$e$ (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

e=\sum \limits _{n=0}^{\infty }{\frac {1}{n!}}=1+{\frac {1}{1}}+{\frac {1}{1\cdot 2}}+{\frac {1}{1\cdot 2\cdot 3}}+\cdots .

To je takođe jedinstveni pozitivni broj $a$ takav da grafik funkcije $y = a x$ ima nagib od 1 na $x = 0$ .

(prirodna) eksponencijalna funkcija $f (x) = e x$ 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 $\gamma$ ), 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 uredi

Broj $e$ se može predstaviti kao:

Granična vrednost beskonačnog niza:
$e=\lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}$
Suma beskonačnog niza:
$e=\sum _{n=0}^{\infty }{\frac {1}{n!}}={\frac {1}{0!}}+{\frac {1}{1!}}+{\frac {1}{2!}}+{\frac {1}{3!}}+{\frac {1}{4!}}+\cdots$
Gde je n! faktorijel n.
Pozitivna vrednost koja zadovoljava sledeću jednačinu:
$\int _{1}^{e}{\frac {1}{t}}\,dt={1}$

Može se dokazati da su navedena tri iskaza ekvivalentna.
Ovaj broj se sreće i kao deo Ojlerovog identiteta:
$e^{\mathrm {i} \cdot \pi }=-1$

Istorija uredi

Prve 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 $e$ . 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 $e$ ):

\lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}.

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 ${\frac {dc}{c}}={\frac {dyds}{rdx}}$ seu $c=e^{\int {\frac {dyds}{rdx}}}$ ubi e denotat numerum, cuius logarithmus hyperbolicus est 1. (So it [i.e., c, the speed] will be ${\frac {dc}{c}}={\frac {dyds}{rdx}}$ or $c=e^{\int {\frac {dyds}{rdx}}}$ , 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

[:1-1] Weisstein, Eric W. „e”. mathworld.wolfram.com (na jeziku: engleski). Pristupljeno 2020-08-10.

[Pickover-2] 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

[OConnor-3] O'Connor, J J; Robertson, E F. „The number e”. MacTutor History of Mathematics.

[4] Howard Whitley Eves (1969). An Introduction to the History of Mathematics . Holt, Rinehart & Winston. ISBN 978-0-03-029558-4.

[5] 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.

[6] 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.

[Bernoulli,_1690-7] 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.)

[8] Carl Boyer; Uta Merzbach (1991). A History of Mathematics (2nd izd.). Wiley. str. 419. ISBN 978-0-471-54397-8.

[9] Leibniz, Gottfried Wilhelm (2003). „Sämliche Schriften Und Briefe” (PDF) (na jeziku: nemački). „look for example letter nr. 6”

[10] 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) … )

[11] Remmert, Reinhold (1991). Theory of Complex Functions. Springer-Verlag. str. 136. ISBN 978-0-387-97195-7.

[Meditatio-12] 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...")

[13] 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 ${\frac {dc}{c}}={\frac {dyds}{rdx}}$ seu $c=e^{\int {\frac {dyds}{rdx}}}$ ubi e denotat numerum, cuius logarithmus hyperbolicus est 1. (So it [i.e., c, the speed] will be ${\frac {dc}{c}}={\frac {dyds}{rdx}}$ or $c=e^{\int {\frac {dyds}{rdx}}}$ , where e denotes the number whose hyperbolic [i.e., natural] logarithm is 1.)

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e (konstanta)