E (mathematical constant)

# E (mathematical constant)

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The mathematical constant
Mathematical constant
A mathematical constant is a special number, usually a real number, that is "significantly interesting in some way". Constants arise in many different areas of mathematics, with constants such as and occurring in such diverse contexts as geometry, number theory and calculus.What it means for a...

is the unique real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

such that the value of the derivative
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

(slope of the tangent
Tangent
In geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point. More precisely, a straight line is said to be a tangent of a curve at a point on the curve if the line passes through the point on the curve and has slope where f...

line) of the function at the point is equal to 1. The function so defined is called the exponential function
Exponential function
In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...

, and its inverse
Inverse function
In mathematics, an inverse function is a function that undoes another function: If an input x into the function ƒ produces an output y, then putting y into the inverse function g produces the output x, and vice versa. i.e., ƒ=y, and g=x...

is the natural logarithm
Natural logarithm
The natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828...

, or logarithm to base . The number is also commonly defined as the base of the natural logarithm (using an integral
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...

to define the latter), as the limit
Limit of a sequence
The limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...

of a certain sequence, or as the sum of a certain series
Series (mathematics)
A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....

(see the alternative characterizations, below).

The number is sometimes called Euler's number after the Swiss
Switzerland
Switzerland name of one of the Swiss cantons. ; ; ; or ), in its full name the Swiss Confederation , is a federal republic consisting of 26 cantons, with Bern as the seat of the federal authorities. The country is situated in Western Europe,Or Central Europe depending on the definition....

mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

Leonhard Euler
Leonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...

. ( is not to be confused with —the Euler–Mascheroni constant
Euler–Mascheroni constant
The Euler–Mascheroni constant is a mathematical constant recurring in analysis and number theory, usually denoted by the lowercase Greek letter ....

, sometimes called simply Euler's constant.) It is also sometimes known as Napier's constant, but Euler's choice of the symbol is said to have been retained in his honor.

The number is of eminent importance in mathematics, alongside 0
0 (number)
0 is both a numberand the numerical digit used to represent that number in numerals.It fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, 0 is used as a placeholder in place value systems...

, 1,
Pi
' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...

and
{{DISPLAYTITLE:{{math|e}} (mathematical constant)}}
{{Redirect|Euler's number|γ, a constant in number theory|Euler's constant|other uses|List of topics named after Leonhard Euler#Euler—numbers}}
{{Portal:Mathematics/Featured article template}}

{{pp-move-indef|small=yes}}
The
mathematical constant
Mathematical constant
A mathematical constant is a special number, usually a real number, that is "significantly interesting in some way". Constants arise in many different areas of mathematics, with constants such as and occurring in such diverse contexts as geometry, number theory and calculus.What it means for a...

{{math|e}} is the unique real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

such that the value of the derivative
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

(slope of the tangent
Tangent
In geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point. More precisely, a straight line is said to be a tangent of a curve at a point on the curve if the line passes through the point on the curve and has slope where f...

line) of the function {{math|f(x) {{=}} ex}} at the point {{math|x {{=}} 0}} is equal to 1. The function {{math|ex}} so defined is called the exponential function
Exponential function
In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...

, and its inverse
Inverse function
In mathematics, an inverse function is a function that undoes another function: If an input x into the function ƒ produces an output y, then putting y into the inverse function g produces the output x, and vice versa. i.e., ƒ=y, and g=x...

is the natural logarithm
Natural logarithm
The natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828...

, or logarithm to base {{math|e}}. The number {{math|e}} is also commonly defined as the base of the natural logarithm (using an integral
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...

to define the latter), as the limit
Limit of a sequence
The limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...

of a certain sequence, or as the sum of a certain series
Series (mathematics)
A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....

(see the alternative characterizations, below).

The number {{math|e}} is sometimes called Euler's number after the Swiss
Switzerland
Switzerland name of one of the Swiss cantons. ; ; ; or ), in its full name the Swiss Confederation , is a federal republic consisting of 26 cantons, with Bern as the seat of the federal authorities. The country is situated in Western Europe,Or Central Europe depending on the definition....

mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

Leonhard Euler
Leonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...

