Editing E (mathematical constant) (section)

== Representations ==
{{Main|List of representations of e|l1=List of representations of {{mvar|e}}}}

The number {{mvar|e}} can be represented in a variety of ways: as an [[infinite series]], an [[infinite product]], a [[continued fraction]], or a [[limit of a sequence]]. In addition to the limit and the series given above, there is also the [[simple continued fraction]]
<!--move to history section or say <ref>[[Leonhard Euler|Euler]] was the first showed that {{mvar|e}} can be represented as a continued fraction.</ref>-->

:<math>
  e = [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, ..., 1, 2n, 1, ...],
</math><ref>{{cite book |last=Hofstadter|first=D.R.|author-link=Douglas Hofstadter |title= Fluid Concepts and Creative Analogies: Computer Models of the Fundamental Mechanisms of Thought|publisher= Basic Books |date=1995|isbn=0-7139-9155-0}}</ref><ref name="OEIS continued fraction">{{Cite OEIS|A003417|Continued fraction for e}}</ref>

which written out looks like

:<math>e = 2 +
\cfrac{1}
   {1 + \cfrac{1}
      {2 + \cfrac{1}
         {1 + \cfrac{1}
            {1 + \cfrac{1}
               {4 + \cfrac{1}
            {1 + \cfrac{1}
               {1 + \ddots}
                  }
               }
            }
         }
      }
   }
.
</math>

The following infinite product evaluates to {{mvar|e}}:<ref name="Finch-2003-p14"/>
<math display="block">e = \frac{2}{1} \left(\frac{4}{3}\right)^{1/2} \left(\frac{6 \cdot 8}{5 \cdot 7}\right)^{1/4} \left(\frac{10 \cdot 12 \cdot 14 \cdot 16}{9 \cdot 11 \cdot 13 \cdot 15}\right)^{1/8} \cdots.</math>

Many other series, sequence, continued fraction, and infinite product representations of {{mvar|e}} have been proved.

=== Stochastic representations ===
In addition to exact analytical expressions for representation of {{mvar|e}}, there are stochastic techniques for estimating {{mvar|e}}.  One such approach begins with an infinite sequence of independent random variables {{math|''X''<sub>1</sub>}}, {{math|''X''<sub>2</sub>}}..., drawn from the [[uniform distribution (continuous)|uniform distribution]] on [0, 1]. Let {{mvar|V}} be the least number {{mvar|n}} such that the sum of the first {{mvar|n}} observations exceeds 1:

:<math>V = \min\left\{ n \mid X_1 + X_2 + \cdots + X_n > 1 \right\}.</math>

Then the [[expected value]] of {{mvar|V}} is {{mvar|e}}: {{math|E(''V'') {{=}} ''e''}}.<ref>{{cite journal|last=Russell|first= K.G. |jstor=2685243 |title=Estimating the Value of e by Simulation |journal= The American Statistician|volume= 45|issue= 1|date= February 1991|pages= 66–68 |doi=10.1080/00031305.1991.10475769}}</ref><ref>Dinov, ID (2007) ''[http://wiki.stat.ucla.edu/socr/index.php/SOCR_EduMaterials_Activities_LawOfLargeNumbers#Estimating_e_using_SOCR_simulation Estimating e using SOCR simulation]'', SOCR Hands-on Activities (retrieved December 26, 2007).</ref>

=== Known digits ===
The number of known digits of {{mvar|e}} has increased substantially since the introduction of the computer, due both to increasing performance of computers and to algorithmic improvements.<ref>Sebah, P. and Gourdon, X.; [http://numbers.computation.free.fr/Constants/E/e.html The constant {{mvar|e}} and its computation]</ref><ref>Gourdon, X.; [http://numbers.computation.free.fr/Constants/PiProgram/computations.html Reported large computations with PiFast]</ref>

