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Template:Function The factorial is a function applied to whole numbers, defined as[1][2]

$$n! = \prod^n_{i = 1} i = n*(n-1)* ... *4*3*2*1.$$


For example, 6! = 6*5*4*3*2*1 = 720. It is equal to the number of ways n distinct objects can be arranged, because there are \(n\) ways to place the first object, n - 1 ways to place the second object, and so forth. The special case 0! = 1 has been set by definition; there is one way to arrange zero objects.

Before the notation \(n!\) was invented, \(n\) was common. 

The function can be defined recursively as 0! = 1 and n! = n * (n - 1)!. The first few values of \(n!\) for \(n = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\) are 1, 1 , 2, 6, 24, 120, 720, 5,040, 40,320, 362,880, 3,628,800, and 39,916,800.

Properties Edit

The sum of the reciprocals of the factorials is = 0 + 1 + x + 1/(2!) + 1/(3!) ... = 2.71828182845904..., a mathematical constant better known as e. In fact, e^x = + ..., which illustrates the important property that \(\frac{d}{dx}e^x = e^x\).

Because \(n! = \Gamma (n + 1)\) (where \(\Gamma (x)\) is the gamma function), \(n! = \int^{\infty}_0 e^{-t} \cdot t^{n} dt\). This identity gives us factorials of positive real numbers (and negative non-integer real numbers), not limited to integers:

  • \(\left(\frac{1}{2}\right)! = \frac{\sqrt{\pi}}{2}\)
  • \(\left(-\frac{1}{2}\right)! = \sqrt{\pi}\) 

The most well-known approximation of n! is \(n!\approx \sqrt{2\pi n}(\frac{n}{e})^n\), and it's called Stirling's approximation.

In base 10, only two non-trivial numbers are equal to the sum of the factorials of their digits: \(145 = 1! + 4! + 5! = 5 × 29\) and \(40,585 = 4! + 0! + 5! + 8! + 5! = 5 × 8,117\).

The number of zeroes at the end of the decimal expansion of \(n!\) is \(\sum_{k = 1} \lfloor n / 5^k\rfloor\).[3] For example, 10,000! has 2,000 + 400 + 80 + 16 + 3 = 2,499 zeroes.

Specific numbers Edit

  • 479,001,600 is equal to \(12!\), and therefore the number of possible tone rows in the twelve-tone technique.
  • 1,124,000,727,777,607,680,000 is a positive integer equal to \(22!\). It is notable in computer science for being the largest factorial number which can be represented exactly in the double floating-point format (which has a 53-bit significand).
  • 70! is the smallest factorial which is greater than googol, while 69! still has only 99 digits.
  • One hundred factorial's decimal expansion is shown below.
    Template:100!
    • In scientific notation, this is approximately 9.3326215443 × 10157. It seems to be approximately googol3/2, although it is almost 100 million times larger.
  • Lawrence Hollom calls 200! faxul.
  • One thousand factorial is about 4.0238726007 × 102,567.
  • Aarex Tiaokhiao has proposed the name Myriadbang for 10,000!.
  • One million factorial is approximately 8.2639317 × 105,565,708.

Variation Edit

Aalbert Torsius defines a variation on the factorial, where \(x!n = \prod^{x}_{i = 1} i!(n - 1) = 1!(n - 1) \cdot 2!(n - 1) \cdot \ldots \cdot x!(n - 1)\) and \(x!0 = x\).[4]

\(x!n\) is pronounced "nth level factorial of x." \(x!1\) is simply the ordinary factorial and \(x!2\) is Sloane and Plouffe's superfactorial \(x\$\).

The special case \(x!x\) is a function known as the Torian.

Pseudocode Edit

// Standard factorial function
function factorial(z):
    result := 1
    for i from 1 to z:
        result := result * i
    return result

// Generalized factorial, using Lanczos approximation for gamma function

g := 7
coeffs := [0.99999999999980993, 676.5203681218851, -1259.1392167224028, 771.32342877765313, -176.61502916214059, 12.507343278686905, -0.13857109526572012, 9.9843695780195716e-6, 1.5056327351493116e-7]

function factorialReal(z):
    ag := coeffs[0]
    for i from 1 to g + 1:
        ag := ag + coeffs[i] / (z + i)
    zg := z + g + 0.5
    return sqrt(2 * pi) * zgz + 0.5 * e-zg * ag

// Torsius' factorial extension
function factorialTorsius(z, x):
    if x = 0:
        return z
    if x = 1:
        return factorial(z)
    result := 1
    for i from 1 to z:
        result := result * factorialTorsius(i, x - 1)
    return result

Sources Edit

  1. Factorial from Wolfram MathWorld
  2. Factorials from PurpleMath
  3. Factorials and Trailing Zeroes from PurpleMath
  4. [1]

See also Edit

Template:Factorials Template:Combinatorial googologisms de:Fakultät ja:階乗

nl:Faculteit

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