Editing E (mathematical constant) (section)

== Applications ==

=== Compound interest ===
[[File:Compound Interest with Varying Frequencies.svg|right|thumb|The effect of earning 20% annual interest on an {{nowrap|initial $1,000}} investment at various compounding frequencies. The limiting curve on top is the graph <math>y=1000e^{0.2t}</math>, where {{mvar|y}} is in dollars, {{mvar|t}} in years, and 0.2 = 20%.]]

Jacob Bernoulli discovered this constant in 1683, while studying a question about [[compound interest]]:<ref name="OConnor" />
{{Blockquote|An account starts with $1.00 and pays 100 percent interest per year. If the interest is credited once, at the end of the year, the value of the account at year-end will be $2.00. What happens if the interest is computed and credited more frequently during the year?}}

If the interest is credited twice in the year, the interest rate for each 6 months will be 50%, so the initial $1 is multiplied by 1.5 twice, yielding {{nowrap|1=$1.00 × 1.5<sup>2</sup> = $2.25}} at the end of the year. Compounding quarterly yields {{nowrap|1=$1.00 × 1.25<sup>4</sup> = $2.44140625}}, and compounding monthly yields {{nowrap|1=$1.00 × (1 + 1/12)<sup>12</sup> = $2.613035...}}. If there are {{mvar|n}} compounding intervals, the interest for each interval will be {{math|100%/''n''}} and the value at the end of the year will be $1.00&nbsp;×&nbsp;{{math|(1 + 1/''n'')<sup>''n''</sup>}}.<ref name="Gonick"/><ref name=":0" />

Bernoulli noticed that this sequence approaches a limit (the [[force of interest]]) with larger {{mvar|n}} and, thus, smaller compounding intervals.<ref name="OConnor" /> Compounding weekly ({{math|1=''n'' = 52}}) yields $2.692596..., while compounding daily ({{math|1=''n'' = 365}}) yields $2.714567... (approximately two cents more). The limit as {{mvar|n}} grows large is the number that came to be known as {{mvar|e}}. That is, with ''continuous'' compounding, the account value will reach $2.718281828... More generally, an account that starts at $1 and offers an annual interest rate of {{mvar|R}} will, after {{mvar|t}} years, yield {{math|''e''<sup>''Rt''</sup>}} dollars with continuous compounding. Here, {{mvar|R}} is the decimal equivalent of the rate of interest expressed as a ''percentage'', so for 5% interest, {{math|1=''R'' = 5/100 = 0.05}}.<ref name="Gonick">{{cite book
 | last = Gonick
 | first = Larry
 | author-link = Larry Gonick
 | year = 2012
 | title = The Cartoon Guide to Calculus
 | publisher = William Morrow
 | url = https://www.larrygonick.com/titles/science/cartoon-guide-to-calculus-2/
 | isbn = 978-0-06-168909-3
 | pages = 29–32
}}</ref><ref name=":0" />

=== Bernoulli trials ===
[[File:Bernoulli trial sequence.svg|thumb|300px|Graphs of probability {{mvar|P}} of {{em|not}} observing independent events each of probability {{math|1/''n''}} after {{mvar|n}} Bernoulli trials, and {{math|1 − ''P''&thinsp;}} vs {{mvar|n}}&thinsp;; it can be observed that as {{mvar|n}} increases, the probability of a {{math|1/''n''}}-chance event never appearing after ''n'' tries rapidly {{nowrap|converges to {{math|1/''e''}}.}}]]
The number {{mvar|e}} itself also has applications in [[probability theory]], in a way that is not obviously related to exponential growth. Suppose that a gambler plays a slot machine that pays out with a probability of one in {{mvar|n}} and plays it {{mvar|n}} times. As {{mvar|n}} increases, the probability that gambler will lose all {{mvar|n}} bets approaches {{math|1/''e''}}, which is approximately 36.79%. For {{math|1=''n'' = 20}}, this is already 1/2.789509... (approximately 35.85%).

