Factorial | |
---|---|

Type | Combinatorial |

Based on | Multiplication |

Growth rate | \(f_{2}(n)\) |

The **factorial** is a function applicable to any non-negative integer \(n\), 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 \(n! = n \cdot (n - 1)!\) for all positive integer \(n\). 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

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, the only positive integers having the property that each one is equal to the sum of the factorials of its digits are 1, 2, 145 and 40585.^{[3]}

- 1 = 1!
- 2 = 2!
- 145 = 1! + 4! + 5!
- 40585 = 4! + 0! + 5! + 8! + 5!

The number of zeroes at the end of the decimal expansion of \(n!\) is \(\sum_{k = 1} \lfloor n / 5^k\rfloor\).^{[4]} For example, 10,000! has 2,000 + 400 + 80 + 16 + 3 = 2,499 zeroes.

## Specific numbers

**153**is the sum of the factorials of the first five positive numbers, and also the exponent in the short scale quinquagintillion.- 1
^{3}+ 5^{3}+ 3^{3}= 153, thus 153 is a narcissistic number. - The first carrier frequency in the longwave radio band is at
**153**kHz. - It is also the number of fish in the second miraculous catch of fish.

- 1
**154**is the sum of the factorials of the first six nonnegative numbers, and 154! + 1 is a factorial prime.- It is also a central polygonal number and the 7th nonagonal number.
- Its prime factorization is 2 × 7 × 11.
^{[5]} - 153 and 154 are Ruth-Aaron pair.

**720**is an integer equal to 6!, the factorial of 6. Consequently, it is the order of the symmetric group of degree 6, which is isomorphic to B_{2}(2), and has an outer automorphism.- It is also the number of degrees in a hexagon.
- Furthermore, it is the number of hours in a 30-day month (April, June, September or November) not containing a DST transition.
- Some high-definition television services have 720 visible scan lines.
- Finally, it is also the number of pixels in a standard-definition television scan line.

**5,040**is an integer equal to 7!. It is the largest known factorial number which is the predecessor of a square number: 7! = 5,040 = 5,041−1 = 71^{2}−1.- Plato mentioned in his
*Laws*that 5,040 is a convenient number to use for dividing many things (including both the citizens and the land of a city-state or*polis*) into lesser parts, making it an ideal number for the number of citizens (heads of families) making up a*polis*. He remarks that this number can be divided by all the (natural) numbers from 1 to 12 with the single exception of 11 (however, it is not the smallest number to have this property; 2,520 is). - This is because 5040 is a highly composite number, where a number has more factors than any number less than it. Not all factorials are highly composite.

- Plato mentioned in his
**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).- In the short scale, this number is written as 1 sextillion 124 quintillion 727 trillion 777 billion 607 million 680 thousand.
- In the long scale, this number is written as 1 trilliard 124 trillion 727 billion 777 milliard 607 million 680 thousand.

- 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 .
- 93,326,215,443,944,152,681,699,238,856,266,700,490,715,968,264,381,621,468,592,963,895,217,599,993,229,915,608,941,463,976,156,518,286,253,697,920,827,223,758,251,185,210,916,864,000,000,000,000,000,000,000,000

- Lawrence Hollom calls 200! faxul.
- One thousand factorial is about 4.0238726007 × 10
^{2,567}. - Aarex Tiaokhiao has proposed the name Myriadbang for 10,000!.
- One million factorial is approximately 8.2639317 × 10
^{5,565,708}. - One billion factorial is approximately 1.57637137 × 10
^{8,565,705,531}.

## Approximations for these numbers

For 5040:

Notation | Lower bound | Upper bound |
---|---|---|

Scientific notation | \(5.04\times10^3\) | |

Arrow notation | \(17\uparrow3\) | \(71\uparrow2\) |

Steinhaus-Moser Notation | 5[3] | 6[3] |

Copy notation | 4[4] | 5[4] |

Chained arrow notation | \(17\rightarrow3\) | \(71\rightarrow2\) |

Taro's multivariable Ackermann function | A(3,9) | A(3,10) |

Pound-Star Notation | #*(50)*2 | #*(51)*2 |

PlantStar's Debut Notation | [2] | [3] |

BEAF | {17,3} | {71,2} |

Bashicu matrix system | (0)[70] | (0)[71] |

Hyperfactorial array notation | 7! | |

Bird's array notation | {17,3} | {71,2} |

Strong array notation | s(17,3) | s(71,2) |

Fast-growing hierarchy | \(f_2(9)\) | \(f_2(10)\) |

Hardy hierarchy | \(H_{\omega^2}(9)\) | \(H_{\omega^2}(10)\) |

Slow-growing hierarchy | \(g_{\omega^3}(17)\) | \(g_{\omega^2}(71)\) |

## Variation

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 \(x!0 = x\).^{[6]}

\(x!n\) is pronounced "*n*th 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

// Standard factorial functionfunctionfactorial(z):result:= 1forifrom1toz:result:=result*ireturnresult// Generalized factorial, using Lanczos approximation for gamma functiong:= 7coeffs:= [0.99999999999980993, 676.5203681218851, -1259.1392167224028, 771.32342877765313, -176.61502916214059, 12.507343278686905, -0.13857109526572012, 9.9843695780195716e-6, 1.5056327351493116e-7]functionfactorialReal(z):ag:=coeffs[0]forifrom1tog+ 1:ag:=ag+coeffs[i] / (z+i)zg:=z+g+ 0.5 return sqrt(2 *pi) *zg^{z + 0.5}*e^{-zg}*ag// Torsius' factorial extensionfunctionfactorialTorsius(z,x):ifx= 0:returnzifx= 1:returnfactorial(z)result:= 1forifrom1toz:result:=result*factorialTorsius(i,x- 1)returnresult

## Sources

- ↑ Factorial from Wolfram MathWorld
- ↑ Factorials from PurpleMath
- ↑ Gupta, S. S. (2004). 88.31 Sum of the Factorials of the Digits of Integers. The Mathematical Gazette, 88(512), 258–261. http://www.jstor.org/stable/3620841
- ↑ Factorials and Trailing Zeroes from PurpleMath
- ↑ https://www.wolframalpha.com/input/?i=154
- ↑ [1]

## See also

**Factorials**

**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 · Primorial · Compositorial · Semiprimorial

**Tetrational growth:**Exponential factorial · Expostfacto function · Superfactorial by Clifford Pickover

**Nested Factorials:**Tetorial · Petorial · Ectorial · Zettorial · Yottorial

**Array-based extensions:**Hyperfactorial array notation · Nested factorial notation

**Other googological variants:**· Tetrofactorial · Superfactorial by Sloane and Plouffe · Torian · Factorexation · Mixed factorial · Bouncing Factorial