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Pengguna:Dedhert.Jr/Uji halaman 01/13

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Revisi sejak 1 April 2022 05.42 oleh Dedhert.Jr (bicara | kontrib)


Halaman ini berisi artikel tentang hasil kali bilangan bulat berurutan. Untuk statistical experiments over all combinations of values, lihat factorial experiment. Untuk data representation by independent components, lihat factorial code.
Selected factorials; nilai dalam notasi ilmiah dibulatkan
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 6,412337688×1010.000
10000 2,846259681×1035.659
25206 1,205703438×10100.000
100000 2,824229408×10456.573
205023 2,503898932×101.000.004
1000000 8,263931688×105.565.708
10100 1010101,9981097754820

Dalam matematika, faktorial dari bilangan bulat, dilambang sebagai , merupakan hasil kali dari semua bilangan bulat positif yang lebih kecil atau sama dengan . Faktorial dari juga merupakan hasil kali dari dengan faktorial lebih kecil berikutnya:Contohnya,Catatan bahwa untuk nilai adalah 1, menurut konvensi darab kosong.[1]

Faktorial ditemukan dalam beberapa budaya kuno, khususnya di matematika India dalam tulisan karya sastra Jain, dan mistisisme Yahudi dalam buku Talmud yang berjudul Sefer Yetzirah. Operasi faktorial biasanya ditemukan dalam banyak cabang matematika, khususnya kombinatorik. Dalam kombinatorik, faktorial merupakan operasi paling dasar yang dipakai untuk menghitung kemungkinan barisan yang berbeda, sebagai contoh, permutasi dari benda yang berbeda ada . Dalam analisis matematika, faktorial dipakai dalam deret kuasa fungsi eksponensial dan fungsi lain. Faktorial juga memiliki aplikasi terhadap aljabar, teori bilangan, teori peluang, dan ilmu komputer.

Fungsi faktorial dalam matematika dikembangkan pada akhir abad ke-18 dan awal abad ke-19. Aproksimasi Stirling menyediakan sebuah hampiran yang akurat mengenai faktorial dari bilangan yang besar, yang memperlihatkan bahwa pertumbuhan nilainya lebih cepat daripada pertumbuhan eksponensial. Adapula rumus Legendre yang menjelaskan eksponen bilangan prima dalam faktorisasi bilangan prima melalui faktorial, dan rumus tersebut dapat dipakai untuk menghitung jejak nol melalui faktorial. Daniel Bernoulli dan Leonhard Euler menginterpolasi fungsi faktorial menjadi sebuah fungsi kontinu pada bilangan kompleks, kecuali pada bilangan bulat negatif. Fungsi tersebut ialah fungsi gamma (ofset).

Ada banyak fungsi khusus dan barisan bilangan lainnya terkait erat dengan faktorial, di antaranya koefisien binomial, faktorial ganda, faktorial turun, primorial, dan subfaktorial. Implementasi fungsi faktorial biasanya dipakai sebagai contoh tentang tampialn pemrograman komputer yang berbeda, di antaranya dalam kalkulator ilmiah dan scientific computing software libraries. Walaupun faktorial yang besar dihitung secara langsung melalui rumus hasil kali atau rekurensi tidak efisien, faster algorithms are known, matching to within a constant factor the time for fast multiplication algorithms for numbers with the same number of digits.

Sejarah

Konsep faktorial muncul secara terpisah dalam banyak budaya. In Indian mathematics, one of the earliest known descriptions of factorials comes from the Anuyogadvāra-sūtra,[2] one of the canonical works of Jain literature, which has been assigned dates varying from 300 BCE to 400 CE.[3] 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.[2] 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.[4]

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.[5][6] Factorials were also studied for similar reasons by 8th-century Arab grammarian Al-Khalil ibn Ahmad al-Farahidi.[5] 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.[7]

In Europe, although Greek mathematics included some combinatorics, and Plato famously used 5040 (a factorial) as the population of an ideal community, in part because of its divisibility properties,[8] 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.[9] In 1677, British author Fabian Stedman described the application of factorials to change ringing, a musical art involving the ringing of several tuned bells.[10][11]

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.[12] 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.[13] 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.[14] 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   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.[15] 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.[16]

The notation   for factorials was introduced by the French mathematician Christian Kramp in 1808.[17] 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.[17] The word "factorial" (originally French: factorielle) was first used in 1800 by Louis François Antoine Arbogast,[18] in the first work on Faà di Bruno's formula,[19] 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.[20]