This is an example of a [[Bernoulli trial]] process. Each time the gambler plays the slots, there is a one in {{mvar|n}} chance of winning. Playing {{mvar|n}} times is modeled by the [[binomial distribution]], which is closely related to the [[binomial theorem]] and [[Pascal's triangle]]. The probability of winning {{mvar|k}} times out of {{mvar|n}} trials is:<ref>{{cite book|first=Mehran |last=Kardar |author-link=Mehran Kardar |title=Statistical Physics of Particles |title-link=Statistical Physics of Particles |year=2007 |publisher=[[Cambridge University Press]] |isbn=978-0-521-87342-0 |oclc=860391091 |page=41}}</ref>
:<math>\Pr[k~\mathrm{wins~of}~n] = \binom{n}{k} \left(\frac{1}{n}\right)^k\left(1 - \frac{1}{n}\right)^{n-k}.</math>

In particular, the probability of winning zero times ({{math|1=''k'' = 0}}) is
:<math>\Pr[0~\mathrm{wins~of}~n] = \left(1 - \frac{1}{n}\right)^{n}.</math>

The limit of the above expression, as {{mvar|n}} tends to infinity, is precisely {{math|1/''e''}}.

=== Exponential growth and decay ===
{{Further|Exponential growth|Exponential decay}}
[[Exponential growth]] is a process that increases quantity over time at an ever-increasing rate. It occurs when the instantaneous [[Rate (mathematics)#Of change|rate of change]] (that is, the [[derivative]]) of a quantity with respect to time is [[proportionality (mathematics)|proportional]] to the quantity itself.<ref name=":0">{{cite book|chapter-url=https://openstax.org/books/college-algebra-2e/pages/6-1-exponential-functions |chapter=6.1 Exponential Functions |title=College Algebra 2e |publisher=OpenStax |first1=Jay |last1=Abramson |display-authors=etal |year=2023 |isbn=978-1-951693-41-1}}</ref> Described as a function, a quantity undergoing exponential growth is an [[Exponentiation#Power functions|exponential function]] of time, that is, the variable representing time is the exponent (in contrast to other types of growth, such as [[quadratic growth]]). If the constant of proportionality is negative, then the quantity decreases over time, and is said to be undergoing [[exponential decay]] instead. The law of exponential growth can be written in different but mathematically equivalent forms, by using a different [[exponentiation|base]], for which the number {{mvar|e}} is a common and convenient choice:
<math display="block">x(t) = x_0\cdot e^{kt} = x_0\cdot e^{t/\tau}.</math>
Here, <math>x_0</math> denotes the initial value of the quantity {{mvar|x}}, {{mvar|k}} is the growth constant, and <math>\tau</math> is the time it takes the quantity to grow by a factor of {{mvar|e}}.

=== Standard normal distribution ===
{{Main|Normal distribution}}

The normal distribution with zero mean and unit standard deviation is known as the ''standard normal distribution'',{{r|openstax}} given by the [[probability density function]]<!--{{r|nordistr}}-->
<math display="block"> \phi(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2} x^2}. </math>

The constraint of unit standard deviation (and thus also unit variance) results in the {{frac2|1|2}} in the exponent, and the constraint of unit total area under the curve <math>\phi(x)</math> results in the factor <math>\textstyle 1/\sqrt{2\pi}</math>. This function is symmetric around {{math|1=''x'' = 0}}, where it attains its maximum value <math>\textstyle 1/\sqrt{2\pi}</math>, and has [[inflection point]]s at {{math|1=''x'' = ±1}}.

