e (konstanta)
e, poznat kao Ojlerov broj ili Neperova konstanta, osnova je prirodnog logaritma i jedan od najznačajnijih brojeva u savremenoj matematici, pored neutrala sabiranja i množenja 0 i 1, imaginarne jedinice broj i i broja pi. Osim što je iracionalan i realan, ovaj broj je još i transcedentan. Do tridesetog decimalnog mesta, ovaj broj iznosi:
- e ≈ 2,71828 18284 59045 23536 02874 71352...
To je osnova prirodnih logaritama. To je granica (1 + 1/n)n kako se n približava beskonačnosti, izraz koji se javlja u proučavanju složenog interesa. Takođe se može izračunati kao zbir beskonačnog niza
To je takođe jedinstveni pozitivni broj a takav da grafik funkcije y = ax ima nagib od 1 na x = 0.
(prirodna) eksponencijalna funkcija f(x) = ex je jedinstvena funkcija f koja je jednaka sopstvenom izvodu i zadovoljava jednačinu f(0) = 1; stoga se e takođe može definisati kao f(1). Prirodni logaritam, ili logaritam bazi e, je inverzna funkcija prirodnoj eksponencijalnoj funkciji. Prirodni logaritam broja k > 1 može se direktno definisati kao površina ispod krive y = 1/x između x = 1 i x = k, u kom slučaju je e vrednost k za koju je ova površina jednaka jedan (pogledajte sliku). Postoje razne druge karakteristike.
Broj e se ponekad naziva Ojlerovim brojem (ne treba ga mešati sa Ojlerovom konstantom ), po švajcarskom matematičaru Leonhardu Ojleru, ili Napijerovom konstantom, po Džonu Napijeru.[1] Konstantu je otkrio švajcarski matematičar Jakob Bernuli dok je proučavao složenu kamatu.[2][3]
Broj e je od velikog značaja u matematici,[4] pored 0, 1, π i i. Svih pet se pojavljuju u jednoj formulaciji Ojlerovog identiteta i igraju važne i ponavljajuće uloge u matematici.[5][6] Kao i konstanta π, e je iracionalno (to jest, ne može se predstaviti kao odnos celih brojeva) i transcendentno (to jest, nije koren nijednog polinoma različitog od nule sa racionalnim koeficijentima).[1]
Definicije
urediBroj e se može predstaviti kao:
- Granična vrednost beskonačnog niza:
- Suma beskonačnog niza:
-
- Gde je n! faktorijel n.
-
- Pozitivna vrednost koja zadovoljava sledeću jednačinu:
- Može se dokazati da su navedena tri iskaza ekvivalentna.
- Ovaj broj se sreće i kao deo Ojlerovog identiteta:
Istorija
urediPrve reference na konstantu objavljene su 1618. godine u tabeli dodatka dela o logaritmima Džona Napijera. Međutim, ovo nije sadržalo samu konstantu, već jednostavno listu logaritama za osnovu . Pretpostavlja se da je tabelu napisao Vilijam Outred.[3]
Otkriće same konstante pripisuje se Jakobu Bernuliju 1683,[7][8] koji je pokušao da pronađe vrednost sledećeg izraza (koji je jednak e):
Prva poznata upotreba konstante, predstavljene slovom b, bila je u prepisci Gotfrida Lajbnica sa Kristijanom Hajgensom 1690. i 1691. godine.[9] Leonhard Ojler je uveo slovo e kao osnovu za prirodne logaritme, pišući u pismu Kristijanu Goldbahu 25. novembra 1731.[10][11] Ojler je počeo da koristi slovo e za konstantu 1727. ili 1728. godine, u neobjavljenom radu o eksplozivnim silama u topovima,[12] dok je prvo pojavljivanje e u jednoj publikaciji bilo u Ojlerovoj Mehanici (1736).[13] Iako su neki istraživači koristili slovo c u narednim godinama, slovo e je bilo češće i na kraju je postalo standardno.
Reference
uredi- ^ a b Weisstein, Eric W. „e”. mathworld.wolfram.com (na jeziku: engleski). Pristupljeno 2020-08-10.
- ^ Pickover, Clifford A. (2009). The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics (illustrated izd.). Sterling Publishing Company. str. 166. ISBN 978-1-4027-5796-9. Extract of page 166
- ^ a b O'Connor, J J; Robertson, E F. „The number e”. MacTutor History of Mathematics.
- ^ Howard Whitley Eves (1969). An Introduction to the History of Mathematics . Holt, Rinehart & Winston. ISBN 978-0-03-029558-4.
- ^ Wilson, Robinn (2018). Euler's Pioneering Equation: The most beautiful theorem in mathematics (illustrated izd.). Oxford University Press. str. (preface). ISBN 978-0-19-251405-9.
