The Factorial of 100: Exploring Large Numbers and Stirling's Approximation

Factorials are an essential concept in mathematics and computer science, often appearing in algorithms, combinatorics, and statistics. One of the most intriguing and often-studied factorials is the factorial of 100, denoted as 100!. This number, while not explicitly required for daily calculations, holds a unique place in the world of mathematics and computer programming.

The Exact Value of 100!

The exact value of 100! is a massive number that can be represented as:

93326215443944152681699238856266700490715968264381621468592963895217599993229915608941463976156518286253697920827223758251185210916864000000000000000000000000

This value is calculated using modern programming languages such as Python 3.5.2, as demonstrated in the provided code snippet. The use of a factorial function in Python's math library simplifies this computation significantly. When computed, the value of 100! approximates to:

9.332621544 times; 10157

Stirling's Approximation for 100!

For practical applications, it is often useful to approximate the value of large factorials without performing the full multiplication, especially when the exact value is not necessary. Stirling's approximation is a powerful tool for this purpose and can provide a close estimate of the factorial value. The approximation for n! is given by:

(n! approx sqrt{2 pi n} left( frac{n}{e} right)^n)

Let's apply this to 100:

Calculate the square root of (2 * π * n): For 100, this is approximately (sqrt{2 pi 100} approx 17.7245 Calculate (left( frac{n}{e} right)^n: For 100, this is approximately (left( frac{100}{2.71828} right)^{100} approx 2.688 times 10^{157}

Multiplying these two values gives:

(17.7245 times; 2.688 times; 10^{157} approx 9.34 times; 10^{157}

This approximation is very close to the exact value calculated earlier, demonstrating the utility of Stirling's approximation for large factorials.

Computer Implementation of Factorial Calculation

For those interested in computing the factorial of 100 using a program, the following C code snippet provides a straightforward example:

#include iostream
#include string
#include vector
using namespace std;
string multiply(string num1, string num2) {
    int n1  num1.length();
    int n2  num2.length();
    if (n1  0 || n2  0) return ;
    vectorint result(n1   n2, 0);
    int i_n1  0;
    int i_n2  0;
    int carry  0;
    for (int i  n1 - 1; i  0; i--) {
        int n1  num1[i] - '0';
        i_n2  0;
        for (int j  n2 - 1; j  0; j--) {
            int n2  num2[j] - '0';
            int sum  n1 * n2   result[i_n1   i_n2]   carry;
            carry  sum / 10;
            result[i_n1   i_n2]  sum % 10;
            i_n2  ;
        }
        if (carry  0) {
            result[i_n1   i_n2]   carry;
            carry  0;
        }
        i_n1  ;
    }
    // To ignore leading zeros
    int i  () - 1;
    while (i  0  result[i]  0) {
        i--;
    }
    if (i  -1) {
        return 0;
    }
    string s  ;
    for (int j  i; j  0; j--) {
        s   to_string(result[j]);
    }
    return s;
}
string factorial(int n) {
    string result  1;
    for (int i  2; i  n; i  ) {
        result  multiply(result, to_string(i));
    }
    return result;
}
int main() {
    int n  100;
    cout  factorial(n)  endl;
    return 0;
}

This C program uses string manipulation to multiply large numbers and compute the factorial. The `multiply` function takes two strings, multiplies them, and returns the result as a string. The `factorial` function then uses this `multiply` function to compute the factorial of the given integer.

Conclusion

The factorial of 100 is an impressive number that showcases the power and complexity of large number computations. Understanding and approximating such large factorials can be crucial in various fields of mathematics, computer science, and engineering. Whether you need the exact value or an estimate, tools like Stirling's approximation and robust programming techniques can be invaluable.