=== Derangements ===
{{Main|Derangement}}
Another application of {{mvar|e}}, also discovered in part by Jacob Bernoulli along with [[Pierre Remond de Montmort]], is in the problem of [[derangement]]s, also known as the ''hat check problem'':<ref>{{cite book| last1=Grinstead|first1= Charles M. |last2= Snell|first2= James Laurie |date=1997 |url=http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/book.html |title= Introduction to Probability |archive-url=https://web.archive.org/web/20110727200156/http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/book.html |archive-date=2011-07-27 |type=published online under the [[GFDL]]|publisher=American Mathematical Society |page= 85 |isbn=978-0-8218-9414-9 |author2-link=J. Laurie Snell }}</ref> {{mvar|n}} guests are invited to a party and, at the door, the guests all check their hats with the butler, who in turn places the hats into {{mvar|n}} boxes, each labelled with the name of one guest. But the butler has not asked the identities of the guests, and so puts the hats into boxes selected at random. The problem of de Montmort is to find the probability that ''none'' of the hats gets put into the right box. This probability, denoted by <math>p_n\!</math>, is:

:<math>p_n = 1 - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \cdots + \frac{(-1)^n}{n!} = \sum_{k = 0}^n \frac{(-1)^k}{k!}.</math>

As {{mvar|n}} tends to infinity, {{math|''p''<sub>''n''</sub>}} approaches {{math|1/''e''}}. Furthermore, the number of ways the hats can be placed into the boxes so that none of the hats are in the right box is {{math|''n''!/''e'',}} [[Rounding|rounded]] to the nearest integer, for every positive&nbsp;{{mvar|n}}.<ref>{{cite book | last=Knuth |first=Donald |author-link=Donald Knuth |date= 1997 | title=The Art of Computer Programming |title-link=The Art of Computer Programming | volume =I|publisher= Addison-Wesley|page= 183 |isbn=0-201-03801-3}}</ref>

=== Optimal planning problems ===
The maximum value of <math>\sqrt[x]{x}</math> occurs at <math>x = e</math>.  Equivalently, for any value of the base {{math|''b'' > 1}}, it is the case that the maximum value of <math>x^{-1}\log_b x</math> occurs at <math>x = e</math> ([[Steiner's calculus problem|Steiner's problem]], discussed [[#Exponential-like functions|below]]).

This is useful in the problem of a stick of length {{mvar|L}} that is broken into {{mvar|n}} equal parts.  The value of {{mvar|n}} that maximizes the product of the lengths is then either<ref name="Finch-2003-p14">{{cite book|title=Mathematical constants|url=https://archive.org/details/mathematicalcons0000finc|url-access=registration|author=Steven Finch|year=2003|publisher=Cambridge University Press|page=[https://archive.org/details/mathematicalcons0000finc/page/14 14]|isbn=978-0-521-81805-6}}</ref>
:<math>n = \left\lfloor \frac{L}{e} \right\rfloor</math> or <math>\left\lceil \frac{L}{e} \right\rceil.</math>

The quantity <math>x^{-1}\log_b x</math> is also a measure of [[Shannon information|information]] gleaned from an event occurring with probability <math>1/x</math> (approximately <math>36.8\%</math> when <math>x=e</math>), so that essentially the same optimal division appears in optimal planning problems like the [[secretary problem]].

=== Asymptotics ===
The number {{mvar|e}} occurs naturally in connection with many problems involving [[asymptotics]]. An example is [[Stirling's formula]] for the [[Asymptotic analysis|asymptotics]] of the [[factorial function]], in which both the numbers {{mvar|e}} and [[pi|{{pi}}]] appear:<ref name="greg">{{cite book|first=Greg |last=Gbur |author-link=Greg Gbur |year=2011 |title=Mathematical Methods for Optical Physics and Engineering |isbn=978-0-521516-10-5 |publisher=Cambridge University Press |page=779}}</ref>
<math display="block>n! \sim \sqrt{2\pi n} \left(\frac{n}{e}\right)^n.</math>

As a consequence,<ref name="greg"/>
<math display="block>e = \lim_{n\to\infty} \frac{n}{\sqrt[n]{n!}} .</math>