. ({{math|e}} is not to be confused with {{math|γ}}—the Euler–Mascheroni constant
Euler–Mascheroni constant
The Euler–Mascheroni constant is a mathematical constant recurring in analysis and number theory, usually denoted by the lowercase Greek letter ....

, sometimes called simply Euler's constant.) It is also sometimes known as Napier's constant, but Euler's choice of the symbol {{math|e}} is said to have been retained in his honor.

The number {{math|e}} is of eminent importance in mathematics, alongside 0
0 (number)
0 is both a numberand the numerical digit used to represent that number in numerals.It fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, 0 is used as a placeholder in place value systems...

, 1, {{pi}}
Pi
' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...

and
{{DISPLAYTITLE:{{math|e}} (mathematical constant)}}
{{Redirect|Euler's number|γ, a constant in number theory|Euler's constant|other uses|List of topics named after Leonhard Euler#Euler—numbers}}
{{Portal:Mathematics/Featured article template}}

{{pp-move-indef|small=yes}}
The
mathematical constant
Mathematical constant
A mathematical constant is a special number, usually a real number, that is "significantly interesting in some way". Constants arise in many different areas of mathematics, with constants such as and occurring in such diverse contexts as geometry, number theory and calculus.What it means for a...

{{math|e}} is the unique real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

such that the value of the derivative
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

(slope of the tangent
Tangent
In geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point. More precisely, a straight line is said to be a tangent of a curve at a point on the curve if the line passes through the point on the curve and has slope where f...

line) of the function {{math|f(x) {{=}} ex}} at the point {{math|x {{=}} 0}} is equal to 1. The function {{math|ex}} so defined is called the exponential function
Exponential function
In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...

, and its inverse
Inverse function
In mathematics, an inverse function is a function that undoes another function: If an input x into the function ƒ produces an output y, then putting y into the inverse function g produces the output x, and vice versa. i.e., ƒ=y, and g=x...

is the natural logarithm
Natural logarithm
The natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828...

, or logarithm to base {{math|e}}. The number {{math|e}} is also commonly defined as the base of the natural logarithm (using an integral
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...

to define the latter), as the limit
Limit of a sequence
The limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...

of a certain sequence, or as the sum of a certain series
Series (mathematics)
A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....

(see the alternative characterizations, below).

The number {{math|e}} is sometimes called Euler's number after the Swiss
Switzerland
Switzerland name of one of the Swiss cantons. ; ; ; or ), in its full name the Swiss Confederation , is a federal republic consisting of 26 cantons, with Bern as the seat of the federal authorities. The country is situated in Western Europe,Or Central Europe depending on the definition....

mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

Leonhard Euler
Leonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...

. ({{math|e}} is not to be confused with {{math|γ}}—the Euler–Mascheroni constant
Euler–Mascheroni constant
The Euler–Mascheroni constant is a mathematical constant recurring in analysis and number theory, usually denoted by the lowercase Greek letter ....

, sometimes called simply Euler's constant.) It is also sometimes known as Napier's constant, but Euler's choice of the symbol {{math|e}} is said to have been retained in his honor.

The number {{math|e}} is of eminent importance in mathematics, alongside 0
0 (number)
0 is both a numberand the numerical digit used to represent that number in numerals.It fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, 0 is used as a placeholder in place value systems...

, 1, {{pi}}
Pi
' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...

and {{math
Imaginary unit
In mathematics, the imaginary unit allows the real number system ℝ to be extended to the complex number system ℂ, which in turn provides at least one root for every polynomial . The imaginary unit is denoted by , , or the Greek...

. All five of these numbers play important and recurring roles across mathematics, and are the five constants appearing in one formulation of Euler's identity.

The number {{math|e}} is irrational
Irrational number
In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b non-zero, and is therefore not a rational number....

; it is not a ratio of integers. Furthermore, it is transcendental
Transcendental number
In mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a non-constant polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e...

; it is not a root of any non-zero polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...

with rational coefficients. The numerical value of {{math|e}} truncated to 50 decimal places
Decimal
The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....

is
{{gaps|2.71828|18284|59045|23536|02874|71352|66249|77572|47093|69995...}} {{OEIS|A001113}}.

