { return(1); } else { while(x>0) { u32fact = u32fact *x; x--; } } }


Factorial value from do-while loop?

double Factorial(int i) { double x = 1; if (i <= 1) return 1; do { x *= i--; } while (i > 0); return x; }


Why one factorial and zero factorial is same?

As we know product of no numbers at all is 1 and for this reason factorial of zero =1and we know factorial of 1=1


A Write a program that reads a nonnegative integer and computes and prints its factorial?

int main() { // Variable declarations. unsigned long int factorial = 1 , number = 1; // reads a number for finding its factorial. cout > number; while( number > 1 ) { factorial *= number * ( number - 1 ); number -= 2; } cout


Write a function to calculate the factorial value of any integer entered through the keyboard?

#include <iostream> using namespace std; int main() { int i, number=0, factorial=1; // User input must be an integer number between 1 and 10 while(number<1 number>10) { cout << "Enter integer number (1-10) = "; cin >> number; } // Calculate the factorial with a FOR loop for(i=1; i<=number; i++) { factorial = factorial*i; } // Output result cout << "Factorial = " << factorial << endl;


Write a program that will input the value of N compute and display the factorial of N?

/* gcc -ansi -Wall -Wextra -pedantic -s -static 0.c -o 0 */ #include <stdio.h> int main ( ) { int n , factorial = 1 ; printf ( "enter the value of n\n") ; scanf ( "%i" , & n ) ; while ( n != 0 ) { factorial *= n ; n -- ; } printf ( "The factorial of n is\n%i\n" , factorial ) ; return 0; }


Factorial of a given number using functions?

double factorial(double N){double total = 1;while (N > 1){total *= N;N--;}return total; // We are returning the value in variable title total//return factorial;}int main(){double myNumber = 0;cout > myNumber;cout


Write a program which employs Recursion?

#include int factorial(int n){if(n 1)return 1;return n*factorial(n-1);}int main(int argc, char *argv[]){if(argc != 2){printf("%s number", argv[0]);exit(1);}printf("%s! = %d", argv[0], factorial(atoi(argv[1])));return 0;}


What is factorial of 998?

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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