
The number is a
mathematical constant
A mathematical constant is a number whose value is fixed by an unambiguous definition, often referred to by a special symbol (e.g., an Letter (alphabet), alphabet letter), or by mathematicians' names to facilitate using it across multiple mathem ...
approximately equal to 2.71828 that is the
base of the
natural logarithm
The natural logarithm of a number is its logarithm to the base of a logarithm, base of the e (mathematical constant), mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approxima ...
and
exponential function. It is sometimes called Euler's number, after the Swiss mathematician
Leonhard Euler
Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...
, though this can invite confusion with
Euler numbers
Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...
, or with
Euler's constant, a different constant typically denoted
. Alternatively, can be called Napier's constant after
John Napier
John Napier of Merchiston ( ; Latinisation of names, Latinized as Ioannes Neper; 1 February 1550 – 4 April 1617), nicknamed Marvellous Merchiston, was a Scottish landowner known as a mathematician, physicist, and astronomer. He was the 8 ...
.
The Swiss mathematician
Jacob Bernoulli
Jacob Bernoulli (also known as James in English or Jacques in French; – 16 August 1705) was a Swiss mathematician. He sided with Gottfried Wilhelm Leibniz during the Leibniz–Newton calculus controversy and was an early proponent of Leibniz ...
discovered the constant while studying compound interest.
[Extract of page 166]
/ref>
The number is of great importance in mathematics, alongside 0, 1, , and . All five appear in one formulation of Euler's identity and play important and recurring roles across mathematics. Like the constant , is irrational
Irrationality is cognition, thinking, talking, or acting without rationality.
Irrationality often has a negative connotation, as thinking and actions that are less useful or more illogical than other more rational alternatives. The concept of ...
, meaning that it cannot be represented as a ratio of integers, and moreover it is transcendental, meaning that it is not a root of any non-zero polynomial
In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
with rational coefficients. To 30 decimal places, the value of is:
Definitions
The number is the limit
an expression that arises in the computation of compound interest
Compound interest is interest accumulated from a principal sum and previously accumulated interest. It is the result of reinvesting or retaining interest that would otherwise be paid out, or of the accumulation of debts from a borrower.
Compo ...
.
It is the sum of the infinite series
It is the unique positive number such that the graph of the function has a slope
In mathematics, the slope or gradient of a Line (mathematics), line is a number that describes the direction (geometry), direction of the line on a plane (geometry), plane. Often denoted by the letter ''m'', slope is calculated as the ratio of t ...
of 1 at .
One has where is the (natural) exponential function, the unique function that equals its own derivative
In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
and satisfies the equation Since the exponential function is commonly denoted as one has also
The logarithm
In mathematics, the logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number. For example, the logarithm of to base is , because is to the rd power: . More generally, if , the ...
of base can be defined as the inverse function
In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ .
For a function f\colon ...
of the function Since one has The equation implies therefore that is the base of the natural logarithm.
The number can also be characterized in terms of an integral
In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental oper ...
:
For other characterizations, see .
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 of Merchiston ( ; Latinisation of names, Latinized as Ioannes Neper; 1 February 1550 – 4 April 1617), nicknamed Marvellous Merchiston, was a Scottish landowner known as a mathematician, physicist, and astronomer. He was the 8 ...
. However, this did not contain the constant itself, but simply a list of logarithms to the base . It is assumed that the table was written by William Oughtred. In 1661, Christiaan Huygens
Christiaan Huygens, Halen, Lord of Zeelhem, ( , ; ; also spelled Huyghens; ; 14 April 1629 – 8 July 1695) was a Dutch mathematician, physicist, engineer, astronomer, and inventor who is regarded as a key figure in the Scientific Revolution ...
studied how to compute logarithms by geometrical methods and calculated a quantity that, in retrospect, is the base-10 logarithm of , but he did not recognize itself as a quantity of interest.
The constant itself was introduced by Jacob Bernoulli
Jacob Bernoulli (also known as James in English or Jacques in French; – 16 August 1705) was a Swiss mathematician. He sided with Gottfried Wilhelm Leibniz during the Leibniz–Newton calculus controversy and was an early proponent of Leibniz ...
in 1683, for solving the problem of continuous compounding of interest.[Jacob Bernoulli considered the problem of continuous compounding of interest, which led to a series expression for . 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 sum of money tinterest, let it accumulate, so that tevery moment twere to receive proportional part of tsannual interest; how much would be owing t theend of heyear?) Bernoulli constructs a power series to calculate the answer, and then writes: " … quæ nostra serie athematical expression for a geometric series&c. major est. … si , debebitur plu quam & minus quam ." ( … which our series geometric seriesis larger han … if , he lenderwill be owed more than and less than .) If , the geometric series reduces to the series for , so . (** The reference is to a problem which Jacob Bernoulli posed and which appears in the ''Journal des Sçavans'' of 1685 at the bottom o
page 314.
In his solution, the constant occurs as the limit
where represents the number of intervals in a year on which the compound interest is evaluated (for example, for monthly compounding).