{{E (mathematical constant)}}

## History

The first references to the constant were published in 1618 in the table of an appendix of a work on logarithms by John Napier
John Napier
John Napier of Merchiston – also signed as Neper, Nepair – named Marvellous Merchiston, was a Scottish mathematician, physicist, astronomer & astrologer, and also the 8th Laird of Merchistoun. He was the son of Sir Archibald Napier of Merchiston. John Napier is most renowned as the discoverer...

. However, this did not contain the constant itself, but simply a list of logarithms calculated from the constant. It is assumed that the table was written by William Oughtred
William Oughtred
William Oughtred was an English mathematician.After John Napier invented logarithms, and Edmund Gunter created the logarithmic scales upon which slide rules are based, it was Oughtred who first used two such scales sliding by one another to perform direct multiplication and division; and he is...

. The discovery of the constant itself is credited to Jacob Bernoulli, who attempted to find the value of the following expression (which is in fact {{math|e}}):

The first known use of the constant, represented by the letter {{math|b}}, was in correspondence from Gottfried Leibniz
Gottfried Leibniz
Gottfried Wilhelm Leibniz was a German philosopher and mathematician. He wrote in different languages, primarily in Latin , French and German ....

to Christiaan Huygens in 1690 and 1691. Leonhard Euler
Leonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...

introduced the letter {{math|e}} as the base for natural logarithms, writing in a letter to Christian Goldbach
Christian Goldbach
Christian Goldbach was a German mathematician who also studied law. He is remembered today for Goldbach's conjecture.-Biography:...

of 25 November 1731. Euler started to use the letter {{math|e}} for the constant in 1727 or 1728, in an unpublished paper on explosive forces in cannons, and the first appearance of {{math|e}} in a publication was Euler's Mechanica
Mechanica
Mechanica is a two-volume work published by mathematician Leonhard Euler, which describes analytically the mathematics governing movement...

(1736). While in the subsequent years some researchers used the letter {{math|c}}, {{math|e}} was more common and eventually became the standard.

### The compound-interest problem

Jacob Bernoulli discovered this constant by studying a question about compound interest
Compound interest
Compound interest arises when interest is added to the principal, so that from that moment on, the interest that has been added also itself earns interest. This addition of interest to the principal is called compounding...

:
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 \$1.00×1.5² = \$2.25. Compounding quarterly yields \$1.00×1.254 = \$2.4414..., and compounding monthly yields \$1.00×(1+1/12)12 = \$2.613035... If there are {{math|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×

Bernoulli noticed that this sequence approaches a limit (the force of interest) with larger {{math|n}} and, thus, smaller compounding intervals. Compounding weekly ({{math|n}}=52) yields \$2.692597..., while compounding daily ({{math|n}}=365) yields \$2.714567..., just two cents more. The limit as {{math|n}} grows large is the number that came to be known as {{math|e}}; with continuous compounding, the account value will reach \$2.7182818.... More generally, an account that starts at \$1 and offers an annual interest rate of R will, after t years, yield {{math|eRt}} dollars with continuous compounding. (Here R is a fraction, not a per cent, so for 5% interest, R = 0.05)

### Bernoulli trials

The number {{math|e}} itself also has applications to probability theory
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

, where it arises in a way not obviously related to exponential growth. Suppose that a gambler plays a slot machine that pays out with a probability of one in {{math|n}} and plays it {{math|n}} times. Then, for large {{math|n}} (such as a million) the probability
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...

that the gambler will win nothing at all is (approximately) {{math|1/e}}.

This is an example of a Bernoulli trials process. Each time the gambler plays the slots, there is a one in one million chance of winning. Playing one million times is modelled by the binomial distribution, which is closely related to the binomial theorem
Binomial theorem
In elementary algebra, the binomial theorem describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the power n into a sum involving terms of the form axbyc, where the exponents b and c are nonnegative integers with , and the coefficient a of...

