E (mathematical constant) (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "E (mathematical constant)" in English language version.

refsWebsite

Global rank English rank

11books.google.com

3^rd place

6web.archive.org

1^st place

6doi.org

2^nd place

6worldcat.org

5^th place

5archive.org

6^th place

4oeis.org

742^nd place

538^th place

4openstax.org

6,581^st place

3,967^th place

4jstor.org

26^th place

20^th place

3ams.org

451^st place

277^th place

2st-andrews.ac.uk

1,547^th place

1,410^th place

2wolfram.com

513^th place

537^th place

2free.fr

321^st place

724^th place

2dartmouth.edu

2,242^nd place

1,513^th place

1nist.gov

355^th place

454^th place

1bas.bg

3,024^th place

low place

1bnf.fr

124^th place

544^th place

1uni-goettingen.de

2,594^th place

2,546^th place

1pacific.edu

low place

1larrygonick.com

low place

1vanilla47.com

low place

1springer.com

274^th place

309^th place

1imj-prg.fr

low place

1claymath.org

8,402^nd place

8,241^st place

1ihes.fr

low place

1arxiv.org

69^th place

59^th place

1colorado.edu

2,526^th place

1,796^th place

1ucla.edu

782^nd place

585^th place

1archive.today

14^th place

1numberworld.org

low place

1ntg.nl

low place

1sagepub.com

731^st place

638^th place

1braintags.com

low place

1mkaz.com

low place

1explorepdx.com

low place

1npr.org

92^nd place

72^nd place

1lwn.net

4,423^rd place

2,925^th place

ams.org

mathscinet.ams.org

Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., eds. (2010), "E (mathematical constant)", NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.
Gelfond, A. O. (2015) [1960]. Transcendental and Algebraic Numbers. Dover Books on Mathematics. Translated by Boron, Leo F. New York: Dover Publications. p. 41. ISBN 978-0-486-49526-2. MR 0057921.

ams.org

Daniel Shanks; John W Wrench (1962). "Calculation of Pi to 100,000 Decimals" (PDF). Mathematics of Computation. 16 (77): 76–99. doi:10.2307/2003813. JSTOR 2003813. p. 78: We have computed e on a 7090 to 100,265D by the obvious program

archive.org

Sawyer, W. W. (1961). Mathematician's Delight. Penguin. p. 155.
Carl Boyer; Uta Merzbach (1991). A History of Mathematics (2nd ed.). Wiley. p. 419. ISBN 978-0-471-54397-8.
Steven Finch (2003). Mathematical constants. Cambridge University Press. p. 14. ISBN 978-0-521-81805-6.
Leonhard Euler, Introductio in Analysin Infinitorum (Lausanne, Switzerland: Marc Michel Bousquet & Co., 1748), volume 1, page 90.
Wozniak, Steve (June 1981). "The Impossible Dream: Computing $e$ to 116,000 Places with a Personal Computer". BYTE. Vol. 6, no. 6. McGraw-Hill. p. 392. Retrieved 18 October 2013.

archive.today

Roger Cotes (1714) "Logometria," Philosophical Transactions of the Royal Society of London, 29 (338) : 5–45; see especially the bottom of page 10. From page 10: "Porro eadem ratio est inter 2,718281828459 &c et 1, … " (Furthermore, by the same means, the ratio is between 2.718281828459… and 1, … )

arxiv.org

Milla, Lorenz (2020). "The Transcendence of π and the Squaring of the Circle". arXiv:2003.14035 [math.HO].

bas.bg

math.bas.bg

Bruins, E. M. (1983). "The Computation of Logarithms by Huygens" (PDF). Constructive Function Theory: 254–257.

bnf.fr

gallica.bnf.fr

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 be owing [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 $2½ a$ & minus quam $3 a$ ." ( … which our series [a geometric series] is larger [than]. … if $a = b$ , [the lender] will be owed more than $2½ a$ and less than $3 a$ .) If $a = b$ , the geometric series reduces to the series for $a \times e$ , 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.)

