Pengguna:Dedhert.Jr/Uji halaman 01/13

Selected factorials; nilai dalam notasi ilmiah dibulatkan
$n$	$n!$
0	1
1	1
2	2
3	6
4	24
5	120
6	720
7	5040
8	40.320
9	362.880
10	3.628.800
11	39.916.800
12	479.001.600
13	6.227.020.800
14	87.178.291.200
15	1.307.674.368.000
16	20.922.789.888.000
17	355.687.428.096.000
18	6.402.373.705.728.000
19	121.645.100.408.832.000
20	2.432.902.008.176.640.000
25	1,551121004×10²⁵
50	3,041409320×10⁶⁴
70	1,197857167×10¹⁰⁰
100	9,332621544×10¹⁵⁷
450	1.733368733×10¹⁰⁰⁰
1000	4.023872601×10²⁵⁶⁷
3249	6,412337688×10^10.000
10000	2,846259681×10^35.659
25206	1,205703438×10^100.000
100000	2,824229408×10^456.573
205023	2,503898932×10^1.000.004
1000000	8,263931688×10^5.565.708
10¹⁰⁰	10^{10^101,9981097754820}

Dalam matematika, faktorial dari $n$ bilangan bulat, dilambang sebagai $n!$ , merupakan hasil kali dari semua bilangan bulat positif yang lebih kecil atau sama dengan $n$ . Faktorial dari $n$ juga merupakan hasil kali dari $n$ dengan faktorial lebih kecil berikutnya: ${\begin{aligned}n!&=n\times (n-1)\times (n-2)\times (n-3)\times \cdots \times 3\times 2\times 1\\&=n\times (n-1)!\\\end{aligned}}$ Contohnya, $5!=5\times 4\times 3\times 2\times 1=5\times 24=120.$ Nilai $0!$ adalah 1, menurut konvensi darab kosong.^[1]

Factorials have been discovered in several ancient cultures, notably in Indian mathematics in the canonical works of Jain literature, and by Jewish mystics in the Talmudic book Sefer Yetzirah. The factorial operation is encountered in many areas of mathematics, notably in combinatorics, where its most basic use counts the possible distinct sequences – the permutations – of $n$ distinct objects: there are $n!$ . In mathematical analysis, factorials are used in power series for the exponential function and other functions, and they also have applications in algebra, number theory, probability theory, and computer science.

Much of the mathematics of the factorial function was developed beginning in the late 18th and early 19th centuries. Stirling's approximation provides an accurate approximation to the factorial of large numbers, showing that it grows more quickly than exponential growth. Legendre's formula describes the exponents of the prime numbers in a prime factorization of the factorials, and can be used to count the trailing zeros of the factorials. Daniel Bernoulli and Leonhard Euler interpolated the factorial function to a continuous function of complex numbers, except at the negative integers, the (offset) gamma function.

Many other notable functions and number sequences are closely related to the factorials, including the binomial coefficients, double factorials, falling factorials, primorials, and subfactorials. Implementations of the factorial function are commonly used as an example of different computer programming styles, and are included in scientific calculators and scientific computing software libraries. Although directly computing large factorials using the product formula or recurrence is not efficient, faster algorithms are known, matching to within a constant factor the time for fast multiplication algorithms for numbers with the same number of digits.

^ Graham, Ronald L.; Knuth, Donald E.; Patashnik, Oren (1988). Concrete Mathematics. Reading, MA: Addison-Wesley. hlm. 111. ISBN 0-201-14236-8.

[gkp-1] Graham, Ronald L.; Knuth, Donald E.; Patashnik, Oren (1988). Concrete Mathematics. Reading, MA: Addison-Wesley. hlm. 111. ISBN 0-201-14236-8.

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