- ^ Posamentier, Alfred S.; Lehmann, Ingmar (2004). Pi: A Biography of the World's Most Mysterious Number (illustrated izd.). Prometheus Books. str. 68. ISBN 978-1-59102-200-8.
- ^ Jacob Bernoulli considered the problem of continuous compounding of interest, which led to a series expression for e. See: Jacob Bernoulli (1690) "Quæstiones nonnullæ de usuris, cum solutione problematis de sorte alearum, propositi in Ephem. Gall. A. 1685" (Some questions about interest, with a solution of a problem about games of chance, proposed in the Journal des Savants (Ephemerides Eruditorum Gallicanæ), in the year (anno) 1685.**), Acta eruditorum, pp. 219–23. On page 222, Bernoulli poses the question: "Alterius naturæ hoc Problema est: Quæritur, si creditor aliquis pecuniæ summam fænori exponat, ea lege, ut singulis momentis pars proportionalis usuræ annuæ sorti annumeretur; quantum ipsi finito anno debeatur?" (This is a problem of another kind: The question is, if some lender were to invest [a] sum of money [at] interest, let it accumulate, so that [at] every moment [it] were to receive [a] proportional part of [its] annual interest; how much would he be owed [at the] end of [the] year?) Bernoulli constructs a power series to calculate the answer, and then writes: " … quæ nostra serie [mathematical expression for a geometric series] &c. major est. … si a=b, debebitur plu quam 2½a & minus quam 3a." ( … which our series [a geometric series] is larger [than]. … if a=b, [the lender] will be owed more than 2½a and less than 3a.) If a=b, the geometric series reduces to the series for a × e, so 2.5 < e < 3. (** The reference is to a problem which Jacob Bernoulli posed and which appears in the Journal des Sçavans of 1685 at the bottom of page 314.)
- ^ Carl Boyer; Uta Merzbach (1991). A History of Mathematics (2nd izd.). Wiley. str. 419. ISBN 978-0-471-54397-8.
- ^ Leibniz, Gottfried Wilhelm (2003). „Sämliche Schriften Und Briefe” (PDF) (na jeziku: nemački). „look for example letter nr. 6”
- ^ Lettre XV. Euler à Goldbach, dated November 25, 1731 in: P.H. Fuss, ed., Correspondance Mathématique et Physique de Quelques Célèbres Géomètres du XVIIIeme Siècle … (Mathematical and physical correspondence of some famous geometers of the 18th century), vol. 1, (St. Petersburg, Russia: 1843), pp. 56–60, see especially. Fuss, Paul Heinrich (1843). Correspondance mathématique et physique de quelques célèbres géomètres du XVIIIème siècle: Précédé d'une notice sur les travaux de Léonard Euler, tant imprimés qu'inédits et publiée sous les auspices de l'Académie impériale des sciences de Saint-Pétersbourg. str. 58. From p. 58: " … ( e denotat hic numerum, cujus logarithmus hyperbolicus est = 1), … " ( … (e denotes that number whose hyperbolic [i.e., natural] logarithm is equal to 1) … )
- ^ Remmert, Reinhold (1991). Theory of Complex Functions. Springer-Verlag. str. 136. ISBN 978-0-387-97195-7.
- ^ Euler, Meditatio in experimenta explosione tormentorum nuper instituta. Scribatur pro numero cujus logarithmus est unitas, e, qui est 2,7182817… (English: Written for the number of which the logarithm has the unit, e, that is 2,7182817...")
- ^ Leonhard Euler, Mechanica, sive Motus scientia analytice exposita (St. Petersburg (Petropoli), Russia: Academy of Sciences, 1736), vol. 1, Chapter 2, Corollary 11, paragraph 171, p. 68. From page 68: Erit enim seu ubi e denotat numerum, cuius logarithmus hyperbolicus est 1. (So it [i.e., c, the speed] will be or , where e denotes the number whose hyperbolic [i.e., natural] logarithm is 1.)
Literatura
uredi- Maor, Eli; e: The Story of a Number. ISBN 978-0-691-05854-2.
- Commentary on Endnote 10 of the book Prime Obsession for another stochastic representation
- McCartin, Brian J. (2006). „e: The Master of All” (PDF). The Mathematical Intelligencer. 28 (2): 10—21. S2CID 123033482. doi:10.1007/bf02987150.
- Aldrich, John; Miller, Jeff. „Earliest Uses of Symbols in Probability and Statistics”.
- Aldrich, John; Miller, Jeff. „Earliest Known Uses of Some of the Words of Mathematics”. In particular, the entries for "bell-shaped and bell curve", "normal (distribution)", "Gaussian", and "Error, law of error, theory of errors, etc.".
- Amari, Shun-ichi; Nagaoka, Hiroshi (2000). Methods of Information Geometry. Oxford University Press. ISBN 978-0-8218-0531-2.
