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Factorial

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Factorial
TypeCombinatorial
Based onMultiplication
Growth rate\(f_{2}(n)\)

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

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


For example, \(6! = 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 720\). 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 \(n! = n \cdot (n - 1)!\). 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, 5040, 40320, 362880, 3628800, and 39916800.

Properties Edit

The sum of the reciprocals of the factorials is \(\sum^{\infty}_{i = 0} \frac{1}{i!} = \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \cdots = 2.71828182845904\ldots\), a mathematical constant better known as \(e\). In fact, \(e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots\), 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: \(145 = 1! + 4! + 5!\) and \(40585 = 4! + 0! + 5! + 8! + 5!\).

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, 10000! has 2000 + 400 + 80 + 16 + 3 = 2499 zeroes.

Specific numbers Edit

One hundred factorial's decimal expansion is shown below.

93326215443944152681699238856266700490715968264381621468592963895217599993229915608941463976156518286253697920827223758251185210916864000000000000000000000000

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, and Aarex Tiaokhiao has proposed the name Myriadbang for 10,000!

One thousand factorial is about 4.0238726007 × 102567.

70! is the smallest factorial which is greater than googol, while 69! still has only 99 digits.

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

Factorials
Main article: Factorial
Multifactorials: Double factorial · Multifactorial
Falling and rising: Falling factorial · Rising factorial
Other mathematical variants: Alternating factorial · Hyperfactorial · q-factorial · Roman factorial · Subfactorial · Weak factorial
Tetrational growth: Exponential factorial · Expostfacto function · Superfactorial by Clifford Pickover
Array-based extensions: Hyperfactorial array notation · Nested factorial notation
Other googological variants: · Superfactorial by Sloane and Plouffe · Torian · Factorexation · Mixed factorial


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