عاملي

هذا المقال هو عن products of consecutive integers. إذا كنت تريد statistical experiments over all combinations of values، انظر factorial experiment. إذا كنت تريد data representation by independent components، انظر factorial code.

Selected factorials; values in scientific notation are rounded

$n$	$n!$
0	1
1	1
2	2
3	6
4	24
5	120
6	720
7	5040
8	40320
9	362880
10	3628800
11	39916800
12	479001600
13	6227020800
14	87178291200
15	1307674368000
16	20922789888000
17	355687428096000
18	6402373705728000
19	121645100408832000
20	2432902008176640000
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	6.412337688×1010000
10000	2.846259681×1035659
25206	1.205703438×10100000
100000	2.824229408×10456573
205023	2.503898932×101000004
1000000	8.263931688×105565708
10100	1010101.9981097754820

في الرياضيات، المضروب أو العاملي لعدد صحيح طبيعي n ، و الذي يكتب n!، و الذي يقرأ "عاملي n"، هو جذاء الأعداد الصحيحة الموجبة قطعا و الأصغر أو تساوي n. ويكتب:

$${\begin{aligned}n!&=n\times (n-1)\times (n-2)\times (n-3)\times \cdots \times 3\times 2\times 1\\&=n\times (n-1)!\\\end{aligned}}$$

For example,

$$5!=5\times 4!=5\times 4\times 3\times 2\times 1=120.$$

The value of 0! is 1, according to the convention for an empty product.

• 1! = 1
• 2! = 1 x 2 = 2
• 3! = 1 x 2 x 3 = 6
• 10! = 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 = 3628800

و تعريف العاملي على شكل جذاء يترتب عنه كون 0! = 1 ذلك أن 0! جذاء مفرغ، و بمعنى آخر مقتصر على العنصر المحايد في عملية الضرب.

و يلعب العاملي دورا أساسيا في علم الإحتمالات و التراتيب بما أنه يوجد n! طريقة مختلفة لتوزيع n شيئا. و يظهر العاملي في عدة معادلات رياضية، مثل سيغة الثنائي لنيوتن و صيغة تايلور.

و تعطينا صيغة سترلنج مساويا لـ n! عندما تكون n كبيرة :

$\lim _{n\to +\infty }{\frac {n!}{{\sqrt {2\pi n}}(n/e)^{n}}}=1.$

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.

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 التاريخ

The concept of factorials has arisen independently in many cultures:

• In Indian mathematics, one of the earliest known descriptions of factorials comes from the Anuyogadvāra-sūtra, one of the canonical works of Jain literature, which has been assigned dates varying from 300 BCE to 400 CE. It separates out the sorted and reversed order of a set of items from the other ("mixed") orders, evaluating the number of mixed orders by subtracting two from the usual product formula for the factorial. The product rule for permutations was also described by 6th-century CE Jain monk Jinabhadra. Hindu scholars have been using factorial formulas since at least 1150, when Bhāskara II mentioned factorials in his work Līlāvatī, in connection with a problem of how many ways Vishnu could hold his four characteristic objects (a conch shell, discus, mace, and lotus flower) in his four hands, and a similar problem for a ten-handed god.
• In the mathematics of the Middle East, the Hebrew mystic book of creation Sefer Yetzirah, from the Talmudic period (200 to 500 CE), lists factorials up to 7! as part of an investigation into the number of words that can be formed from the Hebrew alphabet. Factorials were also studied for similar reasons by 8th-century Arab grammarian Al-Khalil ibn Ahmad al-Farahidi. Arab mathematician Ibn al-Haytham (also known as Alhazen, c. 965 – c. 1040) was the first to formulate Wilson's theorem connecting the factorials with the prime numbers.
• In Europe, although Greek mathematics included some combinatorics, and Plato famously used 5,040 (a factorial) as the population of an ideal community, in part because of its divisibility properties, there is no direct evidence of ancient Greek study of factorials. Instead, the first work on factorials in Europe was by Jewish scholars such as Shabbethai Donnolo, explicating the Sefer Yetzirah passage. In 1677, British author Fabian Stedman described the application of factorials to change ringing, a musical art involving the ringing of several tuned bells.