{| class="wikitable" style="margin: 1em auto 1em auto"
|+ Number of known decimal digits of {{mvar|e}}
! Date || Decimal digits || Computation performed by
|-
| 1690 ||align=right| 1 || [[Jacob Bernoulli]]<ref name="Bernoulli, 1690" />
|-
| 1714 ||align=right| 13 || [[Roger Cotes]]<ref>Roger Cotes (1714) "Logometria," ''Philosophical Transactions of the Royal Society of London'', '''29''' (338) : 5–45; [https://archive.today/20140410203227/http://babel.hathitrust.org/cgi/pt?id=ucm.5324351035;view=2up;seq=16 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, … )</ref>
|-
| 1748 ||align=right| 23 || [[Leonhard Euler]]<ref>Leonhard Euler, ''Introductio in Analysin Infinitorum'' (Lausanne, Switzerland: Marc Michel Bousquet & Co., 1748), volume 1, [https://archive.org/details/bub_gb_jQ1bAAAAQAAJ/page/n115 page 90.]</ref>
|-
| 1853 ||align=right| 137 || [[William Shanks]]<ref>William Shanks, ''Contributions to Mathematics'', ... (London, England: G. Bell, 1853), [https://books.google.com/books?id=d-9ZAAAAcAAJ&pg=PA89 page 89.]</ref>
|-
| 1871 ||align=right| 205 || William Shanks<ref>William Shanks (1871) [https://books.google.com/books?id=sclTAAAAcAAJ&pg=PA27 "On the numerical values of {{mvar|e}}, {{math|log<sub>''e''</sub> 2}}, {{math|log<sub>''e''</sub> 3}}, {{math|log<sub>''e''</sub> 5}}, and {{math|log<sub>''e''</sub> 10}}, also on the numerical value of {{mvar|M}} the modulus of the common system of logarithms, all to 205 decimals,"] ''Proceedings of the Royal Society of London'', '''20''' : 27–29.</ref>
|-
| 1884 ||align=right| 346 || J. Marcus Boorman<ref>J. Marcus Boorman (October 1884) [https://books.google.com/books?id=mG8yAQAAMAAJ&pg=PA204 "Computation of the Naperian base,"] ''Mathematical Magazine'', '''1''' (12) : 204–205.</ref>
|-
| 1949 ||align=right| 2,010 || [[John von Neumann]] (on the [[ENIAC]])
|-
| 1961 ||align=right| 100,265 || [[Daniel Shanks]] and [[John Wrench]]<ref name="We have computed e on a 7090 to 100,265D by the obvious program.">{{cite journal|author1=Daniel Shanks |author2= John W Wrench|quote=We have computed e on a 7090 to 100,265D by the obvious program|title=Calculation of Pi to 100,000 Decimals|journal =Mathematics of Computation|volume= 16 |year=1962| issue =77| pages =76–99 |author1-link=Daniel Shanks |author2-link= John Wrench |quote-page=78|url=https://www.ams.org/journals/mcom/1962-16-077/S0025-5718-1962-0136051-9/S0025-5718-1962-0136051-9.pdf|doi=10.2307/2003813|jstor=2003813}}</ref>
|-
| 1978 ||align=right| 116,000 || [[Steve Wozniak]] on the [[Apple II]]<ref name="wozniak198106">{{cite magazine | url=https://archive.org/stream/byte-magazine-1981-06/1981_06_BYTE_06-06_Operating_Systems#page/n393/mode/2up | title=The Impossible Dream: Computing {{mvar|e}} to 116,000 Places with a Personal Computer | magazine=BYTE |volume=6 |issue=6 |author-link=Steve Wozniak |publisher=McGraw-Hill | date=June 1981 | access-date=18 October 2013 | last=Wozniak |first= Steve | page=392}}</ref>
|}

Since around 2010, the proliferation of modern high-speed [[desktop computer]]s has made it feasible for amateurs to compute trillions of digits of {{mvar|e}} within acceptable amounts of time. On Dec 24, 2023, a record-setting calculation was made by Jordan Ranous, giving {{mvar|e}} to 35,000,000,000,000 digits.<ref>{{cite web
 | title= y-cruncher - A Multi-Threaded Pi Program
 | editor= Alexander Yee
 | work= Numberworld
 | date= 15 March 2025
 | url=http://www.numberworld.org/y-cruncher/#Records
}}</ref>