. The probability of winning {{math|k}} times out of a million trials is;
In particular, the probability of winning zero times ({{math|k}} = 0) is
This is very close to the following limit for {{math|1/e}}:

### Derangements

Another application of {{math|e}}, also discovered in part by Jacob Bernoulli along with Pierre Raymond de Montmort
Pierre Raymond de Montmort
Pierre Rémond de Montmort, a French mathematician, was born in Paris on 27 October 1678, and died there on 7 October 1719. His name was originally just Pierre Rémond or Raymond...

is in the problem of derangement
Derangement
In combinatorial mathematics, a derangement is a permutation of the elements of a set such that none of the elements appear in their original position....

s, also known as the hat check problem: {{math|n}} guests are invited to a party, and at the door each guest checks his hat with the butler who then places them into {{math|n}} boxes, each labelled with the name of one guest. But the butler does not know the identities of the guests, and so he 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. The answer is:

As the number {{math|n}} of guests tends to infinity, {{math|pn}} approaches {{math|1/e}}. Furthermore, the number of ways the hats can be placed into the boxes so that none of the hats is in the right box is {{math|n!/e}} rounded to the nearest integer, for every positive {{math|n}}.

### Asymptotics

The number {{math|e}} occurs naturally in connection with many problems involving asymptotics. A prominent example is Stirling's formula for the asymptotics
Asymptotic analysis
In mathematical analysis, asymptotic analysis is a method of describing limiting behavior. The methodology has applications across science. Examples are...

of the factorial function, in which both the numbers {{math|e}} and {{pi}}
Pi
' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...

enter:
A particular consequence of this is.

## {{math|e}} in calculus

The principal motivation for introducing the number {{math|e}}, particularly in calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

, is to perform differential and integral calculus with exponential function
Exponential function
In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...

s and logarithm
Logarithm
The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...

s. A general exponential function {{math|y {{=}} ax}} has derivative given as the limit
Limit of a function
In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input....

:
The limit on the right-hand side is independent of the variable {{math|x}}: it depends only on the base {{math|a}}. When the base is {{math|e}}, this limit is equal to one, and so {{math|e}} is symbolically defined by the equation:

Consequently, the exponential function with base {{math|e}} is particularly suited to doing calculus. Choosing {{math|e}}, as opposed to some other number, as the base of the exponential function makes calculations involving the derivative much simpler.

Another motivation comes from considering the base-{{math|a}} logarithm
Logarithm
The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...

. Considering the definition of the derivative of {{math|log}}a{{math|x}} as the limit:
where the substitution {{math|u {{=}} h/x}} was made in the last step. The last limit appearing in this calculation is again an undetermined limit that depends only on the base {{math|a}}, and if that base is {{math|e}}, the limit is one. So symbolically,
The logarithm in this special base is called the natural logarithm
Natural logarithm
The natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828...

and is represented as {{math|ln}}; it behaves well under differentiation since there is no undetermined limit to carry through the calculations.

There are thus two ways in which to select a special number {{math|a {{=}} e}}. One way is to set the derivative of the exponential function {{math|ax}} to {{math|ax}}, and solve for {{math|a}}. The other way is to set the derivative of the base {{math|a}} logarithm to {{math|1/x}} and solve for {{math|a}}. In each case, one arrives at a convenient choice of base for doing calculus. In fact, these two solutions for {{math|a}} are actually the same, the number {{math|e}}.

### Alternative characterizations

Other characterizations of {{math|e}} are also possible: one is as the limit of a sequence
Limit of a sequence
The limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...

, another is as the sum of an infinite series, and still others rely on integral calculus. So far, the following two (equivalent) properties have been introduced:

1. The number {{math|e}} is the unique positive real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

such that

2. The number {{math|e}} is the unique positive real number such that

The following three characterizations can be proven equivalent:

3. The number {{math|e}} is the limit
Limit of a sequence
The limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...

Similarly:

4. The number {{math|e}} is the sum of the infinite series
where {{math|n!}} is the factorial
Factorial
In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n...

of {{math|n}}.

5. The number {{math|e}} is the unique positive real number such that

### Calculus

As in the motivation, the exponential function
Exponential function
In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...