The first symbol used for this constant was the letter by Gottfried Leibniz
Gottfried Wilhelm Leibniz (or Leibnitz; – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat who is credited, alongside Isaac Newton, Sir Isaac Newton, with the creation of calculus in ad ...
in letters to Christiaan Huygens
Christiaan Huygens, Halen, Lord of Zeelhem, ( , ; ; also spelled Huyghens; ; 14 April 1629 – 8 July 1695) was a Dutch mathematician, physicist, engineer, astronomer, and inventor who is regarded as a key figure in the Scientific Revolution ...
in 1690 and 1691.
Leonhard Euler
Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...
started to use the letter for the constant in 1727 or 1728, in an unpublished paper on explosive forces in cannons,[Euler, ]
Meditatio in experimenta explosione tormentorum nuper instituta
'. (English: Written for the number of which the logarithm has the unit, e, that is 2,7182817...") and in a letter to Christian Goldbach on 25 November 1731. The first appearance of in a printed publication was in Euler's '' Mechanica'' (1736). It is unknown why Euler chose the letter . Although some researchers used the letter in the subsequent years, the letter was more common and eventually became standard.
Euler proved that is the sum of the infinite series
In mathematics, a series is, roughly speaking, an addition of infinitely many terms, one after the other. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathemati ...
where is the factorial
In mathematics, the factorial of a non-negative denoted is the Product (mathematics), product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial:
\begin
n! &= n \times ...
of . The equivalence of the two characterizations using the limit and the infinite series can be proved via the binomial theorem
In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, the power expands into a polynomial with terms of the form , where the exponents and a ...
.
Applications
Compound interest
Jacob Bernoulli discovered this constant in 1683, while studying a question about compound interest
Compound interest is interest accumulated from a principal sum and previously accumulated interest. It is the result of reinvesting or retaining interest that would otherwise be paid out, or of the accumulation of debts from a borrower.
Compo ...
:
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 at the end of the year. Compounding quarterly yields , and compounding monthly yields . If there are compounding intervals, the interest for each interval will be 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 and, thus, smaller compounding intervals. Compounding weekly () yields $2.692596..., while compounding daily () yields $2.714567... (approximately two cents more). The limit as grows large is the number that came to be known as . 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 will, after years, yield dollars with continuous compounding. Here, is the decimal equivalent of the rate of interest expressed as a ''percentage'', so for 5% interest, .
Bernoulli trials
The number itself also has applications in probability theory
Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
, 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 and plays it times. As increases, the probability that gambler will lose all bets approaches , which is approximately 36.79%. For , this is already 1/2.789509... (approximately 35.85%).
This is an example of a Bernoulli trial
In the theory of probability and statistics, a Bernoulli trial (or binomial trial) is a random experiment with exactly two possible outcomes, "success" and "failure", in which the probability of success is the same every time the experiment is ...
process. Each time the gambler plays the slots, there is a one in chance of winning. Playing times is modeled by the binomial distribution
In probability theory and statistics, the binomial distribution with parameters and is the discrete probability distribution of the number of successes in a sequence of statistical independence, independent experiment (probability theory) ...
, which is closely related to the binomial theorem
In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, the power expands into a polynomial with terms of the form , where the exponents and a ...
and Pascal's triangle
In mathematics, Pascal's triangle is an infinite triangular array of the binomial coefficients which play a crucial role in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Bla ...
. The probability of winning times out of trials is:
:
In particular, the probability of winning zero times () is
:
The limit of the above expression, as tends to infinity, is precisely .
Exponential growth and decay
Exponential growth
Exponential growth occurs when a quantity grows as an exponential function of time. The quantity grows at a rate directly proportional to its present size. For example, when it is 3 times as big as it is now, it will be growing 3 times as fast ...
is a process that increases quantity over time at an ever-increasing rate. It occurs when the instantaneous rate of change (that is, the derivative
In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
) of a quantity with respect to time is proportional to the quantity itself. Described as a function, a quantity undergoing exponential growth is an 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
A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where is the quantity and (lambda
Lambda (; uppe ...
instead. The law of exponential growth can be written in different but mathematically equivalent forms, by using a different base, for which the number is a common and convenient choice:
Here, denotes the initial value of the quantity , is the growth constant, and is the time it takes the quantity to grow by a factor of .
Standard normal distribution
The normal distribution with zero mean and unit standard deviation is known as the ''standard normal distribution'', given by the probability density function
In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the s ...
The constraint of unit standard deviation (and thus also unit variance) results in the in the exponent, and the constraint of unit total area under the curve results in the factor . This function is symmetric around , where it attains its maximum value , and has inflection points at .
Derangements
Another application of , also discovered in part by Jacob Bernoulli along with Pierre Remond de Montmort, is in the problem of derangements, also known as the ''hat check problem'': 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 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 , is:
:
As tends to infinity, approaches . 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 rounded to the nearest integer, for every positive .
Optimal planning problems
The maximum value of