books.google.com

Pickover, Clifford A. (2009). The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics (illustrated ed.). Sterling Publishing Company. p. 166. ISBN 978-1-4027-5796-9. Extract of page 166
Wilson, Robinn (2018). Euler's Pioneering Equation: The most beautiful theorem in mathematics (illustrated ed.). Oxford University Press. p. (preface). ISBN 978-0-19-251405-9.
Posamentier, Alfred S.; Lehmann, Ingmar (2004). Pi: A Biography of the World's Most Mysterious Number (illustrated ed.). Prometheus Books. p. 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 be owing [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 $2½ a$ & minus quam $3 a$ ." ( … which our series [a geometric series] is larger [than]. … if $a = b$ , [the lender] will be owed more than $2½ a$ and less than $3 a$ .) If $a = b$ , the geometric series reduces to the series for $a \times e$ , 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.)
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 p. 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) … )
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.)
Kline, M. (1998). Calculus: An intuitive and physical approach. Dover Publications. p. 337 ff. ISBN 0-486-40453-6.
Gelfond, A. O. (2015) [1960]. Transcendental and Algebraic Numbers. Dover Books on Mathematics. Translated by Boron, Leo F. New York: Dover Publications. p. 41. ISBN 978-0-486-49526-2. MR 0057921.
William Shanks, Contributions to Mathematics, ... (London, England: G. Bell, 1853), page 89.
William Shanks (1871) "On the numerical values of $e$ , $log e 2$ , $log e 3$ , $log e 5$ , and $log e 10$ , also on the numerical value of $M$ the modulus of the common system of logarithms, all to 205 decimals," Proceedings of the Royal Society of London, 20 : 27–29.
J. Marcus Boorman (October 1884) "Computation of the Naperian base," Mathematical Magazine, 1 (12) : 204–205.

braintags.com

"First 10-digit prime found in consecutive digits of $e$ ". Brain Tags. Archived from the original on 2013-12-03. Retrieved 2012-02-24.

claymath.org

Khoshnevisan, Davar (2006). "Normal numbers are normal" (PDF). Clay Mathematics Institute Annual Report 2006. Clay Mathematics Institute. pp. 15, 27–31.

colorado.edu

math.colorado.edu

Hines, Robert. "e is transcendental" (PDF). University of Colorado. Archived (PDF) from the original on 2021-06-23.

dartmouth.edu

Grinstead, Charles M.; Snell, James Laurie (1997). Introduction to Probability (published online under the GFDL). American Mathematical Society. p. 85. ISBN 978-0-8218-9414-9. Archived from the original on 2011-07-27.

math.dartmouth.edu

Euler, L. "De serie Lambertina Plurimisque eius insignibus proprietatibus." Acta Acad. Scient. Petropol. 2, 29–51, 1783. Reprinted in Euler, L. Opera Omnia, Series Prima, Vol. 6: Commentationes Algebraicae. Leipzig, Germany: Teubner, pp. 350–369, 1921. (facsimile)

doi.org

Knoebel, R. Arthur (1981). "Exponentials Reiterated". The American Mathematical Monthly. 88 (4): 235–252. doi:10.2307/2320546. ISSN 0002-9890. JSTOR 2320546.
Anderson, Joel (2004). "Iterated Exponentials". The American Mathematical Monthly. 111 (8): 668–679. doi:10.2307/4145040. ISSN 0002-9890. JSTOR 4145040.
Murty, M. Ram; Rath, Purusottam (2014). Transcendental Numbers. Springer. doi:10.1007/978-1-4939-0832-5. ISBN 978-1-4939-0831-8.
Russell, K.G. (February 1991). "Estimating the Value of e by Simulation". The American Statistician. 45 (1): 66–68. doi:10.1080/00031305.1991.10475769. JSTOR 2685243.
Daniel Shanks; John W Wrench (1962). "Calculation of Pi to 100,000 Decimals" (PDF). Mathematics of Computation. 16 (77): 76–99. doi:10.2307/2003813. JSTOR 2003813. p. 78: We have computed e on a 7090 to 100,265D by the obvious program
Roberge, Jonathan; Melançon, Louis (June 2017). "Being the King Kong of algorithmic culture is a tough job after all: Google's regimes of justification and the meanings of Glass". Convergence: The International Journal of Research into New Media Technologies. 23 (3): 306–324. doi:10.1177/1354856515592506. ISSN 1354-8565.