From the late 15th century onward, factorials became the subject of study by western mathematicians. In a 1494 treatise, Italian mathematician Luca Pacioli calculated factorials up to 11!, in connection with a problem of dining table arrangements. Christopher Clavius discussed factorials in a 1603 commentary on the work of Johannes de Sacrobosco, and in the 1640s, French polymath Marin Mersenne published large (but not entirely correct) tables of factorials, up to 64!, based on the work of Clavius. The power series for the exponential function, with the reciprocals of factorials for its coefficients, was first formulated in 1676 by Isaac Newton in a letter to Gottfried Wilhelm Leibniz. Other important works of early European mathematics on factorials include extensive coverage in a 1685 treatise by John Wallis, a study of their approximate values for large values of $n$ by Abraham de Moivre in 1721, a 1729 letter from James Stirling to de Moivre stating what became known as Stirling's approximation, and work at the same time by Daniel Bernoulli and Leonhard Euler formulating the continuous extension of the factorial function to the gamma function. Adrien-Marie Legendre included Legendre's formula, describing the exponents in the factorization of factorials into prime powers, in an 1808 text on number theory.

The notation $n!$ for factorials was introduced by the French mathematician Christian Kramp in 1808. Many other notations have also been used. Another later notation, in which the argument of the factorial was half-enclosed by the left and bottom sides of a box, was popular for some time in Britain and America but fell out of use, perhaps because it is difficult to typeset. The word "factorial" (originally French: factorielle) was first used in 1800 by Louis François Antoine Arbogast, in the first work on Faà di Bruno's formula, but referring to a more general concept of products of arithmetic progressions. The "factors" that this name refers to are the terms of the product formula for the factorial.

 عاملي عدد غير صحيح

لكل عدد صحيح n، لدينا $\Gamma (n+1)=n!$ حيث Γ هي دالة أولير (دالة غاما) و xcxzczxccعها ليونارد أولير. و تمكننا دالة أولير من تعميم العاملي على مجموعة الأعداد المركّبة باستثناء الأعداد السالبة قطعا. و في النهاية نجد :

$\Gamma (z+1)=\int _{0}^{\infty }t^{z}e^{-t}\,\mathrm {d} t=z!\ \forall z>-1$

 البرمجة

يمكن حساب عاملي عدد ما باستعمال خوارزميات الرجوع. فلنكتب باستعمال لغة Scheme، القريبة من لغة Lisp، برنامجا رجعيّا يعطينا عاملي عدد صحيح :

(define fact
  (lambda (x)
    (if (= x 0) 1
      (* x (fact (- x 1))))))

و هذا البرنامج السابق غير مفيد في حالة الاعداد الكبيرة.

و بنفس الطريقة في Caml :

let rec fact n = 
  match n with
    | 0 -> 1
    | _ -> n * fact(n-1)
;;

و بطريقة أخرى:

let fact n =
  let rec aux n r =
    match n with
      | 0 -> r
      | _ -> aux (n-1) (n*r)
  in
  aux n 1
;;

و في لغة سي:

int factorielle_recursive(int n)
{
  if (n == 0)
    return 1;
  else
    return n * factorielle_recursive(n-1);
}

و بطريقة أخرى:

int factorielle_iterative(int n)
{
  int res;
   
  for (res = 1; n > 1; n--)
    res *= n;
   
  return res;
}

و في لغة Python:

fact = lambda x : x>0 and x*fact(x-1) or 1

----------------------------------------------------

الاستعمال :
for i in range(10):
    print "fact %d = %d" %(i, fact(i))

و يظهر على الشاشة :
fact 0 = 1
fact 1 = 1
fact 2 = 2
fact 3 = 6
fact 4 = 24
fact 5 = 120
fact 6 = 720
fact 7 = 5040
fact 8 = 40320
fact 9 = 362880

هذه الدوال (البرامج) لا تمكننا من حساب عملي أعداد أكبر من 12 إذا كانت الاعداد الصحيحة محدودة بـ 32 بت، لأن النتيجة تتعدى المساحة المتوفرة.

 وصلات خارجية

• قالب:OEIS el
• Hazewinkel, Michiel, ed. (2001), "Factorial", Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104 
• Eric W. Weisstein, Factorial at MathWorld.