{{math|ex}} is important in part because it is the unique nontrivial function (up to multiplication by a constant) which is its own derivative
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

and therefore its own antiderivative
Antiderivative
In calculus, an "anti-derivative", antiderivative, primitive integral or indefinite integralof a function f is a function F whose derivative is equal to f, i.e., F ′ = f...

as well:

### Exponential-like functions

The global maximum for the function

occurs at {{math|x {{=}} e}}. Similarly, {{math|x {{=}} 1/e}} is where the global minimum occurs for the function

defined for positive {{math|x}}. More generally, {{math|x {{=}} e−1/n}} is where the global minimum occurs for the function

for any {{math|n > 0}}. The infinite tetration
Tetration
In mathematics, tetration is an iterated exponential and is the next hyper operator after exponentiation. The word tetration was coined by English mathematician Reuben Louis Goodstein from tetra- and iteration. Tetration is used for the notation of very large numbers...

or

converges if and only if {{math|eexe1/e}} (or approximately between 0.0660 and 1.4447), due to a theorem of Leonhard Euler
Leonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...

.

### Number theory

The real number {{math|e}} is irrational
Irrational number
In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b non-zero, and is therefore not a rational number....

. Euler
Leonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...

proved this by showing that its simple continued fraction expansion is infinite. (See also Fourier
Joseph Fourier
Jean Baptiste Joseph Fourier was a French mathematician and physicist best known for initiating the investigation of Fourier series and their applications to problems of heat transfer and vibrations. The Fourier transform and Fourier's Law are also named in his honour...

's proof that {{math
Proof that e is irrational
In mathematics, the series representation of Euler's number ecan be used to prove that e is irrational. Of the many representations of e, this is the Taylor series for the exponential function evaluated at y = 1.-Summary of the proof:...

.) Furthermore, {{math|e}} is transcendental
Transcendental number
In mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a non-constant polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e...

(Lindemann–Weierstrass theorem
Lindemann–Weierstrass theorem
In mathematics, the Lindemann–Weierstrass theorem is a result that is very useful in establishing the transcendence of numbers. It states that if 1, ...,  are algebraic numbers which are linearly independent over the rational numbers ', then 1, ...,  are algebraically...

). It was the first number to be proved transcendental without having been specifically constructed for this purpose (compare with Liouville number); the proof was given by Charles Hermite
Charles Hermite
Charles Hermite was a French mathematician who did research on number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra....

in 1873. It is conjectured that {{math|e}} is normal
Normal number
In mathematics, a normal number is a real number whose infinite sequence of digits in every base b is distributed uniformly in the sense that each of the b digit values has the same natural density 1/b, also all possible b2 pairs of digits are equally likely with density b−2,...

.

### Complex numbers

The exponential function
Exponential function
In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...

{{math|ex}} may be written as a Taylor series
Taylor series
In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....

Because this series keeps many important properties for {{math|ex}} even when {{math|x}} is complex
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

, it is commonly used to extend the definition of {{math|ex}} to the complex numbers. This, with the Taylor series for sin and cos {{math, allows one to derive Euler's formula
Euler's formula
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the deep relationship between the trigonometric functions and the complex exponential function...

:

which holds for all {{math|x}}. The special case with {{math|x {{=}} π
Pi
' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...

}} is Euler's identity:

from which it follows that, in the principal branch
Principal branch
In mathematics, a principal branch is a function which selects one branch, or "slice", of a multi-valued function. Most often, this applies to functions defined on the complex plane: see branch cut....

of the logarithm,

Furthermore, using the laws for exponentiation,

which is de Moivre's formula
De Moivre's formula
In mathematics, de Moivre's formula , named after Abraham de Moivre, states that for any complex number x and integer n it holds that...

.

The expression

is sometimes referred to as {{math|cis(x)}}.

### Differential equations

The general function

is the solution to the differential equation:

## Representations

{{Main|Representations of e}}

The number {{math|e}} can be represented as a real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

in a variety of ways: as an infinite series, an infinite product, a continued fraction
Continued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on...

, or a limit of a sequence
Limit of a sequence
The limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...