explorepdx.com

The first 10-digit prime in $e$ Archived 2021-04-11 at the Wayback Machine. Explore Portland Community. Retrieved on 2020-12-09.

free.fr

numbers.computation.free.fr

Sebah, P. and Gourdon, X.; The constant $e$ and its computation
Gourdon, X.; Reported large computations with PiFast

ihes.fr

Kontsevich, Maxim; Zagier, Don (2001). "Periods" (PDF).

imj-prg.fr

webusers.imj-prg.fr

Waldschmidt, Michel (2021). "Schanuel's Conjecture: algebraic independence of transcendental numbers" (PDF).

jstor.org

Knoebel, R. Arthur (1981). "Exponentials Reiterated". The American Mathematical Monthly. 88 (4): 235–252. doi:10.2307/2320546. ISSN 0002-9890. JSTOR 2320546.
Anderson, Joel (2004). "Iterated Exponentials". The American Mathematical Monthly. 111 (8): 668–679. doi:10.2307/4145040. ISSN 0002-9890. JSTOR 4145040.
Russell, K.G. (February 1991). "Estimating the Value of e by Simulation". The American Statistician. 45 (1): 66–68. doi:10.1080/00031305.1991.10475769. JSTOR 2685243.
Daniel Shanks; John W Wrench (1962). "Calculation of Pi to 100,000 Decimals" (PDF). Mathematics of Computation. 16 (77): 76–99. doi:10.2307/2003813. JSTOR 2003813. p. 78: We have computed e on a 7090 to 100,265D by the obvious program

larrygonick.com

Gonick, Larry (2012). The Cartoon Guide to Calculus. William Morrow. pp. 29–32. ISBN 978-0-06-168909-3.

lwn.net

Peterson, Benjamin (20 April 2020). "Python 2.7.18, the end of an era". LWN.net.

mkaz.com

Kazmierczak, Marcus (2004-07-29). "Google Billboard". mkaz.com. Archived from the original on 2010-09-23. Retrieved 2007-06-09.

nist.gov

dlmf.nist.gov

Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., eds. (2010), "E (mathematical constant)", NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.

npr.org

Shea, Andrea. "Google Entices Job-Searchers with Math Puzzle". NPR. Retrieved 2007-06-09.

ntg.nl

Knuth, Donald (1990-10-03). "The Future of TeX and Metafont" (PDF). TeX Mag. 5 (1): 145. Retrieved 2017-02-17.

numberworld.org

Alexander Yee, ed. (15 March 2025). "y-cruncher - A Multi-Threaded Pi Program". Numberworld.

oeis.org

Sloane, N. J. A. (ed.). "Sequence A001113 (Decimal expansion of $e$ )". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
Sloane, N. J. A. (ed.). "Sequence A073230 (Decimal expansion of (1/e)^e)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
Sloane, N. J. A. (ed.). "Sequence A073229 (Decimal expansion of e^(1/e))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
Sloane, N. J. A. (ed.). "Sequence A003417 (Continued fraction for e)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

openstax.org

Abramson, Jay; et al. (2023). "6.1 Exponential Functions". College Algebra 2e. OpenStax. ISBN 978-1-951693-41-1.
Illowsky, Barbara; Dean, Susan; et al. (2023). "6.1 The Standard Normal Distribution". Statistics. OpenStax. ISBN 978-1-951693-22-0.
Strang, Gilbert; Herman, Edwin; et al. (2023). "6.3 Taylor and Maclaurin Series". Calculus, volume 2. OpenStax. ISBN 978-1-947172-14-2.
Strang, Gilbert; Herman, Edwin; et al. (2023). "4.10 Antiderivatives". Calculus, volume 2. OpenStax. ISBN 978-1-947172-14-2.

pacific.edu

scholarlycommons.pacific.edu

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...")

