The number is a mathematical constant that is the base of the natural logarithm and exponential function. It is approximately equal to
plays important and recurring roles across mathematics. It is comparable in significance to other mathematical constants like or the imaginary unit . Like , the constant is irrational (it cannot be represented as a ratio of integers) and transcendental (it is not a root of any non-zero polynomial with rational coefficients).
The constant is sometimes called Euler's number, after the Swiss mathematician Leonhard Euler, though this can invite confusion with other numbers named after Euler. Alternatively, can be called Napier's constant after John Napier. The Swiss mathematician Jacob Bernoulli introduced the constant while studying compound interest. Extract of page 166
Definitions
The number is the limit
an expression that arises in the computation of compound interest.
It is the sum of the infinite series
It is the unique positive number such that the graph of the function has a slope of 1 at .
One has where is the (natural) exponential function, the unique function that equals its own derivative and satisfies the equation Therefore, is also the base of the natural logarithm, the inverse of the natural exponential function.
The number can also be characterized in terms of an integral:
History The first calculations entailing this constant were published in 1618 in the table of an appendix of a work on logarithms by John Napier. However, this did not contain the constant itself, but simply a list of natural logarithms (in the form of the original Napierian logarithms). "[T]hat these were logarithms to base was not recognised since the base to which logarithms are computed did not arise in the way that logarithms were thought about at this time. Although we now think of logarithms as the exponents to which one must raise the base to get the required number, this is a modern way of thinking." It is assumed "almost certainly" that the table was written by William Oughtred. In 1661, Christiaan Huygens 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 in 1683, for solving the problem of continuous compounding of interest. 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 in letters to Christiaan Huygens in 1690 and 1691.
Leonhard Euler 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.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) … ) The first appearance of in a printed publication was in Euler's Mechanica (1736).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 seu ubi denotat numerum, cuius logarithmus hyperbolicus est 1. (So it [i.e., , the speed] will be or , where denotes the number whose hyperbolic [i.e., natural] logarithm is 1.) It is unknown why Euler chose the letter . p. 124. 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
where is the factorial of . The equivalence of the two characterizations using the limit and the infinite series can be proved via the binomial theorem.
Applications Compound interest
Jacob Bernoulli discovered this constant in 1683, while studying a question about compound interest:
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, 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 process. Each time the gambler plays the slots, there is a one in chance of winning. Playing times is modeled by the binomial distribution, which is closely related to the binomial theorem and Pascal's triangle. 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 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) 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 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
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 occurs at . Equivalently, for any value of the base , it is the case that the maximum value of occurs at (Steiner's problem, discussed below).
This is useful in the problem of a stick of length that is broken into equal parts. The value of that maximizes the product of the lengths is then either
or
The quantity is the contribution to the entropy gleaned from an event occurring with probability (approximately when ), so that essentially the same optimal division appears in optimal planning problems like the secretary problem.
Asymptotics The number occurs naturally in connection with many problems involving asymptotics. An example is Stirling's formula for the asymptotics of the factorial function, in which both the numbers and appear:
As a consequence,
Properties Calculus
The principal motivation for introducing the number , particularly in calculus, is to perform differential and integral calculus with exponential functions and logarithms. A general exponential has a derivative, given by a limit:
The parenthesized limit on the right is independent of the Its value turns out to be the logarithm of to base . Thus, when the value of is set this limit is equal and so one arrives at the following simple identity:
Consequently, the exponential function with base is particularly suited to doing calculus. (as opposed to some other number) as the base of the exponential function makes calculations involving the derivatives much simpler.
Another motivation comes from considering the derivative of the base- logarithm (i.e., ), for :
where the substitution was made. The base- logarithm of is 1, if equals . So symbolically,
The logarithm with this special base is called the natural logarithm, and is usually denoted as ; it behaves well under differentiation since there is no undetermined limit to carry through the calculations.
Thus, there are two ways of selecting such special numbers . One way is to set the derivative of the exponential function equal to , and solve for . The other way is to set the derivative of the base logarithm to and solve for . In each case, one arrives at a convenient choice of base for doing calculus. It turns out that these two solutions for are actually the same: the number .
The Taylor series for the exponential function can be deduced from the facts that the exponential function is its own derivative and that it equals 1 when evaluated at 0:
Setting recovers the definition of as the sum of an infinite series.
The natural logarithm function can be defined as the integral from 1 to of , and the exponential function can then be defined as the inverse function of the natural logarithm. The number is the value of the exponential function evaluated at , or equivalently, the number whose natural logarithm is 1. It follows that is the unique positive real number such that
Because is the unique function (up to multiplication by a constant ) that is equal to its own derivative,
it is therefore its own antiderivative as well:
Equivalently, the family of functions
where is any real or complex number, is the full solution to the differential equation
Inequalities
The number is the unique real number such that
for all positive .