- Bernardo, José M.; Smith, Adrian F. M. (2000). Bayesian Theory. Wiley. ISBN 978-0-471-49464-5.
- Bryc, Wlodzimierz (1995). The Normal Distribution: Characterizations with Applications. Springer-Verlag. ISBN 978-0-387-97990-8.
- Casella, George; Berger, Roger L. (2001). Statistical Inference (2nd izd.). Duxbury. ISBN 978-0-534-24312-8.
- Cody, William J. (1969). „Rational Chebyshev Approximations for the Error Function”. Mathematics of Computation. 23 (107): 631—638. doi:10.1090/S0025-5718-1969-0247736-4 .
- Cover, Thomas M.; Thomas, Joy A. (2006). Elements of Information Theory. John Wiley and Sons.
- de Moivre, Abraham (1738). The Doctrine of Chances. American Mathematical Society. ISBN 978-0-8218-2103-9.
- Fan, Jianqing (1991). „On the optimal rates of convergence for nonparametric deconvolution problems”. The Annals of Statistics. 19 (3): 1257—1272. JSTOR 2241949. doi:10.1214/aos/1176348248 .
- Galton, Francis (1889). Natural Inheritance (PDF). London, UK: Richard Clay and Sons.
- Galambos, Janos; Simonelli, Italo (2004). Products of Random Variables: Applications to Problems of Physics and to Arithmetical Functions . Marcel Dekker, Inc. ISBN 978-0-8247-5402-0.
- Gauss, Carolo Friderico (1809). Theoria motvs corporvm coelestivm in sectionibvs conicis Solem ambientivm [Theory of the Motion of the Heavenly Bodies Moving about the Sun in Conic Sections] (na jeziku: latinski). Hambvrgi, Svmtibvs F. Perthes et I. H. Besser. English translation.
- Gould, Stephen Jay (1981). The Mismeasure of Man (first izd.). W. W. Norton. ISBN 978-0-393-01489-1.
- Halperin, Max; Hartley, Herman O.; Hoel, Paul G. (1965). „Recommended Standards for Statistical Symbols and Notation. COPSS Committee on Symbols and Notation”. The American Statistician. 19 (3): 12—14. JSTOR 2681417. doi:10.2307/2681417.
- Hart, John F.; et al. (1968). Computer Approximations. New York, NY: John Wiley & Sons, Inc. ISBN 978-0-88275-642-4.
- Hazewinkel Michiel, ур. (2001). „Normal Distribution”. Encyclopaedia of Mathematics. Springer. ISBN 978-1556080104.
- Herrnstein, Richard J.; Murray, Charles (1994). The Bell Curve: Intelligence and Class Structure in American Life. Free Press. ISBN 978-0-02-914673-6.
- Huxley, Julian S. (1932). Problems of Relative Growth. London. ISBN 978-0-486-61114-3. OCLC 476909537.
- Johnson, Norman L.; Kotz, Samuel; Balakrishnan, Narayanaswamy (1994). Continuous Univariate Distributions, Volume 1. Wiley. ISBN 978-0-471-58495-7.
- Johnson, Norman L.; Kotz, Samuel; Balakrishnan, Narayanaswamy (1995). Continuous Univariate Distributions, Volume 2. Wiley. ISBN 978-0-471-58494-0.
- Karney, C. F. F. (2016). „Sampling exactly from the normal distribution”. ACM Transactions on Mathematical Software. 42 (1): 3:1—14. S2CID 14252035. arXiv:1303.6257 . doi:10.1145/2710016.
- Kinderman, Albert J.; Monahan, John F. (1977). „Computer Generation of Random Variables Using the Ratio of Uniform Deviates”. ACM Transactions on Mathematical Software. 3 (3): 257—260. S2CID 12884505. doi:10.1145/355744.355750.
- Krishnamoorthy, Kalimuthu (2006). Handbook of Statistical Distributions with Applications. Chapman & Hall/CRC. ISBN 978-1-58488-635-8.
- Kruskal, William H.; Stigler, Stephen M. (1997). Spencer, Bruce D., ur. Normative Terminology: 'Normal' in Statistics and Elsewhere. Statistics and Public Policy. Oxford University Press. ISBN 978-0-19-852341-3.
- Laplace, Pierre-Simon de (1774). „Mémoire sur la probabilité des causes par les événements”. Mémoires de l'Académie Royale des Sciences de Paris (Savants étrangers), Tome 6: 621—656. JSTOR 2245476. Translated by Stephen M. Stigler in Statistical Science 1 (3), 1986: .
Spoljašnje veze
uredi- The number e to 1 million places and NASA.gov 2 and 5 million places
- e Approximations – Wolfram MathWorld
- Earliest Uses of Symbols for Constants Jan. 13, 2008
- "The story of e", by Robin Wilson at Gresham College, 28 February 2007 (available for audio and video download)
- e Search Engine 2 billion searchable digits of e, π and √2