sagepub.com

journals.sagepub.com

Roberge, Jonathan; Melançon, Louis (June 2017). "Being the King Kong of algorithmic culture is a tough job after all: Google's regimes of justification and the meanings of Glass". Convergence: The International Journal of Research into New Media Technologies. 23 (3): 306–324. doi:10.1177/1354856515592506. ISSN 1354-8565.

springer.com

link.springer.com

Murty, M. Ram; Rath, Purusottam (2014). Transcendental Numbers. Springer. doi:10.1007/978-1-4939-0832-5. ISBN 978-1-4939-0831-8.

st-andrews.ac.uk

mathshistory.st-andrews.ac.uk

Miller, Jeff. "Earliest Uses of Symbols for Constants". MacTutor. University of St. Andrews, Scotland. Retrieved 31 October 2023.
O'Connor, John J.; Robertson, Edmund F. (September 2001). "The number $e$ ". MacTutor History of Mathematics Archive. University of St Andrews.

ucla.edu

wiki.stat.ucla.edu

Dinov, ID (2007) Estimating e using SOCR simulation, SOCR Hands-on Activities (retrieved December 26, 2007).

uni-goettingen.de

leibniz.uni-goettingen.de

Leibniz, Gottfried Wilhelm (2003). "Sämliche Schriften Und Briefe" (PDF) (in German). look for example letter nr. 6

vanilla47.com

Sandifer, Ed (Feb 2006). "How Euler Did It: Who proved $e$ is Irrational?" (PDF). MAA Online. Archived from the original (PDF) on 2014-02-23. Retrieved 2010-06-18.

web.archive.org

Grinstead, Charles M.; Snell, James Laurie (1997). Introduction to Probability (published online under the GFDL). American Mathematical Society. p. 85. ISBN 978-0-8218-9414-9. Archived from the original on 2011-07-27.
Sandifer, Ed (Feb 2006). "How Euler Did It: Who proved $e$ is Irrational?" (PDF). MAA Online. Archived from the original (PDF) on 2014-02-23. Retrieved 2010-06-18.
Hines, Robert. "e is transcendental" (PDF). University of Colorado. Archived (PDF) from the original on 2021-06-23.
"First 10-digit prime found in consecutive digits of $e$ ". Brain Tags. Archived from the original on 2013-12-03. Retrieved 2012-02-24.
Kazmierczak, Marcus (2004-07-29). "Google Billboard". mkaz.com. Archived from the original on 2010-09-23. Retrieved 2007-06-09.
The first 10-digit prime in $e$ Archived 2021-04-11 at the Wayback Machine. Explore Portland Community. Retrieved on 2020-12-09.

wolfram.com

mathworld.wolfram.com

Weisstein, Eric W. "e". mathworld.wolfram.com. Retrieved 2020-08-10.
Weisstein, Eric W. "Irrationality Measure". mathworld.wolfram.com. Retrieved 2024-09-14.

worldcat.org

search.worldcat.org

Kardar, Mehran (2007). Statistical Physics of Particles. Cambridge University Press. p. 41. ISBN 978-0-521-87342-0. OCLC 860391091.
Knoebel, R. Arthur (1981). "Exponentials Reiterated". The American Mathematical Monthly. 88 (4): 235–252. doi:10.2307/2320546. ISSN 0002-9890. JSTOR 2320546.
Anderson, Joel (2004). "Iterated Exponentials". The American Mathematical Monthly. 111 (8): 668–679. doi:10.2307/4145040. ISSN 0002-9890. JSTOR 4145040.
Finch, Steven R. (2005). Mathematical constants. Cambridge Univ. Press. ISBN 978-0-521-81805-6. OCLC 180072364.
Roberge, Jonathan; Melançon, Louis (June 2017). "Being the King Kong of algorithmic culture is a tough job after all: Google's regimes of justification and the meanings of Glass". Convergence: The International Journal of Research into New Media Technologies. 23 (3): 306–324. doi:10.1177/1354856515592506. ISSN 1354-8565.

worldcat.org

Finch, Steven R. (2005). Mathematical constants. Cambridge Univ. Press. ISBN 978-0-521-81805-6. OCLC 180072364.