. The chief among these representations, particularly in introductory calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

courses is the limit
given above, as well as the series
given by evaluating the above power series for {{math|ex}} at {{math|x {{=}} 1}}.

Still other less common representations are also available. For instance, {{math|e}} can be represented as an infinite simple
Continued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on...

or generalized continued fraction
Generalized continued fraction
In complex analysis, a branch of mathematics, a generalized continued fraction is a generalization of regular continued fractions in canonical form, in which the partial numerators and partial denominators can assume arbitrary real or complex values....

due to Euler
Leonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...

:

or, in a more compact form {{OEIS|id=A003417}}:

which can be written more harmoniously by allowing zero:

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

### Stochastic representations

In addition to exact analytical expressions for representation of {{math|e}}, there are stochastic techniques
Stochastic process
In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...

for estimating {{math|e}}. One such approach begins with an infinite sequence of independent random variables {{math|X1}}, {{math|X2}}, ..., drawn from the uniform distribution
Uniform distribution (continuous)
In probability theory and statistics, the continuous uniform distribution or rectangular distribution is a family of probability distributions such that for each member of the family, all intervals of the same length on the distribution's support are equally probable. The support is defined by...

on [0, 1]. Let {{math|V}} be the least number {{math|n}} such that the sum of the first {{math|n}} samples exceeds 1:
Then the expected value
Expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...

of {{math|V}} is {{math|e}}: {{math|E(V) {{=}} e}}.

### Known digits

The number of known digits of {{math|e}} has increased dramatically during the last decades. This is due both to the increase of performance of computers as well as algorithmic improvements.
 Date Decimal digits Computation performed by 1748 23 Leonhard EulerLeonhard EulerLeonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion... 1853 137 William ShanksWilliam ShanksWilliam Shanks was a British amateur mathematician.Shanks is famous for his calculation of π to 707 places, accomplished in 1873, which, however, was only correct up to the first 527 places. This error was highlighted in 1944 by D. F... 1871 205 William ShanksWilliam ShanksWilliam Shanks was a British amateur mathematician.Shanks is famous for his calculation of π to 707 places, accomplished in 1873, which, however, was only correct up to the first 527 places. This error was highlighted in 1944 by D. F... 1884 346 J. Marcus Boorman 1946 808 Unknown 1949 2,010 John von NeumannJohn von NeumannJohn von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,... (on the ENIACENIACENIAC was the first general-purpose electronic computer. It was a Turing-complete digital computer capable of being reprogrammed to solve a full range of computing problems....) 1961 100,265 Daniel ShanksDaniel ShanksDaniel Shanks was an American mathematician who worked primarily in numerical analysis and number theory. He is best known as the first to compute π to 100,000 decimal places, and for his book Solved and Unsolved Problems in Number Theory.-Life and education:Dan Shanks was born on January 17,... and John WrenchJohn WrenchJohn William Wrench, Jr. was an American mathematician who worked primarily in numerical analysis. He was a pioneer in using computers for mathematical calculations, and is noted for work done with Daniel Shanks to calculate the mathematical constant pi to 100,000 decimal places.-Life and... 1978 116,000 Stephen Gary Wozniak (on the Apple IIApple IIThe Apple II is an 8-bit home computer, one of the first highly successful mass-produced microcomputer products, designed primarily by Steve Wozniak, manufactured by Apple Computer and introduced in 1977...) 1994 April 1 10,000,000 Robert Nemiroff & Jerry Bonnell 1997 May 18,199,978 Patrick Demichel 1997 August 20,000,000 Birger Seifert 1997 September 50,000,817 Patrick Demichel 1999 February 200,000,579 Sebastian Wedeniwski 1999 October 869,894,101 Sebastian Wedeniwski 1999 November 21 1,250,000,000 Xavier Gourdon 2000 July 10 2,147,483,648 Shigeru Kondo & Xavier Gourdon 2000 July 16 3,221,225,472 Colin Martin & Xavier Gourdon 2000 August 2 6,442,450,944 Shigeru Kondo & Xavier Gourdon 2000 August 16 12,884,901,000 Shigeru Kondo & Xavier Gourdon 2003 August 21 25,100,000,000 Shigeru Kondo & Xavier Gourdon 2003 September 18 50,100,000,000 Shigeru Kondo & Xavier Gourdon 2007 April 27 100,000,000,000 Shigeru Kondo & Steve Pagliarulo 2009 May 6 200,000,000,000 Shigeru Kondo & Steve Pagliarulo 2010 February 21 500,000,000,000 Alexander J. Yee 2010 July 5 1,000,000,000,000 Shigeru Kondo & Alexander J. Yee

## In computer culture

In contemporary internet culture, individuals and organizations frequently pay homage to the number {{math|e}}.