Referensi

  1. ^ Graham, Ronald L.; Knuth, Donald E.; Patashnik, Oren (1988). Concrete Mathematics. Reading, MA: Addison-Wesley. hlm. 111. ISBN 0-201-14236-8. 
  2. ^ a b Datta, Bibhutibhusan; Singh, Awadhesh Narayan (2019). "Use of permutations and combinations in India". Dalam Kolachana, Aditya; Mahesh, K.; Ramasubramanian, K. Studies in Indian Mathematics and Astronomy: Selected Articles of Kripa Shankar Shukla. Sources and Studies in the History of Mathematics and Physical Sciences. Springer Singapore. hlm. 356–376. doi:10.1007/978-981-13-7326-8_18.  . Revised by K. S. Shukla from a paper in Indian Journal of History of Science 27 (3): 231–249, 1992, MR1189487. See p. 363.
  3. ^ Jadhav, Dipak (August 2021). "Jaina Thoughts on Unity Not Being a Number". History of Science in South Asia. University of Alberta Libraries. 9: 209–231. doi:10.18732/hssa67.  . See discussion of dating on p. 211.
  4. ^ Biggs, Norman L. (May 1979). "The roots of combinatorics". Historia Mathematica. 6 (2): 109–136. doi:10.1016/0315-0860(79)90074-0  . MR 0530622. 
  5. ^ a b Katz, Victor J. (June 1994). "Ethnomathematics in the classroom". For the Learning of Mathematics. 14 (2): 26–30. JSTOR 40248112. 
  6. ^ Sefer Yetzirah at Wikisource, Chapter IV, Section 4
  7. ^ Rashed, Roshdi (1980). "Ibn al-Haytham et le théorème de Wilson". Archive for History of Exact Sciences (dalam bahasa Prancis). 22 (4): 305–321. doi:10.1007/BF00717654. MR 0595903. 
  8. ^ Acerbi, F. (2003). "On the shoulders of Hipparchus: a reappraisal of ancient Greek combinatorics". Archive for History of Exact Sciences. 57 (6): 465–502. doi:10.1007/s00407-003-0067-0. JSTOR 41134173. MR 2004966. 
  9. ^ Katz, Victor J. (2013). "Chapter 4: Jewish combinatorics". Dalam Wilson, Robin; Watkins, John J. Combinatorics: Ancient & Modern. Oxford University Press. hlm. 109–121. ISBN 978-0-19-965659-2.  See p. 111.
  10. ^ Hunt, Katherine (May 2018). "The Art of Changes: Bell-Ringing, Anagrams, and the Culture of Combination in Seventeenth-Century England". Journal of Medieval and Early Modern Studies. 48 (2): 387–412. doi:10.1215/10829636-4403136. 
  11. ^ Stedman, Fabian (1677). Campanalogia. London. hlm. 6–9.  The publisher is given as "W.S." who may have been William Smith, possibly acting as agent for the Society of College Youths, to which society the "Dedicatory" is addressed.
  12. ^ Knobloch, Eberhard (2013). "Chapter 5: Renaissance combinatorics". Dalam Wilson, Robin; Watkins, John J. Combinatorics: Ancient & Modern. Oxford University Press. hlm. 123–145. ISBN 978-0-19-965659-2.  See p. 126.
  13. ^ Knobloch 2013, hlm. 130–133.
  14. ^ Ebbinghaus, H.-D.; Hermes, H.; Hirzebruch, F.; Koecher, M.; Mainzer, K.; Neukirch, J.; Prestel, A.; Remmert, R. (1990). Numbers. Graduate Texts in Mathematics. 123. New York: Springer-Verlag. hlm. 131. doi:10.1007/978-1-4612-1005-4. ISBN 0-387-97202-1. MR 1066206. 
  15. ^ Dutka, Jacques (1991). "The early history of the factorial function". Archive for History of Exact Sciences. 43 (3): 225–249. doi:10.1007/BF00389433. JSTOR 41133918. MR 1171521. 
  16. ^ Dickson, Leonard E. (1919). "Chapter IX: Divisibility of factorials and multinomial coefficients". History of the Theory of Numbers. 1. Carnegie Institution of Washington. hlm. 263–278.  See in particular p. 263.
  17. ^ a b Cajori, Florian (1929). "448–449. Factorial "n"". A History of Mathematical Notations, Volume II: Notations Mainly in Higher Mathematics. The Open Court Publishing Company. hlm. 71–77. 
  18. ^ Miller, Jeff. "Earliest Known Uses of Some of the Words of Mathematics (F)". MacTutor History of Mathematics archive. University of St Andrews. 
  19. ^ Craik, Alex D. D. (2005). "Prehistory of Faà di Bruno's formula". The American Mathematical Monthly. 112 (2): 119–130. doi:10.1080/00029890.2005.11920176. JSTOR 30037410. MR 2121322. 
  20. ^ Arbogast, Louis François Antoine (1800). Du calcul des dérivations (dalam bahasa Prancis). Strasbourg: L'imprimerie de Levrault, frères. hlm. 364–365. 

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