For example, in the IPO filing for Google
Google Inc. is an American multinational public corporation invested in Internet search, cloud computing, and advertising technologies. Google hosts and develops a number of Internet-based services and products, and generates profit primarily from advertising through its AdWords program...

, in 2004, rather than a typical round-number amount of money, the company announced its intention to raise \$2,718,281,828, which is {{math|e}} billion dollars
United States dollar
The United States dollar , also referred to as the American dollar, is the official currency of the United States of America. It is divided into 100 smaller units called cents or pennies....

to the nearest dollar. Google was also responsible for a billboard that appeared in the heart of Silicon Valley
Silicon Valley
Silicon Valley is a term which refers to the southern part of the San Francisco Bay Area in Northern California in the United States. The region is home to many of the world's largest technology corporations...

, and later in Cambridge, Massachusetts
Cambridge, Massachusetts
Cambridge is a city in Middlesex County, Massachusetts, United States, in the Greater Boston area. It was named in honor of the University of Cambridge in England, an important center of the Puritan theology embraced by the town's founders. Cambridge is home to two of the world's most prominent...

; Seattle, Washington
Seattle, Washington
Seattle is the county seat of King County, Washington. With 608,660 residents as of the 2010 Census, Seattle is the largest city in the Northwestern United States. The Seattle metropolitan area of about 3.4 million inhabitants is the 15th largest metropolitan area in the country...

; and Austin, Texas
Austin, Texas
Austin is the capital city of the U.S. state of :Texas and the seat of Travis County. Located in Central Texas on the eastern edge of the American Southwest, it is the fourth-largest city in Texas and the 14th most populous city in the United States. It was the third-fastest-growing large city in...

. It read {first 10-digit prime found in consecutive digits of {{math|e}}}.com (now defunct). Solving this problem and visiting the advertised web site led to an even more difficult problem to solve, which in turn led to Google Labs
Google Labs was a page created by Google to demonstrate and test new Google projects. Google calls Google Labs,Google also uses an invitation-only phase for trusted testers to test projects including Gmail, Google Calendar and Google Wave and many of these have their own "labs" webpages for...

where the visitor was invited to submit a resume. The first 10-digit prime in {{math|e}} is 7427466391, which starts as late as at the 99th digit.

In another instance, the computer scientist
Computer scientist
A computer scientist is a scientist who has acquired knowledge of computer science, the study of the theoretical foundations of information and computation and their application in computer systems....

Donald Knuth
Donald Knuth
Donald Ervin Knuth is a computer scientist and Professor Emeritus at Stanford University.He is the author of the seminal multi-volume work The Art of Computer Programming. Knuth has been called the "father" of the analysis of algorithms...

let the version numbers of his program Metafont
METAFONT
Metafont is a programming language used to define vector fonts. It is also the name of the interpreter that executes Metafont code, generating the bitmap fonts that can be embedded into e.g. PostScript...

approach {{math|e}}. The versions are 2, 2.7, 2.71, 2.718, and so forth. Similarly, the version numbers of his TeX
TeX
TeX is a typesetting system designed and mostly written by Donald Knuth and released in 1978. Within the typesetting system, its name is formatted as ....

program approach {{pi}}.

• Maor, Eli; {{math|e}}: The Story of a Number, ISBN 0-691-05854-7
• Commentary on Endnote 10 of the book Prime Obsession
Prime Obsession
Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics is a historical book on mathematics by John Derbyshire, detailing the history of the Riemann hypothesis, named for Bernhard Riemann, and some of its applications...

for another stochastic representation